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Higher order QMC Petrov-Galerkin discretization for affine parametric operator equations with random field inputs. (English) Zbl 1326.65013

The efficient numerical computation of statistical quantities for solutions of partial differential and integral equations with random imputs is a key task in uncertainty quantification in engineering and in science. The quantity of interest of this paper is exposed as a mathematical expectation and the efficient computation of these quantities involves two basic steps – the approximate (numerical) solutions of the operator equation and the approximate evaluation of the mathematical expectation by numerical integration. The realization of these two aims is based on the using Petrov-Galerkin discretization of the operator equation and on quasi-Monte Carlo (QMC) integration.
In the present paper, the regularity of the solutions with respect to parameters in the terms of the rate of decay of the fluctuations of the input fields is analysed. It is proved that deterministic interlaced polynomial lattice rules of order \(\displaystyle \alpha = \left[ {1 \over p} \right] + 1\) in dimension \(s\) can be computed by using a fast component-by-component algorithm in \({\mathcal O}(\alpha s N \log N + \alpha^2 s^2 N)\) operations, to achieve a convergence rate \({\mathcal O}(N ^{-{1 \over p}}).\) Here \(p \in (0,1]\) denotes the “summability exponent”.
In the introduction, the main problem of the paper is described. Here \({\mathbf y} = (y_j)_{j \geq 1}\) denotes a set of parameters from a domain \(U \subset \mathbb R^N\), \(A({\mathbf y})\) denotes a \({\mathbf y}\)-parametric bounded linear operator between the spaces \(\chi\) and \({\mathcal Y}^\prime\). For a given \(f \in {\mathcal Y}^\prime\) and for every \({\mathbf y}\) one must find \(u({\mathbf y}) \in \chi\) such that \(A({\mathbf y}) u({\mathbf y}) = f\). It is assumed that \(A({\mathbf y})\) has an affine parameter dependence. For a given bounded linear functional \(G(\cdot) : \chi \to \mathbb R\), the integral of the functional \(G(\cdot)\) of the parametric solution \(I(G({\mathbf u})) = \int_{U} G(U({\mathbf y}))d {\mathbf y}\) is considered. The techniques of solving the above problem – truncation, using the Petrov-Galerkin discretization and the QMC quadrature are described.
In Section 2, a class of parametric operator equations is presented. The parametric and spatial regularities of their solutions are given. The concept of the Petrov-Galerkin discretization is exposed. By truncation of an infinite sum the problem is transformed to a parametric weak problem.
In Section 3, an analysis of the error of higher-ordered QMC methods is given. In Theorem 3.1, an order \({\mathcal O}(N^{-{1 \over p}})\) of the error of the QMC integration by using an interlaced polynomial lattice rule is obtained. In Subsection 3.1, the numerical integration for smooth integrands is considered. A class of Sobolev spaces is defined. The constructive principle of digital nets in prime base \(b\) is given. In Theorem 3.5, an upper bound of the worst case error is presented. It is shown that Theorem 3.5 also holds for any digitally shifted digital net. In Subsection 3.2, the polynomial lattice rules are interlaced. The concept of the interlaced polynomial lattice rules is given. In Subsection 3.3, a component-by-component (CBC) construction of interlaced polynomial lattice rules is introduced and analysed. In Theorem 3.9, it is shown that a generating vector can be constructed using a CBC approach which minimizes the error in each step. In Theorem 3.10, the worst case error of the integration over the introduced functional class by using a CBC algorithm is estimated from above. Two specific choices of weights – SPOD weights and product weights are discussed. The fast CBC implementations for SPOD weights and for product weights are given as algorithms.
In Section 4, the obtained theoretical results of the paper are applied to partial differential equation problems. In Theorem 4.1, under the presented assumptions, if the considered integrals are approximated by using interlaced polynomial lattice rules of order \(\alpha\) with \(b^m\) points in dimension \(s\), combined with a Petrov-Galerkin method in the domain \(D\), an upper bound of the integration in the introduced functional class is obtained.

MSC:

65C05 Monte Carlo methods
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs
35R60 PDEs with randomness, stochastic partial differential equations
45R05 Random integral equations
65C30 Numerical solutions to stochastic differential and integral equations
60H25 Random operators and equations (aspects of stochastic analysis)
60G60 Random fields
65J10 Numerical solutions to equations with linear operators
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