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**2022-02-02: Ingrid Lacroix-Violet, Institut Élie Cartan de Lorraine**

Linearly implicit numerical methods

*Abstract:*

*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

*Abstract*

*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?**

*Abstract:*

*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

*Abstract:*

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.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

*Abstract:*

*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

*Abstract:*

*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

*Abstract:*

2021-11-17: Charles Elliott, University of Warwick

**Evolving finite elements**

*Abstract:*

*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

*Abstract:*

*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

*Abstract:*

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.

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

*Abstract:*

*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**

*Abstract:*

*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).

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**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

*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)

This is joint work with Emil Engström (Lund University)

2021-09-22: Mikhail Roop, Chalmers & GU

**Singularities of solutions to barotropic Euler equations**

*Abstract:*

*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**

*Abstract:*

*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).

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

*Abstract:*

*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**

*Abstract:*

*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 (*

*'*

*'p*

*ollution 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

*Abstract:*

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).

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**

*Abstract:*

*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**

*Abstract:*

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***Abstract:*

*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.*

2021-05-12:

2021-05-12:

**Des Higham, The University of Edinburgh**

**Should We Be Perturbed About Deep Learning?**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

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**

*Abstract:*

*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**

****

*Abstract:*

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**

*Abstract:*

I

*n 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**

*Abstract:*

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**

*Abstract:*

*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*

*(https://arxiv.org/abs/2011.02341).*

2021-02-03:

**Larisa Beilina, Chalmers & GU****An adaptive finite element method for solution of an ill-posed problem with applications**

**in microwave thermometry**

*Abstract:*

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**

*Abstract:*

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.

References:

[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**

*Abstract:*

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**

*Abstract:*

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**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

*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**

*Abstract:*

*In this talk I will consider composition methods for the time inte*

*gration 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 composi*

*tions 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**

*Abstract:*

*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.

Joint work with Oleg Butkovsky and Konstantinos Dareiotis.

2020-10-14:

**Chus Sanz-Serna, Universidad Carlos III de Madrid**:**Vibrational resonance: A study with word series***Abstract:*

*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***Abstract:*

*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.*

*References:*

*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**

*Abstract:*

*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**

*Abstract:*

*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.*