Sebastian Matera

Sandra Döpking, Sebastian Matera

Many problems require to approximate an expected value by some kind of Monte Carlo (MC) sampling, e.g. molecular dynamics (MD) or simulation of stochastic reaction models (also termed kinetic Monte Carlo (kMC)). Often, we are furthermore interested in some integral of the MC model's output over the input parameters. We present a Multilevel Adaptive Sparse Grid strategy for the numerical integration of such problems where the integrand is implicitly defined by a Monte Carlo model. In this approach, we exploit different levels of sampling accuracy in the Monte Carlo model to reduce the overall computational costs compared to a single level approach. Unlike existing approaches for Multilevel Numerical Quadrature, our approach is not based on a telescoping sum, but we rather utilize the intrinsic multilevel structure of the sparse grids and the employed locally supported, piecewise linear basis functions. Besides illustrative toy models, we demonstrate the methodology on a realistic kMC model for CO oxidation. We find significant savings compared to the single level approach - often orders of magnitude.

Patrick Gelß, Sebastian Matera, Reinhold Schneider et al.

Low-rank tensor methods are an important tool in the numerical treatment of equations with a high-dimensional state space. Nearest neighbor interaction systems like the Ising model or more general Markov jump processes, as well as 1D finite-state quantum systems are examples of such problems. While low-rank tensor train/matrix product state models have been shown to be highly efficient for the simulation of such systems, providing theoretical justification for this remains a challenging task. One approach for obtaining estimates on required ranks for certain accuracies is to investigate the rank increase in Krylov subspace methods for solving the problem at hand. In the context of area laws for ground states of 1D spin systems, nontrivial results on rank-increasing properties of nearest neighbor operator polynomials have been obtained in work of Arad et al. [arXiv:1301.1162] by studying the partial commutativity of local operators. In the present work, this technique is applied to polynomial methods for definite linear equations and dissipative linear ODEs with nearest neighbor structure. This allows to derive corresponding low-rank approximability statements for solutions of such problems which are independent of the system size. Numerical simulations of high-dimensional nearest neighbor systems illustrate the theoretical findings.