During the last decades (numerical) simulations based on partial differential equations have considerably gained importance in engineering applications, life sciences, environmental issues, and finance. However, especially when multiple simulation requests or a real-time simulation response are desired, standard methods such as finite elements are prohibitive. Model order reduction approaches have been developed to tackle such situations. Here, the key concept is to prepare a problem-adapted low-dimensional subspace of the high-dimensional discretization space in a possibly expensive offline stage to realize a fast simulation response by Galerkin projection on that low-dimensional space in the subsequent online stage. In this talk we show how randomization as used say in randomized linear algebra or compressed sensing can be exploited both for constructing reduced order models and deriving bounds for the approximation error. We also demonstrate those techniques for the generation of local reduced approximation spaces that can be used within domain decomposition or multiscale methods.
We consider the numerical approximation of boundary conditions in radiative transfer problems by a perfectly matched layer approach. The main idea is to extend the computational domain by an absorbing layer and to use an appropriate reflection boundary condition at the boundary of the extended domain. A careful analysis shows that the consistency error introduced by this approach can be made arbitrarily small by increasing the size of the extension domain or the magnitude of the artificial absorption in the surrounding layer. A particular choice of the reflection boundary condition allows us to circumvent the half-space integrals that arise in the variational treatment of the original vacuum boundary conditions and which destroy the sparse coupling observed in numerical approximation schemes based on truncated spherical harmonics expansions. A combination of the perfectly matched layer approach with a mixed variational formulation and a PN-finite element approximation leads to discretization schemes with optimal sparsity pattern and provable quasi-optimal convergence properties. As demonstrated in numerical tests these methods are accurate and very efficient for radiative transfer in the scattering regime.
We consider parabolic stochastic partial differential equations with multiplicative Wiener noise. For the second moment of the mild solution, a deterministic space-time variational problem is derived. Well-posedness is proven on projective and injective tensor product spaces as trial and test spaces. From these results, a deterministic equation for the covariance function is deduced. These deterministic equations in variational form are used to derive numerical methods for approximating the second moment of solutions to stochastic ordinary and partial differential equations. For the canonical example of a stochastic ordinary differential equation with multiplicative noise, the geometric Brownian motion, we introduce and analyze Petrov-Galerkin discretizations based on tensor product piecewise polynomials. For approximating the second moment of solutions to stochastic partial differential equations, we then propose conforming space-time Petrov-Galerkin discretizations. In both cases, we derive stability bounds in the natural tensor product spaces. Numerical experiments validate the theoretical results.
Schwarz iterative methods in Hilbert spaces are a recurring theme in my collaboration with M. Griebel. Lately, we became interested in greedy and stochastic versions of this class of iterative methods, where a variational problem in an infinite-dimensional Hilbert space $H$ is decomposed into infinitely many auxiliary subproblems. Convergence of a particular instance of the Schwarz iteration (called relaxed greedy and averaged stochastic descent, respectively) can be proved, algebraic rates of convergence can be obtained for large subsets of $H$ (this is in contrast to finite decompositions, where rates are exponential in the number of iteration steps). When working on the stochastic versions, we realized a certain connection to the theory of reproducing kernel Hilbert spaces and online learning algorithms. It turns out that the iterative solution of variational problems in a Hilbert space via subproblem solves based on space splittings can always be viewed as online learning of a variable Hilbert space valued function using kernel methods. Even though this connection does not help perfoming the algorithm, it sheds new light on the convergence proofs. The introduction of RKHS of variable Hilbert space valued functions may have some value for statistical learning as well, this hope is, however, not yet substantiated.