Abstracts CAM seminar

2022-12-14: Per Ljung, Chalmers & GU (live)
Multiscale methods for solving wave equations on spatial networks
We present and analyze a multiscale method for wave propagation problems, posed on spatial networks. By introducing a coarse scale, using a finite element space interpolated onto the network, we construct a discrete multiscale space using the localized orthogonal decomposition (LOD) methodology. The spatial discretization is then combined with an energy conserving temporal scheme to form the proposed method. Under the assumption of well-prepared initial data, we derive an a priori error bound of optimal order with respect to the space and time discretization. Finally, we present numerical experiments that confirm our theoretical findings.

2022-12-07: Jan S Hesthaven, EPFL (zoom)
Digital Twins through Reduced Order Models and Machine Learning
The vision of building digital twins for complex infrastructure and systems is old. However, realizing it remains very challenging due to the need to combine advanced computational modeling, reduced order models, data infusion for calibration, updating and uncertainty management, and sensor integration to obtain models with true predictive value for decision support. Nevertheless, the perspectives of using digital twins for predictive maintenance, operational optimization, and risk analysis are very substantial and the potential for impact significant, from safety, planning, and financial points of view.
In this talk we shall first discuss the importance of reduced models in the development of digital twin technologies and continue by discussing different aspects of the challenges associated with developing digital twins through a few examples, combining advanced model and data driven technologies, e.g., classifiers, Gaussian regression and neural networks, to enable failure analysis, optimal sensor placement and, time permitting, multi-fidelity methods and risk analysis for rare events.
These are all elements of the workflow that needs to be realized to address the challenge of building predictive digital twins and we shall demonstrated the value of such technologies through a number of different examples of increasing complexity.

2022-11-30: Charles Henry Alexander Curry, NTNU (live)
Simulating diffusions and frame bundles
We discuss numerical methods for simulation of elliptic diffusions by stochastic development on orthonormal frame bundles, first introduced by Cruzeiro, Malliavin and Thalmeier. We demonstrate a framework for their analysis, discuss their benefits and failings, and demonstrate some extensions to geometric/hypoelliptic settings along with the limits/challenges of such efforts.

2022-11-30: Malin Nilsson, Chalmers & GU (postponed)

2022-11-23: Lukas Einkemmer, University of Innsbruck  (zoom)
Asymptotic preserving dynamical low-rank approximation
Solving high-dimensional kinetic equations (such as the Vlasov equation, gyrokinetic equations, or the Boltzmann equations numerically is extremely challenging. Methods that discretize phase space suffer from the exponential growth of the number of degrees of freedom, the so-called curse of dimensionality, while Monte Carlo methods converge slowly and suffer from numerical noise. In addition, standard complexity reduction techniques (such as sparse grids) usually perform rather poorly due to the lack of smoothness for such problems. Dynamical low-rank techniques approximate the dynamics by a set of lower-dimensional functions. For those low-rank factors, partial differential equations are derived that can then be solved numerically. Finding a good separation between the variables is often driven by physical insight. Due to this flexibility and their capacity to handle non-smooth solutions, dynamical low-rank approximations have been shown to work well for a range of kinetic equations. In this talk, we will discuss some recent advances in constructing asymptotic preserving dynamical low-rank methods. Such methods are useful in situations where a numerical scheme needs to capture a specific limit behavior. For example, a fluid or diffusive limit that arises when a
small parameter tends to zero.

2022-11-16: Siddhartha Mishra, ETHZ  (zoom)
Learning Operators
We introduce the emerging area of operator learning i.e., learning mappings between infinite-dimensional spaces from data. Such operators arise in many different contexts, especially in the numerical solutions of differential equations. We present different frameworks for operator learning and compare them both theoretically and on a suite of numerical experiments.

2022-11-09: Kasper Bågmark, Chalmers & GU (live)
The nonlinear filtering problem: An Energy-based Deep splitting method
In this talk I aim to give intuition and basic understanding of the Bayesian filtering problem in the continuous setting. Signals and observations are modeled by SDEs.  An important task is to estimate the signal given noisy observations, this is known as the filtering problem. We demonstrate classical approaches to estimate the filtering density. In the nonlinear case the classical methods suffer the curse of dimensionality. As an alternative I introduce our sampling based approximative filter. This method approximatively solves the Zakai equation—which models the filtering density—by combining an energy-based approach with a deep splitting method. We benchmark the method on linear and nonlinear settings.
This talk is based on the preprint “An energy-based deep splitting method for the nonlinear filtering problem” by K.B., A. Andersson, S. Larsson  https://arxiv.org/abs/2203.17153.

2022-11-02: Ernst Hairer, University of Geneva  (live)
High order PDE-convergence of ADI-type integrators for parabolic problems
This work considers space-discretised parabolic problems on a rectangular domain
subject to Dirichlet boundary conditions. For the time integration s-stage
AMF-W-methods, which are ADI (Alternating Direction Implicit) type integrators,
are considered. They are particularly efficient when the space dimension of the
problem is large. The classical algebraic conditions for order p (with p<=3) are shown
to be sufficient for PDE-convergence of order p (independently of the spatial resolution)
in the case of time independent Dirichlet boundary conditions. Under additional conditions,
PDE-convergence of order p=3.25-eps for every eps>0 can be obtained.
In the case of time dependent boundary conditions there is an order reduction.
This is joint work with Severiano Gonzalez-Pinto and Domingo Hernandez-Abreu.
Related publications can be downloaded from

Wednesday October 26,  Thursday October 27 (live)
4th Workshop on Scientific Computing in Sweden

Monday October 24, Tuesday October 25 (live)
JSPS Seminar

2022-10-19: Konstantinos C. Zygalakis, The University of Edinburgh  (zoom)
Differential Equations, numerical methods and optimisation algorithms
The ability of calculating the minimum (maximum) of a function lies in the heart of many applied mathematics applications.  In this talk, we will connect such optimization problems to the large time behaviour of solutions to  differential equations. Establishing such a connection allows  to utilise existing knowledge from the field of numerical analysis of differential equations. In particular, numerical stability is key for a good performing optimization  since the larger the time-step used while the limiting behaviour of the underlying differential equation is preserved, the more computationally efficient an algorithm is.  With this in mind we will explore the applicability of explicit stabilised Runge-Kutta methods for optimization. These methods are optimal in terms of their stability properties within the class of explicit integrators and we will show that when used as optimization methods they match the optimal convergence rate of the conjugate gradient method for quadratic optimization problems. Numerical investigations indicate that in the general case they are able to outperform state of the art optimization methods like Nesterov’s accelerated method. Time permitted, we will also discuss an alternative approach to the analysis of the long time behaviour of these differential equations and their numerical discretisations by introducing an appropriate control theoretic formulation of them.

2022-10-18: Roland Maier,  Friedrich-Schiller-Universität Jena (live)
Neural network approximation of coarse-scale surrogates in numerical homogenization
Coarse-scale surrogate models in the context of numerical homogenization of linear elliptic problems with arbitrary rough diffusion coefficients rely on the efficient solution of fine-scale sub-problems on local subdomains whose solutions are then employed to deduce appropriate coarse contributions to the surrogate model. However, in the absence of periodicity and scale separation, the reliability of such models requires the local subdomains to cover the whole domain which may result in high offline costs, in particular for parameter-dependent and stochastic problems.
In this talk, we justify the use of neural networks for the approximation of coarse-scale surrogate models. For a prototypical and representative numerical homogenization technique, the Localized Orthogonal Decomposition method, we show that one single moderately sized neural network is sufficient to approximate the coarse contributions of all occurring coefficient-dependent local sub-problems for a non-trivial class of diffusion coefficients up to arbitrary accuracy. We present rigorous upper bounds on the depth and number of non-zero parameters for such a network to achieve a given accuracy. Further, we analyze the overall error of the resulting neural network enhanced numerical homogenization surrogate model and present numerical examples.

2022-10-12: Georgios Akrivis, University of Ioannina  (live)
Discontinuous Galerkin time-stepping methods: Maximal regularity and a posteriori error estimates
We consider the discretization of differential equations satisfying the maximal parabolic $L^p$-regularity property in Banach spaces by discontinuous Galerkin (dG) methods. We use the maximal regularity framework to establish that the dG methods preserve the maximal $L^p$-regularity and satisfy corresponding a posteriori error estimates.
The a posteriori  estimators are of optimal asymptotic order of convergence. A key point in our approach is a suitable  interpretation  of  the dG methods as modified Radau IIA methods; this interpretation allows us to transfer the known maximal regularity property of Radau IIA methods to dG methods.

