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Sparse quadrature for high-dimensional integration with Gaussian measure. (English) Zbl 06966736

Summary: In this work we analyze the dimension-independent convergence property of an abstract sparse quadrature scheme for numerical integration of functions of high-dimensional parameters with Gaussian measure. Under certain assumptions on the exactness and boundedness of univariate quadrature rules as well as on the regularity assumptions on the parametric functions with respect to the parameters, we prove that the convergence of the sparse quadrature error is independent of the number of the parameter dimensions. Moreover, we propose both an a priori and an a posteriori schemes for the construction of a practical sparse quadrature rule and perform numerical experiments to demonstrate their dimension-independent convergence rates.

MSC:

65C20 Probabilistic models, generic numerical methods in probability and statistics
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs

Software:

HRMSYM; PATSYM
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References:

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