2022-12-14: Per Ljung, Chalmers & GU (live)
Multiscale methods for solving wave equations on spatial networks
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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.
Acknowledgement:
This is joint work with Severiano Gonzalez-Pinto and Domingo Hernandez-Abreu.
Related publications can be downloaded from
http://www.unige.ch/~hairer/preprints.html
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Reference
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
Abstract:
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
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