The talk is based on the papers:
G. A., Ch. G. Makridakis: On maximal regularity estimates for discontinuous Galerkin time-discrete methods,
SIAM J. Numer. Anal. \60 (2022) 180--194.
G. A., Ch. G. Makridakis: A posteriori error estimates for Radau IIA methods via maximal regularity.
Numer. Math. 150 (2022) 691--717.

2022-10-05: Kenneth Aksel Hvistendahl Karlsen, University of Oslo  (zoom)
Stochastic Conservation Laws
Kruzkov's ideas have over the years influenced many areas in nonlinear PDE. In this talk I will give an overview of well-posedness problems for stochastic conservation laws, with techniques taking some inspiration from Kruzkov.

2022-09-28: Moritz Hauck, University of Augsburg  (live)
Super-localization of elliptic multi-scale problems with an extension to spatial networks
Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough (non-periodic) coefficients. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size. This holds even for under-resolved rough coefficients. However, the canonical basis associated with this construction is non-local and, hence, numerically infeasible. This is why the true challenge of numerical homogenization is the identification of a computable local basis for such an operator-dependent approximate solution space. This talk introduces a constructive and near optimal solution to this localization problem for the prototypical elliptic model problem along with possible generalizations. In particular, the construction carries over to the setting of a weighted graph Laplacian on spatial networks fulfilling certain connectivity and homogeneity assumptions which enables near optimal localization also for such problems. A sequence of numerical experiments illustrates the significance of the novel localization technique when compared to other state-of-the-art results.

2022-09-07: Axel Målqvist, Chalmers & GU (live)
Iterative solution of spatial network models by subspace decomposition
We present and analyze a preconditioned conjugate gradient method (PCG) for solving spatial network problems. Primarily, we consider diffusion and structural mechanics simulations for fiber based materials, but the methodology can be applied to a wide range of models, fulfilling a set of abstract assumptions. The proposed method builds on a classical subspace decomposition into a coarse subspace, realized as the restriction of a finite element space to the nodes of the spatial network, and localized subspaces with support on mesh stars. The main contribution of this work is the convergence analysis of the proposed method. The analysis translates results from finite element theory, including interpolation bounds, to the spatial network setting. A convergence rate of the PCG algorithm, only depending on global bounds of the operator and homogeneity, connectivity and locality constants of the network, is established. The theoretical results are confirmed by several numerical experiments.

2022-09-06: Per-Gunnar Martinsson, University of Texas at Austin (live)
Direct solvers for elliptic PDEs
That the linear systems arising upon the discretization of elliptic PDEs can be solved efficiently is well-known, and iterative solvers that often attain linear complexity (multigrid, Krylov methods, etc) have proven very successful. Interestingly, it has recently been demonstrated that it is often possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will argue that such direct solvers have several compelling qualities, including improved stability and robustness, the ability to solve certain problems that have remained intractable to iterative methods, and dramatic improvements in speed in certain environments.

After a general introduction to the field, particular attention will be paid to a set of recently developed randomized algorithms that construct data sparse representations of large dense matrices that arise in scientific computations. These algorithms are entirely black box, and interact with the linear operator to be compressed only via the matrix-vector multiplication.

2022-06-08: Gilles Vilmart, University of Geneva
Superconvergent methods inspired by the Crank-Nicolson scheme in the context of diffusion PDEs
In this talk, we present two different situations where the Crank-Nicolson method is surprisingly more accurate than one could expect and inspires the design of new efficient numerical integrators:
-in the context of splitting methods for deterministic parabolic PDEs with inhomogeneous general oblique boundary conditions, where order reduction phenomena can be avoided,
-in the context of ergodic parabolic stochastic PDEs, where high order can be achieved for sampling the invariant distribution, in spite of the low regularity of the solution.

This talk is based on joint works with Assyr Abdulle, Ibrahim Almuslimani, Guillaume Bertoli, Christophe Besse, and Charles-Edouard Bréhier.

2022-06-01: Andrii Dmytryshyn, Örebro University
Versal deformations of matrices
Jordan canonical form for matrices is well known and studied with various purposes but reduction to this form is an  unstable operation: both the corresponding canonical form and the reduction transformation depend discontinuously on the entries of an original matrix. This issue complicates the use of the canonical form for numerical purposes. Therefore V.I. Arnold introduced a normal form to which an arbitrary family of matrices A' close to a given matrix A can be reduced by similarity transformation smoothly depending on the entries of A’. He called such a normal form a versal deformation of A.
In this presentation we will discuss versal deformations and their use in investigation of possible changes in canonical forms (eigenstructures), reduction of unstructured perturbations to structured perturbations, and codimension computations.

2022-05-25: Mohammad Asadzadeh, Chalmers & GU
On HP-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system
We study stability and convergence of hp-streamline diffusion (SD) finite element, and Nitsche's schemes
for the three dimensional, relativistic (3 spatial dimension and 3 velocities), time dependent Vlasov-Maxwell system
and Maxwell's equations, respectively. For the hp scheme for the Vlasov-Maxwell system, assuming that the exact
solution is in the Sobolev space H^{s+1}, we derive global a priori error bound of order O(h/p)^{s+1/2},
where h(= maxK h_K) is the mesh parameter and p(= max_K p_K) is the spectral order. This estimateis based on the local version with h_K = diam K being the diameter of the phase-space-time element K and p_K is the spectral order (the degree of approximating finite element polynomial) for K. As for the Nitsche's scheme, by a simple calculus of the  field equations, we convert the Maxwell's system to an elliptic type equation. Then, combining the Nitsche's method for the spatial discretization with a second order time scheme, we obtain optimal convergence of O(h^2 +k^2), where h is the spatial mesh size and k is the time step.Here, as in the classical literature, the second order time scheme requires higher order regularity assumptions. Numerical justification of the results, in lower dimensions, is presented.
This is a joint work with P.Kowalczyk and C. Standar (appeared in KRM, 2019)

2022-05-18: Vidar Thomée, Chalmers & GU
On Positivity Preservation in Finite Element Methods for the Heat Equation
We consider  the initial boundary value problem for the homogeneous
heat equation, with homogeneous Dirichlet boundary
conditions. By the maximum principle  the solution is
nonnegative for positive time if the initial data are nonnegative.
We study to what extent this property carries over to some
finite element discretizations.

2022-05-11: Kent-Andre Mardal, Simula
Multi-physics problems related to brain clearance
Recent theories suggest that a fundamental reason for sleep is simply clearance of metabolic waste produced during the activities of the day. In this talk we will present multi-physics problems and numerical schemes that target these applications. In particular, we  will be lead from basic applications of neuroscience into multi-physics problems involving Stokes, Biot and fractional solvers at the brain-fluid interface.

2022-05-04: Sebastian Reich, University of Potsdam
Robust parameter estimation using the ensemble Kalman filter
Estimating the parameters of a stochastic differential equation (SDE) from
continuous time observations of the process constitutes a classical inverse problem.
It is well-known that the maximum likelihood solution to this inverse problem does not
depend continuously on the data, i.e., is not robust, under the standard topology of continuous functions.
Here we revisit this problem from the perspective of continuous-time gradient descent and
continuous-time ensemble Kalman filtering. Both approaches lead SDEs in the unknown parameters
which are driven by the given observations in a multiplicative manner. This perspective allows
us to clearly identify the source of the non-robustness via a rough path analysis as well as to
identify two possible solutions to the non-robustness issue.

2022-04-27: Johan Hoffman, KTH
Matrix Schur factorization and the structure of turbulence
Any matrix is unitary equivalent to an upper triangular matrix, expressed as a Schur factorization. Analogously, a spectral theorem states that any normal matrix is unitary equivalent to a diagonal matrix. Therefore, a matrix can be decomposed into a sum of a normal matrix corresponding to the diagonal part of the upper triangular matrix of the Schur factorization, and a non-normal matrix corresponding to the remaining non-diagonal part of the upper triangular matrix. The normal matrix may then be further decomposed into the sum of a symmetric and a skew-symmetric matrix. Hence, the result is a triple decomposition of a general matrix into a sum of a symmetric matrix with real eigenvalues, a skew-symmetric matrix with purely imaginary eigenvalues, and a non-normal matrix. In fluid mechanics, it is common to separate straining flow from rotating flow by a decomposition of the velocity gradient tensor into a symmetric and a skew-symmetric part. This double decomposition, however, does not distinguish shear flow from any of the two flow components. In contrast, a triple decomposition of the velocity gradient tensor leads to a decomposition of any flow field into pure straining flow, rigid body rotational flow, and shear flow. First proposed as a flow visualization technique for improved vortex identification, recent interest in the triple decomposition has focused e.g. on improved analysis of turbulence simulations, models for thrombosis in blood flow simulations, and stability analysis flow structures in turbulence. 

