A sparse FFT approach for ODE with random coefficients
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A sparse FFT approach for ODE with random coefficients 1 · Daniel Potts1 ¨ Maximilian Bochmann1 · Lutz Kammerer
Received: 16 July 2019 / Accepted: 1 July 2020 / © The Author(s) 2020
Abstract The paper presents a general strategy to solve ordinary differential equations (ODE), where some coefficient depend on the spatial variable and on additional random variables. The approach is based on the application of a recently developed dimensionincremental sparse fast Fourier transform. Since such algorithms require periodic signals, we discuss periodization strategies and associated necessary deperiodization modifications within the occurring solution steps. The computed approximate solutions of the ODE depend on the spatial variable and on the random variables as well. Certainly, one of the crucial challenges of the high-dimensional approximation process is to rate the influence of each variable on the solution as well as the determination of the relations and couplings within the set of variables. The suggested approach meets these challenges in a full automatic manner with reasonable computational costs, i.e., in contrast to already existing approaches, one does not need to seriously restrict the used set of ansatz functions in advance. Keywords Ordinary differential equation with random coefficient · Sparse fast Fourier transform · Sparse FFT · Lattice FFT · Lattice rule · Periodization · Uncertainty quantification · Approximation of moments · High-dimensional approximation Mathematics Subject Classification 2010 42A10 · 60H10 · 60H35 · 65C20 · 65N35 · 65T40 · 65T50
Communicated by: Ivan Oseledets Lutz K¨ammerer
[email protected] Daniel Potts [email protected] 1
Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany
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Adv Comput Math
(2020) 46:65
1 Introduction During the last years, the concept of random variables has become a very popular tool to model uncertain properties mathematically. For instance, diffusion characteristics of inhomogeneous materials can be distinctly more accurately described by functions that additionally depend on random variables. One common application area of these mathematical designs are diffusion coefficients in differential equations. Certainly, the additional random variables affect the solvability and—if exist—the solutions of the differential equations under consideration. Besides investigations on existence, uniqueness and regularity of solutions for specific mathematical problems that involve randomness (cf., e.g., [1, 2, 5, 10]), numerical solution approaches need to be developed in order to compute approximations of the desired solutions. Accordingly, the established numerical solution approaches for differential equations without random coefficients need to be—at least—extended in order to meet the new challenges that are caused by the randomness of the diffusion coefficient. Commonly, discretizations of the domain of the stochastic variables lead to discretized solutions that are used to comput
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