Hierarchical Numerical Methods for PDEs

Abstracts for the talks

Ivan Graham
University of Bath
Convergence of iterative solvers for the Helmholtz equation
The Restricted Additive Schwarz method with impedance transmission
conditions (often called the ORAS method) is a domain decomposition
method which can be used as an iterative solver or as a preconðitioner
for the solution of discretized Helmholtz boundary-value problems. It
is a very simple parallel one-level method, applicable in very general
geometries and it has been successfully combined with coarse spaces to
obtain two- or multi-level methods. Its multiplicative variants are
related to `sweeping methods' which have enjoyed considerable
recent practical interest. To date there is relatively little
convergence analysis for this method. In the talk we present a novel
analysis of the ORAS method. The main components of the talk are : (i)
ORAS is a non-conforming finite element approximation of a classical
parallel Schwarz method formulated at the PDE level; (ii) The
parallel Schwarz method is well-posed in suitable function spaces of
Helmholtz-harmonic functions in general geometries; (iii) The
parallel Schwarz method is proved to be power contractive for domain
decompositions of `strip-type' and is observed to be power contractive
for general domain decompositions in 2D experiments; (iv) Working in
suitable Helmholtz-harmonic finite element spaces and for fine enough
meshes, the ORAS method is proved to enjoy the same convergence
estimates as the parallel Schwarz method; in particular its power
contractive property is independent of the polynomial order of the
finite element spaces used.

The proof of (iv) uses a new finite element error estimate, proving
higher order convergence for certain Helmholtz problems at interior

The work on ORAS is recent joint work with Shihua Gong and Euan Spence
(Bath) while the results on the parallel Schwarz method were obtained
with Shihua Gong, Martin Gander (Geneva), David Lafontaine (Bath) and
Euan Spence.

Sabine Le Borne
TU Hamburg
Potential (&) pitfalls in RBF-FD discretization of PDEs
There exist several discretization techniques for the numerical solution of
partial differential equations (PDEs). In addition to classical finite
finite element and finite volume techniques, a more recent approach employs
radial basis functions to generate differentiation stencils on unstructured
point sets. This approach, abbreviated by RBF-FD (radial basis function -
finite difference), has gained in popularity since it enjoys several
advantages: It is (relatively) straightforward, does not require a mesh and
generalizes easily to higher spatial dimensions. However, its application is
not quite as blackbox as it may appear at first sight. The computed
solution might suffer severely from various sources of errors if RBF-FD
parameters are not selected carefully.

Through comprehensive numerical experiments, we study the influence of
several of
these parameters on the condition numbers of intermediate (local) weight
matrices, on the condition number of the resulting (global) stiffness matrix
and ultimately on the approximation
error of the computed discrete solution to the partial differential
The parameters of investigation include the type of RBF (and its shape or
other parameters if applicable), the degree of polynomial augmentation,
the discretization stencil size, the underlying type of point set
(structured/unstructured), and the total number of (interior and boundary)
points to discretize the PDE, here chosen as a three-dimensional
Poisson's problem with Dirichlet boundary conditions.

The goal is to illustrate and steer away from potential
pitfalls in RBF-FD discretization where a computationally more expensive
set-up not always leads to a more accurate numerical solution, and to guide
toward a compatible selection of RBF-FD parameters.

Mateusz Michalek
University of Konstanz
Matrix Product States from the Point of View of Algebraic Geometry
We will present uniform Matrix Product States (uMPS) and algebraic varieties associated to them. Using tools from algebraic geometry we give answers to questions posed by Hackbusch, Morton and Critch concerning topological properties of these varieties and their defining equations. We will also present some open problems. The talk is based on a joint work with Czaplinski and Seynnaeve.

Anthony Nouy
EC Nantes
Approximation theory of tree tensor networks
Tree Tensor networks (TTNs) are prominent model classes for the approximation of high-dimensional functions in computational and data science. After an introduction to approximation tools based on tensorization of functions and TTNs, 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 TTNs 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 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.

Stefan Sauter
University of Zurich
Variable Order, Directional H2-Matrices for Helmholtz Problems with Complex Frequency
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; hence their sparse approximation is of outstanding importance. In
our talk we will generalize the directional H2-matrix
techniques from the "original" Helmholtz operator (purely imaginary wave number) to
general complex frequencies z with Re(z) >0.
In this case, the fundamental solution decreases exponentially for large
arguments. We will develop a new admissibility condition which contain
Re(z) and Im(z) 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 present an error analysis which is explicit with
respect to the expansion order and with respect to the real and imaginary part
of z. This allows us to choose the variable expansion order in a
quasi-optimal way depending on Re(z) but independent of, possibly large, Im(z). The
complexity analysis is explicit with respect to Re(z) and Im(z) and shows how higher
values of Re(z) reduce the complexity. In certain cases, it even turns out that
the discrete matrix can be replaced by its nearfield part.

Numerical experiments illustrate the sharpness of the derived estimates and
the efficiency of our sparse approximation.

This talk comprises joint work with S. Börm, Christian-Albrechts-Universität Kiel, Germany
and M. Lopez-Fernandez, Sapienza Universita di Roma, Italy

Barbara Wohlmuth
TU Munich
The challenge of non-local operators and mixed dimensions in simulation and modelling
In this talk, we address some mathematical and numerical challenges of non-linear or non-local partial differential equations in applications. We use a variety of applications ranging from porous-media flow systems, fluid and structural mechanics to finance. The special challenges of variational inequalities, mixed-dimensions and non-integer differential operators are illustrated. We show the flexibility of abstract mathematical concepts and discuss limitations in theory and convergence.

Jinchao Xu
Penn State
Neural Networks and Numerical PDEs
I will first give a brief introduction to neural network functions, their applications to image classifications, and their relationship with finite element and multigrid methods. I will then present some recent results on the approximation properties of neural network functions, error analysis for numerical PDEs (in view of generalization accuracy in machine learning) and optimization algorithms for the underlying non-convex problems.

Harry Yserentant
TU Berlin
Iterations, Tensors, and beyond
This talk deals with the equation Δu + μu = f on high-dimensional spaces m, where μ is a positive constant. If the right-hand side f is a rapidly converging series of separable functions, the solution u can be represented in the same way. These constructions are based on approximations of the function 1∕r by sums of exponential functions. I will present results of similar kind for more general right-hand sides f(x) = F(Tx) that are composed of a separable function on a space of a dimension n greater than m and a linear mapping given by a matrix T of full rank. These results are based on the observation that in the high-dimensional case, for ω in most of the n, the euclidean norm of the vector Ttω in the lower dimensional space m behaves like the euclidean norm of ω. This effect has much to do with the random projection theorem, which plays an important role in the data sciences, and can be seen as a concentration of measure phenomenon.


Date and Location

September 28 - 29, 2021
Max Planck Institute for Mathematics in the Sciences
Inselstr. 22
04103 Leipzig

Scientific Organizers

Peter Benner
MPI for Dynamics of Complex Technical Systems, Magdeburg

Lars Grasedyck
RWTH Aachen

André Uschmajew
MPI for Mathematics in the Sciences, Leipzig

Administrative Contact

Katja Heid
MPI for Mathematics in the Sciences
Contact by Email
02.10.2021, 01:27