2022-04-21: Ozan Öktem, KTH
Microlocal analysis and deep learning for tomographic reconstruction
The talk outlines recent progress in developing domain adapted deep  neural networks for the task of (a) extracting the wavefront set of an  image from its shearlet coefficients and (b) inpainting the invisible  part of the wavefront set in limited angle tomography. A key component in both tasks is to represent them as optimal  non-randomised decision rules in statistical decision theory. The talk  will also outline how to combine these two networks with a deep neural  network for reconstruction, whose architecture is obtained by unrolling a suitable iterative scheme. Specifying the visible parts  of the wavefront set relies on characterising the microlocal canonical  relation of the deep neural network for reconstruction, which here  inverts the ray transform. This results in a deep learning based approach for limited angle tomographic reconstruction  that is aware of the microlocal canonical relation for the ray transform  and also on the characterisation of visible part of the wavefront set. 

2022-04-20: Efthymios Karatzas, National Technical University of Athens
Random geometries and a Unified Reduced Order Basis for Parametrized PDEs based on Embedded Finite Element Methods and applications
We consider parametrized PDEs, geometrical randomly deformed systems, and we present a new beneficial approach for reduced basis construction based on level set geometry descriptions and fixed background geometries. This unified Reduced Order Basis employs a background mesh, solves efficiently with less computational cost, and it is independent of any random parameter which affects the physics of the PDE model. We will discuss results related to unfitted finite element methods for parameterized partial differential equations enhanced by a proper orthogonal decomposition method. This approach achievements are twofold. Firstly, we reduce much the computational effort since the unfitted mesh method allows us to avoid remeshing when updating the parametric domain. Secondly, the proposed reduced order model technique gives an implementation advantage considering geometrical parametrization. Computational efforts are even exploited more efficiently since the mesh is computed once and the transformation of each geometry to a reference geometry is not required. These combined advantages allow to solve many PDE problems faster and “cheaper” and provide the capability to find solutions in cases that could not be resolved in the past.
Acknowledgments. This work has received funding from the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under grant agreement No[1115], and the ”First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment” grant 3270.

2022-04-13: Marlis Hochbruck, Karlsruhe Institute of Technology
Exponential integrators for quasilinear hyperbolic evolution equation
In this talk we propose two exponential integrators of first and second order applied to quasilinear hyperbolic evolution equations. We work in an analytical framework which is an extension of the classical Kato framework and covers quasilinear Maxwell’s equations in full space and on a smooth domain as well as a class of quasilinear wave equations.
In contrast to earlier works, we do not assume regularity of the solution but only on the data. From this we deduce a well-posedness result upon which we base our error analysis.

This is joint work with Benjamin Dörich, KIT
B. Dörich and M. Hochbruck. Exponential integrators for quasilinear wave-type equations. CRC 1173 Preprint 2021/12, Karlsruhe Institute of Technology, 2021, to appear in SIAM J. Numer. Anal.
https://www.waves.kit. edu/downloads/CRC1173_Preprint_2021-12.pdf

2022-04-12: Andreas Rosén (Chalmers and GU)
Dirac Integral equations, plasmonics and eddy currents
A well-known problem in Maxwell scattering is how to handle the divergence-free constraints on the fields. We present a recently developed integral equation reformulation of the Maxwell transmission problem, which solves this problem by embedding Maxwell's equations in a more stable 8/8 Dirac system. In the talk we discuss the Dirac integral equation obtained from a Cauchy representation of the fields, and how to optimize its numerical performance by tuning its 12 free parameters. We also demonstrate numerical results: an efficient solver for dielectric, plasmonic, and eddy current scattering without any false eigenwavenumbers or low-frequency breakdown. This is joint work with Johan Helsing and Anders Karlsson, Lund.

2022-04-06: Andreas Petersson, UiO
Numerical approximation of the heat modulated infinite dimensional Heston model
The HEat modulated Infinite DImensional Heston (HEIDEH) model and its numerical approximation are introduced and analysed. This is a special case of the infinite dimensional Heston stochastic volatility model of (F.E. Benth, I. C. Simonsen '18). Therein, the authors consider a potential model for risk-neutral forward prices of commodity-delivering contracts. The model consists of a one-dimensional stochastic advection equation coupled with a stochastic volatility process, defined as a Cholesky-type decomposition of the tensor product of a Hilbert-space valued Ornstein-Uhlenbeck process. In this work the Ornstein-Uhlenbeck process is specified to be the solution to a stochastic heat equation and the resulting HEIDEH model is studied in a fractional Sobolev space setting.

In the talk, a description and motivation of the model will be given. Then, a class of covariance kernels are described that give rise to admissible Q-Wiener processes. Regularity of the model is discussed under this class of kernels. Finally, an approximation based on a combination of an explicit finite difference scheme, a finite element method, the backward Euler scheme and the circulant embedding method is presented. Convergence rates are derived with the error measured pointwise, in a mean square sense, in time and space. The resulting rates are higher than what can be obtained from a standard Sobolev embedding technique.

This is joint work with Fred Espen Benth, Gabriel Lord and Giulia Di Nunno.
2022-03-30: Christian Lubich, University of Tübingen
A large-stepsize modified Boris method for charged-particle dynamics in a strong nonhomogeneous magnetic field
Efficient numerical integrators for charged-particle dynamics are of substantial interest in the context of particle methods for the partial differential equations arising in plasma physics. The standard integrator in this field is the Boris method but that method requires tiny step sizes in the presence of strong magnetic fields. We give an error analysis of a remarkably simple modification of the Boris method that was recently proposed by Xiao and Qin (Computer Physics Comm., 2021). We show that the guiding center motion of a charged particle in a nonhomogeneous magnetic field of size inversely proportional to a small parameter $\eps\ll 1$ is approximated on fixed time intervals with a second-order error $O(h^2)$ for step sizes $h$ that satisfy $h^2 \ge \eps$, as opposed to the standard Boris method that would require tiny step sizes $h \ll \eps$. The proof is based on comparing the modulated Fourier expansions of the exact solution, which was studied by Hairer and Lubich (Numer. Math., 2020), and of the numerical solution of the modified Boris method.
The talk is based on joint work with Yanyan Shi.

2022-03-23: Sergio Blanes, Valencia Polytechnic University
Positivity-preserving methods for ordinary differential equations
Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge--Kutta methods and multistep methods,  face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier:  second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need  be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.

2022-03-16: Max Jensen, Sussex University
Finite Element Approximation of Hamilton-Jacobi-Bellman equations with nonlinear mixed boundary conditions
We show uniform convergence of monotone P1 finite element methods to the viscosity solution of isotropic parabolic Hamilton-Jacobi-Bellman equations with mixed boundary conditions on unstructured meshes and for possibly degenerate diffusions. Boundary operators can generally be discontinuous across face-boundaries and type changes. Robin-type boundary conditions are discretised via a lower Dini derivative. In time the Bellman equation is approximated through IMEX schemes. Existence and uniqueness of numerical solutions follows through Howard’s algorithm. We show how equations of this type naturally appear in models of mathematical finance.

2022-03-09: Tony Stillfjord, Lund University
SRKCD: stabilized Runge-Kutta methods for stochastic optimization
I will introduce a family of stochastic optimization methods that we call SRKCD, which are based on the Runge-Kutta-Chebyshev (RKC) schemes. The RKC methods are explicit methods originally designed for solving stiff ordinary differential equations by ensuring that their stability regions are of maximal size. In the optimization context, this allows for larger step sizes (learning rates) and better robustness compared to e.g. the popular stochastic gradient descent method.

Our main contribution is a convergence proof, not only for SRKCD but for essentially any stochastic optimization method based on an explicit Runge-Kutta scheme. This shows convergence in expectation with an optimal sublinear rate, under standard assumptions of strong convexity and Lipschitz-continuous gradients. For non-convex objectives, we get convergence to zero in expectation of the gradients. I will illustrate the improved stability properties of SRKCD on both a small-scale test example and on a problem arising from an image classification application in machine learning.

2022-03-02: Anna Persson, KTH
Superconvergence of a multiscale method for ground state computations of Bose-Einstein condensates
Bose-Einstein condensates are formed when a gas of Bosons are cooled to ultra-low temperatures. At this point, most of the Bosons occupy the same quantum state and behave like one giant ``macro particle''. This is classified as a new state of matter, where some quantum mechanical phenomena can be observed macroscopically. One such phenomenon is superfluidity, which describes a flow without inner friction.

