The present paper is dedicated to the application of the pivoted Cholesky decomposition to compute low-rank approximations of dense, positive semi-definite matrices. The resulting approximation error is rigorously controlled in terms of the trace norm. Exponential convergence rates are proved under the assumption that the eigenvalues of the matrix under consideration exhibit a sufficiently fast exponential decay. By numerical experiments it is demonstrated that the pivoted Cholesky decomposition leads to very efficient algorithms to separate the variables of bi-variate functions.
In this article, we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains. In particular, based on the decay of the Karhunen-Loève expansion of the domain perturbation field, we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension. For the implementation of a related approximation scheme, like quasi-Monte Carlo quadrature, stochastic collocation, etc., we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field. This simplifies the implementation and yields a non-intrusive approach. Having this machinery at hand, we can easily transfer it to stochastic interface problems. The theoretical findings are complemented by numerical examples for both, stochastic interface problems and boundary value problems on random domains.
We consider the rapid computation of separable expansions for the approximation of random fields. We compare approaches based on techniques from the approximation of non-local operators on the one hand and based on the pivoted Cholesky decomposition on the other hand. We provide an a-posteriori error estimate for the pivoted Cholesky decomposition in terms of the trace. Numerical examples validate and quantify the considered methods. The Karhunen-Loève expansionLet (Ω, F, P) be a separable, complete probability space and let D ⊂ R d be a Lipschitz domain. In the sequel, we consider random fields a ∈ L 2 P Ω; L 2 (D) . For a given random field a, we denote the related centered field according to a 0 (ω, x) := a(ω, x) − E[a](x).Moreover, we define the Hilbert-Schmidt operator associated to a 0 asOne can show that the covariance operator C := SS is of trace-class, i.e. Tr C :One readily verifiesAdditionally, C is symmetric and positive semi-definite. Therefore, by the spectral theorem, C exhibits a representation of the formwhere {(λ i , φ i )} i∈I denote the corresponding eigen-pairs.is called Karhunen-Loève expansion with respect to a. Herein, the random variables {X m } m∈I are given according to Finite element approximationFor the approximation of spatial functions in L 2 (D), we employ (parametric) finite elements of order s. To that end, we introduce a family of quasi-uniform triangulations T h for D with mesh width h and define the spacesThen, given a function v ∈ H t (D) with 0 ≤ t ≤ s, it holds due to the Bramble-Hilbert lemma the approximation estimateuniformly in h.In the sequel, we assume that the random field a exhibits additional spatial regularity, i.e. a ∈ L 2 P (Ω; H p (D) for some p > 0. Then, we may consider the spatial approximationIn terms of the trace, we obtain the following approximation result in V s h . Theorem 2.1. Let a ∈ L 2 P Ω; H p (D) . Then, the spatially approximated, centered random field Q h a 0 satisfies the error estimateBy the application of the theorem and the approximation estimate (1) it is straightforward to show the following Corollary 2.2. The trace error satisfiesand its covariance according to C h,M := P h C h P h . We arrive at the subsequent approximation result., then the random field a h,M given by (2) satisfies the error estimateThe theorem indicates that, after fixing the ansatz space V s h , the approximation error of the stochastic field is controllable in terms of the discretized operators C h and C h,M . The optimal choice of P h in terms of minimizing the trace error is the orthogonal projection onto the dominant invariant subspace of C h , i.e. U M,h := span{φ 1,h , . . . , φ M,h } corresponding to the M dominant eigenvalues of C h . The related Karhunen-Loève expansion then readswhere the random variables are given according toIn this setting, the discretization of the stochastic field implies a change of the stochastic model induced by (5). The pivoted Cholesky decompositionBased on the observation in Theorem 2.3 and the subsequent discussion, we ...
We compare fast black-box boundary element methods on parametric surfaces in R 3. These are the adaptive cross approximation, the multipole method based on interpolation, and the wavelet Galerkin scheme. The surface representation by a piecewise smooth parameterization is in contrast to the common approximation of surfaces by panels. Nonetheless, parametric surface representations are easily accessible from Computer Aided Design (CAD) and are recently topic of the studies in isogeometric analysis. Especially, we can apply two-dimensional interpolation in the multipole method. A main feature of this approach is that the cluster bases and the respective moment matrices are independent of the geometry. This results in a superior compression of the far field compared to other cluster methods.
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