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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.
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.
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).
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)
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 SPDEsAbstract:
In this talk we will discuss two different viewpoints on a space-discrete reaction-diffusion equation with noise: First, as an
interacting particle system in a bistable potential, and second, as a lattice differential equation. Each viewpoint sheds light on a different phenomenon, which will be highlighted in the talk.
Based on joint works with A. Bovier (Bonn) and Ch. Kuehn (TU Munich).
2021-09-08: Morgan Görtz, Chalmers & GU
Network Models For Paper Simulations
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 (''pollution error'') minimal, but they are also apt at locally resolving small geometric features or material interfaces. Hence the combined FEM and LF based numerical discretization of second-order wave equations has proved a versatile and highly efficient approach, be it for the simulation of acoustic or elastic waves.
Local mesh refinement, however, can cause a severe bottleneck for the LF method, or any other standard explicit scheme, due to the stringent CFL stability condition on the time-step imposed by the smallest element in the mesh. Even when the locally refined region consists of a few small elements only, those elements will dictate a tiny time-step throughout the entire computational domain for stability. To overcome the crippling effect of a few small elements, without sacrificing its
high efficiency or explicitness elsewhere, a leapfrog based explicit local time-stepping (LF-LTS) method was proposed for the time integration of second-order wave equations. Recently, optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step depends on the smallest elements in the mesh. In general one cannot improve upon that stability constraint, as the LF-LTS method may become unstable at certain
discrete values of the time-step. To remove those critical values for the time-step, we apply a slight modification to the original LF-LTS method which nonetheless preserves its desirable properties: it is fully explicit, second-order accurate,
satisfies a three-term (leapfrog like) recurrence relation, and conserves the energy. The new stabilized LF-LTS method
also yields optimal convergence rates for a standard conforming FE discretization, yet under a CFL condition
where the time-step no longer depends on the mesh size inside the locally refined region.
Joint work with: Simon Michel (Univ. of Basel) and Stefan Sauter (Univ. of Zurich)
2021-06-09: Ulrik Skre Fjordholm, University of Oslo
Numerical methods for conservation laws on graphs
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).
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: 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
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.