To compute ground states of Bose-Einstein condensates the stationary Gross-Pitaevskii equation (GPE) is often used. In this talk we revisit a two-level multiscale technique introduced in (P.Henning, A.Målqvist, D. Petersim '14) for computing numerical approximations to the GPE. The method is based on localized orthogonal decomposition of the solution space, which exhibit high approximation properties and improves the convergence rates compared to classical finite element methods. This reduces the computational cost for computing the ground states significantly.

In the previous paper, high order convergence of the method was proven, but even higher orders were observed numerically. In this talk we show how to improve the analysis to achieve the observed rates, which are as high as O(H^6) for the eigenvalues. We also show some numerical experiments for both smooth and discontinuous potentials.

2022-02-23: Dan Wang, Hong Kong University of Science and Technology
Phase Analysis of MIMO LTI Systems
In this work, we define the phases of a special class of complex matrices, called sectorial matrices. Various properties of matrix phases have been studied. In particular, a majorization relation between the phases of the eigenvalues of AB and the phases of A and B is established. With the concept of matrix phases, we define the phase response of MIMO LTI systems, which, together with magnitudes response, gives a complete MIMO Bode plot. The phase concept subsumes the well-known notions of positive real systems and negative imaginary systems. We also formulate a small phase theorem for feedback stability, as a counterpart to the celebrated small gain theorem.

2022-02-16: Anders Logg, Chalmers and GU
Mesh generation for large-scale city modeling and simulation
Digital Twins are digital replicas of physical things. Digital twins
have traditionally been used in the manufacturing industry to model
products and production processes. In recent years, Digital Twins have
extended to new domains and are now becoming reality for buildings and
whole cities.

At the Digital Twin Cities Centre, we are developing new methods for
large-scale city modeling and simulation. An important first challenge
is how to automatically and efficiently generate computational meshes
from raw data. In this talk, I will present and demonstrate a new
method for 3D mesh generation from publicly available data in the form
of point clouds and cadastral maps.

This is joint work together with Vasilis Naserentin.

2022-02-09: Daniele Boffi, KAUST
On the inf-sup condition for a mixed formulation
The stability and convergence analysis of finite element
discretizations arising from mixed formulations is based on the
validity of suitable inf-sup conditions. It is well known that the
inf-sup conditions are sufficient and necessary, in a suitable sense,
for the optimal behavior of a discrete scheme.
Starting from an example coming from linear elasticity, we discuss a
non standard mixed formulation for the approximation of elliptic
problems.  We will show that, although the inf-sup condition is not
satisfied, the method can be successfully used if certain regularity
assumptions are met.
This is a joint work with Fleurianne Bertrand.

2022-02-02: Ingrid Lacroix-Violet, Institut Élie Cartan de Lorraine
Linearly implicit numerical methods
In this talk, I will present a new class of numerical methods for the time integration of evolution equations. The systematic design of these methods mixes the Runge-Kutta collocation formalism with collocation techniques in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so. After the introduction of the methods, I will set suitable definitions of consistency and stability for these methods, prove their convergence and order on the ODE case and finally I will present some numerical simulations.

2022-01-26: Axel Ringh, Chalmers and GU
Multi-marginal graph-structure optimal transport: Modeling, applications, and computational methods
Optimal transport has seen rapid development over the last decades, and recently there has been a surge of research related to computational methods for numerically solving the problem. The most well-known algorithm is the Sinkhorn algorithm, in which the transport plan is iteratively updated so that its projections match the given marginals.

In this talk I will consider a generalization of optimal transport, namely multi-marginal optimal transport. In particular, I will introduce graph-structured multi-marginal optimal transport and show that this type of structured problems can be used to model and solve a variety of different problems. This includes, e.g., barycenter problems, displacement interpolation problems, and multi-species density control problems. Moreover, while the Sinkhorn algorithm extends to the multi-marginal setting, it only partially alleviates the computational difficulty for the multi-marginal problem. This is because computing the corresponding projections needed in the algorithm still scales exponentially in the number of marginals. However, in this talk I will present how these projections can be efficiently compute when the underlying graph structure is a trees, as well as for some other simple graph structures.

2022-01-19: Stig Larsson, Chalmers & GU
What is DeepONet?
In this talk I will review the paper ``DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators'' by Lu Lu, Pengzhan Jin, and George Em Karniadakis, arXiv:1910.0319

2022-01-12: Adam Andersson, Chalmers & GU and Saab
Robust machine learning methods for coupled FBSDE from stochastic control
Forward Backward Stochastic Differential Equations (FBSDE) are important in mathematical finance, stochastic control and for stochastic representations of second order parabolic PDE. It is a coupled system of one forward and one backward SDE. Contrary to ODE, where a backward problem can be reformulated as a forward problem, and vice versa, by a change of variable, forward and backward SDE equations are fundamentally different from each others. In particular, their numerical methods differs significantly. While solutions to forward SDE are easily approximated with the Euler-Maruyama scheme (or more sophisticated schemes) schemes for backward SDE relies on approximating conditional expectations. In 2017 E, Jan and Jentzen introduced the deep BSDE method. It is a deep learning based numerical scheme for BSDE. In this talk I present ongoing work with Kristoffer Andersson and Cornelis Oosterlee (CWI Amsterdam and Utrecht University). We show numerical examples that indicates that the desired solutions of the deep BSDE and FBSDE methods, which are optimisation problems, for some equations are obtained at local minimas. In such cases, the global minimum approximates the solution poorly and I will explain why this is particularly bad for stochastic control problems. An alternative family of related methods is introduced and a partial error analysis is presented for it. Experimental convergence rates are given for different examples. The problems encountered in the full error analysis are discussed.

2021-12-08: Sonja Cox, University of Amsterdam
An affine infinite-dimensional stochastic volatility model
Affine stochastic processes have received a considerable amount of attention in the past years due to their tractability and (relative) flexibility. For example, in 2011 Cuchiero, Filipovic, Mayerhofer, and Teichmann provided a characterization of all affine processes taking values in the cone of non-negative semi-definite matrices, thus identifying all affine finite-dimensional stochastic covariance models. Inspired by the need of infinite-dimensional stochastic covariance models, we established existence of affine processes in the cone of positive self-adjoint Hilbert-Schmidt operators.
In my talk I will explain what affine processes are, present our infinite-dimensional existence result, and give some examples of infinite-dimensional affine volatility models.

The talk concerns joint work with Sven Karbach and Asma Khedher

2021-12-01: Irina Pettersson, Chalmers & GU
Multiscale analysis of myelinated axons
A neuron is a basic structural unit of the nervous system, and one needs to know how a signal propagates along neurons to be able to simulate the excitation. We focus on the multiscale modeling of a myelinated axon. Considering a microstructure with alternating myelinated parts and nodes Ranvier,  we derive an effective nonlinear cable equation describing the potential propagation along a single axon. Cable equations used in electrophysiology are traditionally formulated based on an equivalent circuit consisting of a capacitor in parallel with a conductor. Such models, however, do not take into account the geometry of the myelin sheath.   
I will also discuss some recent results about the multiscale analysis of a bundle consisting of many myelinated axons.

2021-11-24: Buyang Li, The Hong Kong Polytechnic University
Strong convergence of exponential integrators for the semilinear stochastic heat equation with nonsmooth initial data and additive noise

2021-11-17: Charles Elliott, University of Warwick
Evolving finite elements
We discuss some ideas around the numerical approximation of PDEs on evolving domains. Particular concerns are models arising in cell biology concerning two phase biomembranes and cell motility.

2021-11-15: Carlos Jerez Hanckes, Universidad Adolfo Ibanez, Chile
High Order Galerkin Methods for Helmholtz Scattering in Quasi-Periodic Layered Media and Screens
We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from Helmholtz problems in multi-layered periodic structures or gratings as well as in screens. Employing suitably parametrized Fourier basis we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed schemes. Through several numerical examples, we confirm our findings and show performances competitive to those attained via Nyström methods.

2021-11-10: Josefin Ahlkrona, Stockholm University
Finite Element Ice-Ocean Modelling
Better computer models of ice sheets in contact with the warming ocean are needed in order to reduce the uncertainty in estimates of future sea level rise. In our group we focus on FEM modelling of ice sheets and the adjacent ocean, using mathematical models of high accuracy. In this talk we give and overview of our work on overcoming some of the major challenges of ice-ocean modelling, namely the stability and representation of a free surface problems, coupled FEM modelling of ocean flow in ice cavities, and solvers for the arising linear and non-linear systems.

