Efficient Resolution of Anisotropic Structures

TL;DR

This paper introduces advanced adaptive discretization and variational methods for efficiently solving high-dimensional, anisotropic transport equations, achieving convergence and overcoming the curse of dimensionality in parameter-dependent problems.

Contribution

It develops directionally adaptive discretization techniques inspired by shearlet systems and well-conditioned variational formulations, along with new reduced basis and tensor methods for high-dimensional problems.

Findings

Proves convergence of adaptive schemes in L2 norm.

Introduces a rate-optimal reduced basis method for parameter-dependent problems.

Demonstrates effectiveness of sparse tensor methods in radiative transfer simulations.

Abstract

We highlight some recent new delevelopments concerning the sparse representation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus on the solution of transport equations which exhibit propagation of singularities where, additionally, high-dimensionality enters when the convection field, and hence the solutions, depend on parameters varying over some compact set. Important constituents of our approach are directionally adaptive discretization concepts motivated by compactly supported shearlet systems, and well-conditioned stable variational formulations that support trial spaces with anisotropic refinements with arbitrary directionalities. We prove that they provide tight error-residual relations which are used to contrive rigorously founded adaptive refinement schemes which converge in $L_2$. Moreover, in the context of parameter dependent problems we discuss two approaches serving different purposes and working under different regularity assumptions. For frequent query problems, making essential use of the novel well-conditioned variational formulations, a new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems. For the radiative transfer problem with scattering a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable (so far still isotropic) regularity assumptions. Numerical examples for both methods illustrate the theoretical findings.