2021-12-15: Adam Andersson, Chalmers & GU and Saab
2021-12-08: Irina Pettersson, Chalmers & GU
2021-12-01: Sonja Cox, University of Amsterdam
2021-11-24: Buyang Li, The Hong Kong Polytechnic University
2021-11-17: Charles Elliott, University of Warwick
2021-11-10: Josefin Ahlkrona, Stockholm University
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 minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers 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.
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
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).