2021-11-04: Stefan Horst Sommer, University of Copenhagen
Stochastic shape analysis and probabilistic geometric statistics
Analysis and statistics of shape variation can be formulated in geometric settings with geodesics modelling transitions between shapes. The talk will concern extensions of these smooth geodesic models to account for noise and uncertainty: Stochastic shape processes and stochastic shape matching algorithms. In the stochastic setting, matching algorithms take the form of bridge simulation schemes which also provide approximations of the transition density of the stochastic shape processes. The talk will cover examples of stochastic shape processes and connected bridge simulation algorithms. I will connect these ideas to geometric statistics, the statistical analysis of general manifold valued data, particularly to the diffusion mean.

2021-11-03: Daniel Peterseim, Universität Augsburg​
Energy-adaptive Riemannian Optimization on the Stiefel Manifold
This talk addresses the numerical simulation of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimisation problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimisers we propose a novel Riemannian gradient descent method induced by an energy-adaptive metric. Quantified convergence of the method is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established  schemes.

This is joint work with Robert Altmann (U Augsburg) and Tatjana Stykel (U Augsburg).

2021-10-20: Nick Higham, The University of Manchester
Random Orthogonal Matrices in Numerical Linear Algebra
An important use of random orthogonal matrices is in forming random matrices with
given singular values—so-called “randvsd” matrices. We explain how to speed up the usual
approach to forming a randsvd matrix that employs random orthogonal matrices from
the Haar distribution. We also explain the role that random orthogonal matrices play
in a recently identified class of random, dense n × n matrices for which LU factorization
with any form of pivoting produces a growth factor typically of size at least n/(4 log n)
for large n.

2021-10-14: Steven A. Gabriel, University of Maryland/NTNU
Some Approaches for Solving the Discretely-Constrained Mixed Complementarity Problem
Many interesting equilibrium problems in game theory, engineering, and more generally involving systems of independent players can be modeled via the mixed complementarity problem (MCP) or the related variational inequality problem (VI).  These equilibrium formulations normally assume that the decision variables need to be continuous.  One type of solution method then is to transform the original MCP into a non-smooth, zero-finding problem and then apply approaches that iteratively apply smooth approximations and traditional (smooth) methods. This presentation will provide some examples of new approaches that solve MCPs in which a subset of the variables additionally need to be integer-valued, often binary.  This leads to a host of interesting discretely-constrained MCP applications.  We describe some recent approaches to solve these DC-MCPs and give motivating examples.

2021-10-13: Aurélien Deya, Institut Élie Cartan de Lorraine
A full discretization of the rough fractional linear heat equation

2021-10-06: Elias Jarlebring, KTH
Computation graph for matrix functions
Matrix functions, such as the matrix exponential, matrix square root, the sign-function, are fundamental concepts that appear in wide range of fields in science and technology. In this talk, the evaluation of  matrix functions is viewed as a computational graphs. More precisely, by viewing methods as directed acyclic graphs (DAGs)  we can improve and analyze existing techniques them, and  derive new ones. The accuracy of these matrix techniques can be characterized by the accuracy of their scalar counterparts, thus designing algorithms for matrix functions can be regarded as a scalar-valued optimization problem. The derivatives needed during the optimization can be calculated automatically by exploiting the structure of the DAG, in a fashion analogous to backpropagation. These functions and many more features are implemented in GraphMatFun.jl, a Julia package that offers the means to generate and manipulate computational graphs, optimize their coefficients, and generate Julia, MATLAB, and C code to evaluate them efficiently at a matrix argument. The software also provides tools to estimate the accuracy of a graph-based algorithm and thus obtain numerically reliable methods. For the exponential, for example, using a particular form (degree-optimal) of polynomials produces implementations that in many cases are cheaper, in terms of computational cost, than the Padé-based techniques typically used in mathematical software. The optimized graphs and the corresponding generated code are available online.

2021-09-29: Eskil Hansen, Lund University​
Convergence analysis of the nonoverlapping Robin–Robin method for nonlinear elliptic equations
The nonoverlapping Robin–Robin method is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. In this talk we present a convergence analysis for the Robin–Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. The analysis relies on a new theory for nonlinear Steklov–Poincaré operators based on the p-structure and the Lp-generalization of the Lions–Magenes spaces. This framework allows the reformulation of the Robin–Robin method into a Peaceman–Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. 

This is joint work with Emil Engström (Lund University)

2021-09-22: Mikhail Roop, Chalmers & GU
Singularities of solutions to barotropic Euler equations
Barotropic Euler equations are a particular case of Jacobi type systems, which have a geometrical representation in terms of differential 2-forms on 0-jet space. This observation naturally leads to the notion of a generalized (multivalued) solution of Euler equations understood as an integral manifold of the mentioned forms. We combine this observation with adding a differential constraint to the original system in such a way that integration can be performed explicitly. We will discuss two ways of finding such constraints: in one of them, the mentioned specific geometry of barotropic Euler equations is used, while another one is based on differential invariants of their symmetry group and quotient PDEs. Finally, we will show how all these ideas make it possible to find solutions with singularities.

2021-09-15: Carina Geldhauser, Lund University
Space-discretizations of reaction-diffusion SPDEs
In this talk we will discuss two different viewpoints on a space-discrete reaction-diffusion equation with noise: First, as an
interacting particle system in a bistable potential, and second, as a lattice differential equation. Each viewpoint sheds light on a different phenomenon, which will be highlighted in the talk.
Based on joint works with A. Bovier (Bonn) and Ch. Kuehn (TU Munich).

2021-09-08: Morgan Görtz, Chalmers & GU
Network Models For Paper Simulations
Paper is composed of a substantial amount of paper fibers. These fibers create a weblike structure held together with microscopic forces and mechanical locking. This web lacks uniformity, and modeling typically requires this structure to be represented to be relevant. However, resolving individual paper fibers leads to enormous numerical problems, and simplifications are necessary for efficiency. Our proposed approach uses a network model, where the single paper fibers are interpreted as one-dimensional beams. Paired with classical linearized beam theory, a simple yet effective linear network model may be constructed. The network model is introduced in detail in this presentation, along with experimental validation results for four different mechanical tests on low-density paper and paperboard. The talk will finish with preliminary homogenization results for the network model and a short comment on enabling large-scale simulations.

2021-06-16: Marcus Grote​, University of Basel​
Stabilized leapfrog based local time-stepping method for wave propagation
For the time integration of second-order wave equations, the leapfrog (LF) method probably remains to this day the most popular numerical method. Based on a centered finite difference approximation of the second-order time derivative,
it is second-order accurate, explicit, time-reversible, and, for linear problems, conserves (a discrete version of) the energy for all time. For the spatial discretization of partial differential equations, finite element methods (FEM) provide a flexible approach, which easily accommodates a varying mesh size or polynomial degree. Not only do FEM permit the use of high-order polynomials, necessary to capture the oscillatory nature of wave phenomena and keep numerical dispersion (
''pollution error'') minimal, but they are also apt at locally resolving small geometric features or material interfaces. Hence the combined FEM and LF based numerical discretization of second-order wave equations has proved a versatile and highly efficient approach, be it for the simulation of acoustic or elastic waves.

Local mesh refinement, however, can cause a severe bottleneck for the LF method, or any other standard explicit scheme, due to the stringent CFL stability condition on the time-step imposed by the smallest element in the mesh. Even when the locally refined region consists of a few small elements only, those elements will dictate a tiny time-step throughout the entire computational domain for stability. To overcome the crippling effect of a few small elements, without sacrificing its
high efficiency or explicitness elsewhere, a leapfrog based explicit local time-stepping (LF-LTS) method was proposed for the time integration of second-order wave equations. Recently, optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step depends on the smallest elements in the mesh. In general one cannot improve upon that stability constraint, as the LF-LTS method may become unstable at certain
discrete values of the time-step. To remove those critical values for the time-step, we apply a slight modification to the original LF-LTS method which nonetheless preserves its desirable properties: it is fully explicit, second-order accurate,
satisfies a three-term (leapfrog like) recurrence relation, and conserves the energy. The new stabilized LF-LTS method
also yields optimal convergence rates for a standard conforming FE discretization, yet under a CFL condition
where the time-step no longer depends on the mesh size inside the locally refined region.

Joint work with: Simon Michel (Univ. of Basel)  and  Stefan Sauter (Univ. of Zurich)

2021-06-09: Ulrik Skre Fjordholm​, University of Oslo
Numerical methods for conservation laws on graphs
We consider a set of scalar conservation laws on a graph. Based on a choice of stationary states of the problem – analogous to the constants in Kruzkhov's entropy condition – we establish the uniqueness and stability of entropy solutions. For rather general flux functions we establish the convergence of an easy-to-implement Engquist–Osher-type finite volume method.

This is joint work with Markus Musch and Nils Henrik Risebro (University of Oslo).

2021-06-02: André Massing​, NTNU
Stabilized Cut Discontinuous Galerkin Methods
To ease the burden of mesh generation in finite-element based simulation pipelines, novel so called unfitted finite element methods have gained much attention in recent years. The main idea is to embed the domain of interest into a structured but unfitted background mesh where the domain boundary can cut through the mesh in an arbitrary fashion. Unfitted finite-element based methods typically suffer from stability and conditioning problems caused by the presence of small cut elements. In this talk we develop both theoretically and practically a novel cut Discontinuous Galerkin framework (CutDG) by combining stabilization techniques from the cut finite element method with the classical interior penalty discontinuous Galerkin methods for elliptic and hyperbolic problems. To cope with robustness problems caused by small ​cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi-)norms associated with​ the relevant partial differential operators.  A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the both elliptic and stationary advection-reaction problems which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two-and three-dimensional test problems.

2021-05-26: Sven-Erik Ekström​, Uppsala University
Matrix-less methods: approximating eigenvalues and eigenvectors without the matrix
Discretizing partial differential equations (PDE) typically gives rise to structured matrices. Exploiting these, often hidden, structures can be very beneficial when analysing these matrices or when constructing fast solvers.
In this talk we first present an overview of the theory of generalised locally Toeplitz (GLT) sequences; we introduce the notions of matrix sequences and symbols, and how one can approximate eigenvalues and singular values of matrices belonging to these sequences.
Then, we discuss the so-called matrix-less methods (not to be confused with matrix-free methods). These methods are extremely fast and efficient compared with conventional methods for computing eigenvalues.
Illustrative numerical examples will be presented, from model problems and PDE discretizations, both for Hermitian and non-Hermitian matrix sequences.
Finally, we show preliminary results for approximating the eigenvalues of discretization matrices of variable coefficient problems, and the computation of eigenvectors of Hermitian Toeplitz matrices.
2021-05-19: Sara Zahedi​, KTH 
A cut finite element method for incompressible two-phase flows
I will present a computational technique, a space-time Cut Finite Element Method (CutFEM), that can be used for simulations of two-phase flow problems. With CutFEM we develop a strategy for accurately approximating solutions to Partial Differential Equations (PDEs) in complex geometries that are arbitrarily located with respect to a fixed background mesh. We consider the time-dependent Navier-Stokes equations involving two immiscible incompressible fluids with different viscosities, densities, and with surface tension. Due to surface tension effects at the interface separating the two fluids and different fluid viscosities the pressure may be discontinuous and the velocity field may have a kink across the interface.  I will address several challenges that computational methods for such simulations must handle, such as how to accurately capture discontinuities across an evolving interface and how to compute the mean curvature vector needed for the surface tension force with a convenient implementation allowing the interface to be arbitrary located with respect to a fixed mesh. 

Des Higham​, The University of Edinburgh
Should We Be Perturbed About Deep Learning?​​
Many commentators are asking whether current AI solutions are sufficiently robust, resilient, and trustworthy; and how such issues should be quantified and addressed. I believe that numerical analysts can contribute to the debate. In part 1 of this talk I will look at the common practice of using low precision floating point formats to speed up computation time. I will focus on evaluating the softmax and log-sum-exp functions, which play an important role in many classification tools. Here, across widely used packages we see mathematically equivalent but computationally different formulations; these variations have been designed in an effort to avoid overflow and underflow. I will show that classical rounding error analysis gives insights into their floating point accuracy, and suggests a method of choice. In part 2 I will look at a bigger picture question concerning sensitivity to adversarial attacks in deep learning. Adversarial attacks are deliberate, targeted perturbations to input data that have a dramatic effect on the output; for example, a traffic "Stop" sign on the roadside can be misinterpreted  as a speed limit sign when minimal graffiti is added. The vulnerability of systems to such interventions raises questions around security, privacy and ethics, and there has been a rapid escalation of attack and defence strategies. I will consider a related higher-level question: under realistic assumptions, do adversarial examples always exist with high probability? I will also introduce and discuss the idea of a stealth attack: an undetectable, targeted perturbation to the trained network itself.

Part 1 is joint work with Pierre Blanchard (ARM) and Nick Higham (Manchester).
Part 2 is joint work with Alexander Gorban and Ivan Tyukin (Leicester).

2021-05-05: Erik Jansson​, Chalmers & GU
A recursive SFEM scheme for fractional elliptic PDEs on the sphere
In this talk I consider a class of fractional elliptic partial differential equations on the sphere.  I present a recursive surface finite element scheme for approximation of solutions to these equations.  I derive convergence rates and analyze the properties of the method. I also present how this algorithm can be used for surface finite element-based simulation of spherical Gaussian random fields of Matérn type. 
This talk is based on joint work with Annika Lang and Mihály Kovács.

2021-04-28: Eric Cancès,  Ecole des Ponts ParisTech and INRIA Paris
Iterative methods in quantum chemistry and first-principle materials science
Electronic structure calculation is one of the major application fields of scientific computing. It is used daily in any chemistry or materials science department, and it accounts for a high percentage of machine occupancy in supercomputing centers. Current challenges include the study of complex molecular systems and processes (e.g. photosynthesis, high-temperature superconductivity...), and the building of large, reliable databases for the design of materials and drugs.
The most commonly used models are the Kohn-Sham Density Functional Theory (DFT), and the (post) Hartree-Fock models. The Hartree-Fock and Kohn-Sham models have similar mathematical structures. They consist in minimizing an energy functional on the Sobolev space $(H^1(R^3))^N$ under $L^2$-orthonormality constraints. The associated Euler-Lagrange equations are systems on nonlinear elliptic PDEs. After discretization in a Galerkin basis, one obtains smooth optimization problems on matrix manifolds, or on convex hulls of matrix manifolds.
Solving these problems is easy for small simple molecular systems, but very challenging for large or complex systems. Two classes of numerical methods compete in the field: constrained direct minimization of the energy functional, and self-consistent field (SCF) iterations to solve the Euler-Lagrange equations. In this talk, I will present a comparative study of these two approaches, as well as new efficient algorithms for systems with spin symmetries.

2021-04-21: Adrian Muntean, Karlstad University
Does vesicle micro-dynamics enhance the signalling among plants macro-transport? A modeling with measures approach
We study a transport problem for signalling among plants in the context of measure-valued equations. We report on preliminary results concerning the modelling and mathematical analysis of a reaction-diffusion scenario involving the  macroscopic diffusion of signalling molecules enhanced by the presence of a finite number of microscopic vesicles - pockets with own dynamics able to capture and release signals as a relay system. The coupling between the macroscopic and microscopic spatial scales relies on the use of a two-scale transmission condition and benefits of the posing of the problem in terms of measures. Mild solutions to our problem will turn to exist and will also be positive weak solutions. A couple of open questions at the modeling, mathematical analysis, and simulation levels will be pointed out. This is a joint work with Sander Hille (Leiden, NL) and is supported financially by the KVA’s G. S. Magnussonsfond.

​2021-04-14: Aretha Teckentrup, The University of Edinburgh
Convergence and Robustness of Gaussian Process Regression
We are interested in the task of estimating an unknown function from data, given as a set of point evaluations. In this context, Gaussian process regression is often used as a Bayesian inference procedure, and we are interested in the convergence as the number of data points goes to infinity. Hyper-parameters appearing in the mean and covariance structure of the Gaussian process prior, such as smoothness of the function and typical length scales, are often unknown and learnt from the data, along with the posterior mean and covariance. We work in the framework of empirical Bayes', where a point estimate of the hyper-parameters is computed, using the data, and then used within the standard Gaussian process prior to posterior update. Using results from scattered data approximation, we provide a convergence analysis of the method applied to a fixed, unknown function of interest.

[1] A.L. Teckentrup. Convergence of Gaussian process regression with estimated hyper-parameters and applications in Bayesian inverse problems. SIAM/ASA Journal on Uncertainty Quantification, 8(4), p. 1310-1337, 2020. 

2021-04-07: Monika Eisenmann, Lund University
Sub-linear convergence of stochastic optimization methods in Hilbert space
In order to solve a minimization problem, a possible approach is to find the steady state of the corresponding gradient flow initial value problem through a long time integration. The well-known stochastic gradient descent (SGD) method then corresponds to the forward Euler scheme with a stochastic approximation of the gradient. Our goal is to find more suitable schemes that work well in the stochastic setting.
In the talk, we first present a stochastic version of the proximal point algorithm. This method corresponds to the backward Euler method with a stochastic approximation of the gradient. While it is an implicit method, it has better stability properties than the SGD method and advantages can be seen if the implicit equation can be solved within an acceptable time frame. Secondly, we present a stochastic version of the tamed Euler scheme in this context. This method is fully explicit but it is more stable for larger step sizes. We provide convergence results with a sub-linear rate also in an infinite-dimensional setting. We will illustrate the theoretical results on numerical examples.
A typical application for such optimization problems is supervised learning.

The talk is based on a joint work with Tony Stillfjord and Måns Williamson (both Lund University).

​2021-03-24: Maria Lopez Fernandez​, University of Malaga
Directional H2-matrices for lossy Helmholtz problems
The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated and hence their sparse approximation is of outstanding importance. We generalize the directional H2-matrix techniques from the "pure" Helmholtz operator, with imaginary frequency, to general complex frequencies with a positive real part. In this case, the fundamental solution decreases exponentially for large arguments. We develop a new admissible condition which contains the real part of the frequency in an explicit way and introduce the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent directional expansion functions. We develop an error analysis which is explicit with respect to the expansion order and with respect to the real and the imaginary parts of the frequency. This allows us to choose the variable expansion order in a quasi-optimal way. The complexity analysis shows how higher values of the real part of the frequency reduce the complexity. In certain cases, it even turns out that the discrete matrix can be replaced by its near field part. Numerical experiments illustrate the sharpness of the derived estimates and the efficiency of our sparse approximation.​

2021-03-17: Luigi Brugnano​, Università di Firenze​
Spectral numerical solution of evolutionary problems
In this talk, I will briefly recall recent results on the use of spectral methods for the efficient numerical solution of evolutionary problems. Their main feature is the ability of obtaining full machine accuracy despite the use of (relatively) large time-steps. This approach, at first devised for solving highly-oscillatory problems [1], has then been extended to the numerical solution of Hamiltonan PDEs [2], general ODE-IVPs [3], and fractional ODEs [4]. A theoretical analysis of the methods has been given in [5]. 

[1] L.Brugnano, J.I.Montijano, L.Rández. On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems. Numerical Algorithms 81 (2019) 345-376.
[2] L.Brugnano, F.Iavernaro, J.I.Montijano, L.Randez. Spectrally accurate space-time solution of Hamiltonian PDEs. Numerical Algorithms 81 (2019) 1183-1202.
[3] P.Amodio, L.Brugnano, F.Iavernaro, C.Magherini. Spectral solution of ODE-IVPs using SHBVMs. AIP Conference Proceedings 2293 (2020) 100002.
[4]  P.Amodio, L.Brugnano, F.Iavernaro. Spectrally accurate solutions of nonlinear fractional initial value problems. AIP Conference Proceedings 2116 (2019) 140005.
[5] P.Amodio, L.Brugnano, F.Iavernaro. Analysis of Spectral Hamiltonian Boundary Value Methods (SHBVMs) for the numerical solution of ODE problems. Numerical Algorithms 83 (2020) 1489-1508.
2021-03-10: Patrick Henning​, Ruhr-Universität Bochum
The approximation and conservation of energy in nonlinear Schrödinger equations
In this talk we consider the numerical treatment of nonlinear Schrödinger equations as they appear for example in quantum physics and fluid dynamics. We give numerical examples that demonstrate the influence of the discrete energy on the accuracy of numerical approximations and that a spurious energy can create artificial phenomena such as drifting particles. In order to conserve the exact energy of the equation as accurately as possible, we propose a Crank-Nicolson-type time discretization that is combined with a suitable generalized finite element discretization in space. The space discretization is based on the technique of Localized Orthogonal Decompositions (LOD) and allows to capture general time invariants with a 6th order accuracy with respect to the chosen mesh size H. This accuracy is preserved due to the conservation properties of the time stepping method. The computational efficiency of the method is demonstrated for a numerical benchmark problem with known exact solution, which is however not solvable with traditional methods on long time scales.​
2021-03-03: Raphael Kruse​, Martin-Luther-University Halle-Wittenberg​
Discretization of Elliptic PDEs with the Finite Element Method and Randomized Quadrature Formulas
The implementation of the finite element method for linear elliptic partial differential equations (PDE) requires to assemble the stiffness matrix and the load vector. In general, the entries of this matrix-vector system are not known explicitly but need to be approximated by quadrature rules. However, if the coefficient functions of the differential operator or the forcing term are irregular, then standard quadrature formulas, such as the barycentric quadrature rule, may not be reliable. In this talk we discuss the application of two randomized quadrature formulas to the finite element method for such elliptic PDE with irregular coefficient functions. We derive detailed error estimates for these methods, discuss their implementation in numerical experiments.

This talk is based on joint work with Nick Polydorides (U Edinburgh) and Yue Wu (U Oxford).

2021-02-24: Balázs Kovács, Universität Regensburg
 $L^2$ error estimates for wave equations with dynamic boundary conditions
In this talk we will discuss $L^2$ norm error estimates of
semi- and full discretisations, using bulk--surface finite elements
and Runge--Kutta methods, of wave equations with dynamic boundary
conditions. The presented analysis resides on an abstract formulation
and error estimates, via energy techniques, within this abstract
setting. Four prototypical linear wave equations with dynamic boundary
conditions are analysed which fit into the abstract framework. For
problems with velocity terms, or with acoustic boundary conditions we
prove surprising results: for such problems the spatial convergence
order is shown to be less than two.
These can also be observed in the numerical experiments which we will present.
The talk is based on joint work with D. Hipp (previously KIT, Germany).

2021-02-17: Benoit Dherin, Google Cloud Dublin​
Implicit Gradient Regularization
Large deep neural networks used in modern supervised learning have a large submanifold of interpolating solutions, most of which are not good. However, it has been observed experimentally that gradient descent tends to converge in the vicinity of flat interpolating solutions producing trained models that generalize well to new data points, and the more so
as the learning rate increases. Using backward error analysis, we will show that gradient descent actually follows the exact gradient flow of a modified loss surface, which can be described by a regularized loss preferring optimization paths with shallow slopes, and in which the learning rate plays the role of a regularization rate. 
(This is joint work with David Barrett from DeepMind). 

2021-02-10: Charles-Edouard Bréhier, Université Lyon 1 
Asymptotic preserving schemes for a class of SDEs
I will present a class of numerical methods for SDEs with
multiple time scales. When the time-scale separation parameter goes to
0, the slow component converges to an averaged or homogenized equation.
We design asymptotic preserving schemes: passing to the limit in the
scheme provides a consistent approximation of the limiting equation.
This is a joint work with Shmuel Rakotonirina-Ricquebourg

2021-02-03: Larisa Beilina​, Chalmers & GU
An adaptive finite element method for solution of an ill-posed problem with applications in microwave thermometry​
We will present an adaptive finite element method for solution of a linear Fredholm integral equation of the first kind. We derive a posteriori error estimates in the functional to be minimized and in the regularized solution to
this functional, and formulate corresponding adaptive algorithms. Balancing principle for optimal choice of the regularization parameter will be presented. Finally,  numerical experiments  will show  the efficiency of a posteriori estimates applied  to data measured in microwave thermometry.

2021-01-27: Anthony Nouy​, Centrale Nantes​​
Approximation with tensor networks
Tensor networks (TNs) are prominent model classes for the approximation of high-dimensional functions in computational and data science. Tree-based TNs based, also known as tree-based tensor formats, can be seen as particular feed-forward neural networks. After an introduction to approximation tools based on tree tensor networks, we introduce their  approximation classes and present some recent results on their properties. In particular, we show that classical smoothness (Besov) spaces are continuously embedded in TNs approximation classes. For such spaces, TNs achieve (near to) optimal rate that is usually achieved by classical approximation tools, but without requiring to adapt the tool to the regularity of the function. The use of deep networks with free depth is shown to be essential for obtaining this property. Also, it is shown that exploiting sparsity of tensors  allows to obtain optimal rates achieved by classical nonlinear approximation tools, or to better exploit structured smoothness (anisotropic or mixed) for multivariate approximation. We also show that approximation classes of tensor networks are not contained in any Besov space, unless one restricts the depth of the tensor network. That reveals again the importance of depth and the potential of tensor networks to achieve approximation or learning tasks for functions beyond standard regularity classes. 

[1] M. Ali and A. Nouy. Approximation with Tensor Networks. Part I: Approximation Spaces. arXiv:2007.00118 
[2] M. Ali and A. Nouy. Approximation with Tensor Networks. Part II: Approximation Rates for Smoothness Classes. arXiv:2007.00128 
[3] M. Ali and A. Nouy. Approximation with Tensor Networks. Part III: Multivariate approximation. 
[4] B. Michel and A. Nouy. Learning with tree tensor networks: complexity estimates and model selection. arXiv:2007.01165.​

2021-01-20: Roland Maier, Chalmers & GU​
Multiscale scattering in nonlinear Kerr-type media 
Wave propagation in heterogeneous and nonlinear media has arisen growing interest in the last years since corresponding materials can lead to unusual and interesting effects and therefore come with a wide range of applications. An important example for such materials are Kerr-type media, where the intensity of a wave directly influences the refractive index. In the time-harmonic regime, this effect can be modelled with a nonlinear Helmholtz equation. If underlying material coefficients are highly oscillatory on a microscopic scale, the numerical approximation of corresponding solutions can be a delicate task.
In this talk, a multiscale technique is presented that allows one to deal with microscopic coefficients in a nonlinear Helmholtz equation without the need for global fine-scale computations. The method is based on an iterative and adaptive construction of appropriate multiscale spaces based on the multiscale method known as Localized Orthogonal Decomposition, which works under minimal structural assumptions.
This talk is based on joint work with Barbara Verfürth (KIT, Karlsruhe)

2021-01-13: Yvon Maday, Sorbonne Université​
A few more things I learned about modelling the Covid epidemic 19
After the presentation that I had entitled "Two or three things that came out of the Maths-4-Covid-19 working group" at the seminar of the Laboratoire Jacques-Louis Lions last June 
(see https://www.youtube.com/watch?v=QphZv1kytnQ&list=PL2W1YCsKIaN5g7x9QtirnR14jPxFAWYTV&index=3), 
my knowledge on the subject has benefited from collaboration with several colleagues on various projects, some recent results of which I would like to present. 
The main contribution is about deterministic compression of information derived from the knowledge of epidemiologists and  infectious disease dynamics exerts. This reduction of complexity involves reduced basis methods, in particular in the frame of positive functions where the approximation guarantees the preservation of the positivity.​

2020-12-09: Antoine Gloria, Sorbonne Université​
The structure of fluctuations as a computational tool in stochastic homogenization
Consider an elliptic equation in divergence form with random coefficients. The solution of the equation is itself a random field. When the correlation length of the coefficient field is small with respect to the length scale of the (deterministic) forcing term, homogenization occurs in form of an ergodic theorem for the solution : the latter looks deterministic at large scales (that is, compared to the correlation length). The gradient of the solution, however, both oscillates and fluctuates. In this talk I will describe the so-called path wise structure of fluctuations, and explain how it leads to a reduction of complexity that can be taken advantage of for numerical purposes.
This is based on joint works with Mitia Duerinckx (CNRS, Orsay) and Felix Otto (MPI Leipzig).

2020-12-02: Mike Pereira​, Chalmers and GU​
The STONE Project: A multidisciplinary approach to road traffic modeling
Understanding efficiency and behavior aspects in partially automated (vehicular) technology in large-scale (traffic) context is an unsolved problem nowadays. In this context, the STONE project aims at developing  learning methods for uncertain traffic networks while relying on interdisciplinary approaches between mathematical sciences (stochastic Partial Differential Equations), traffic flow theory (hyperbolic conservation vehicular laws, network efficiency) and probabilistic machine leaning concepts.
In this talk, we will present the motivations of the project and the modeling choices we are making, as well as the technical challenges we wish to overcome.

2020-11-25: Salvador Ortiz-Latorre​, University of Oslo
High order discretizations for the solution of the nonlinear filtering problem
The solution of the continuous time stochastic filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. In this talk I will introduce a class of discretization schemes of these functionals of arbitrary order. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the mth power of the mesh of the partition for arbitrary natural number m. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. Moreover, the result paves the way for constructing high order numerical approximation for the solution of the filtering problem. 

This talk is based in a joint work with Dan Crisan (Imperial College) recently published in Stochastics and Partial Differential equations: Analysis and Computations Vol. 8, Issue 4, December 2020.

2020-11-18:  Kristian Debrabant, University of Southern Denmark​
Order conditions for generalized exponential stochastic partitioned Runge—Kutta methods
In Molecular Physics, Volume 118, 2020 - Issue 8, Grønbech-Jensen presented a complete set of stochastic Verlet-type methods for asymptotically statistically correct Langevin simulations. In this talk, we will discuss how this class of methods can be interpreted as stochastic partitioned Runge—Kutta methods with non-linear coefficients, generalizing exponential methods. Based on B-series, we derive then order conditions both for strong and weak convergence and analyse the order of the Grønbech-Jensen methods.

The content of this talk is based on joint work with Anne Kværnø.

2020-11-11:  Lehel Banjai​, Heriot-Watt
A tensor finite element method for a space fractional wave equation
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers of symmetric, coercive, linear, elliptic, second-order operators in bounded spatial domains. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder. We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space-time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in with a suitable hp-FEM in the extended variable. For both schemes we derive stability and error estimates and present numerical results.

This is joint work with E. Otarola. 

2020-10-28: Fernando Casas, Jaume I University 
Compositions of pseudo-symmetric integrators with complex coefficients in the numerical integration of differential equations
In this talk I will consider composition methods for the time integration of differential equations obtained as double jump compositions with complex coefficients and projection on the real axis. It is shown in particular that the new integrators are time-symmetric and symplectic up to high orders if one uses a time-symmetric and symplectic basic method. This technique requires fewer stages than standard compositions of the same orders and is thus expected to lead to faster methods.

2020-10-21: Máté Gerencsér, IST Austria 
Approximation of SDEs - a stochastic sewing approach
We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Le (2020). As an alternative to earlier PDE-based works, this approach allows one to go beyond Markovian settings. We discuss the first results on convergence rates of the Euler-Maruyama scheme for SDEs driven by additive fractional noise and irregular drift, as well as the derivation of optimal convergence rates for SDEs driven by multiplicative standard Brownian noise and arbitrary Holder-continuous drift.
Joint work with Oleg Butkovsky and Konstantinos Dareiotis.

2020-10-14: Chus Sanz-Serna, Universidad Carlos III de MadridVibrational resonance: A study with word series​
Vibrational resonance occurs when the response of a system to a periodic forcing is enhanced by the presence of an additional fast forcing. In the talk I will study this phenomenon by means of word series: formal series that make it possible to study systematically dynamical systems and also numerical integrators.

2020-09-30: Elena Celledoni​, NTNU: Deep learning as optimal control and structure preserving deep learning
Over the past few years, deep learning has risen to the foreground as a topic of massive interest, mainly as a result of successes obtained in solving large-scale image processing tasks. There are multiple challenging mathematical problems involved in applying deep learning.
We consider recent work of Haber and Ruthotto 2017 and Chang et al. 2018, where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We review the first order conditions for optimality, and the conditions ensuring optimality after discretisation.  There is a growing effort to mathematically understand the structure in existing deep learning methods and to systematically design new deep learning methods to preserve certain types of structure in deep learning. Examples are invertibility, orthogonality constraints, or group equivariance, and new algorithmic frameworks based on conformal Hamiltonian systems and Riemannian manifolds.

Deep learning as optimal control problems: models and numerical methods
Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb

Structure preserving deep learning
Elena Celledoni, Matthias J. Ehrhardt, Christian Etmann, Robert I McLachlan, Brynjulf Owren, Carola-Bibiane Schönlieb, Ferdia Sherry

2020-09-23: Anders Szepessy, KTH: Optimal control estimates of residual networks​  
I will show and explain estimates of the generalization error
to approximate given data by a residual neural network.

2020-09-16. Karl Larsson, Umeå University: Least-Squares Stabilized Nitsche Boundary Conditions for Unfitted Finite Element Methods
Weak enforcement of Dirichlet boundary conditions in finite element methods (FEM) can be done efficiently and with higher order accuracy using so-called Nitsche boundary conditions. This makes Nitsche boundary conditions suitable for unfitted FEM where the domain boundary is allowed to arbitrarily cut through the computational mesh. However, in some cut situations this leads to a method which is non-coercive. The problem is usually dealt with by using a very large penalty parameter in the Nitsche boundary condition or adding some stabilization terms to the method but in this talk we take a different approach. By adding certain consistent least-squares terms to the Nitsche boundary conditions we achieve a method which is proven coercive in every cut situation using only a moderate size penalty parameter.

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