Sparse operator compression of higher-order elliptic operators with rough coefficients

Abstract

We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy minimizing functions on local patches. On a regular mesh with mesh size h, the localized basis functions have supports of diameter O(h log(1/h)) and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter O(h log(1/h)), our method achieves the optimal compression rate of the solution operator. From the perspective of the generalized finite element method to solve elliptic equations, the localized basis functions have the optimal convergence rate O(h k ) for a (2k)th-order elliptic problem in the energy norm. From the perspective of the sparse PCA, our results show that a large set of Matérn covariance functions can be approximated by a rank-n operator with a localized basis and with the optimal accuracy. Background Main objectives and the problem settingThe main purpose of this paper is to develop a general strategy to compress a class of self-adjoint higher-order elliptic operators by localized basis functions that give optimal approximation property of the solution operator. To be more specific, suppose L is a self-adjoint elliptic operator in the divergence form Lu = 0≤|σ |,|γ |≤k

Compression, Inversion, and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity

“…We note that many covariance functions with finite smoothness are the Green's functions of such partial differential operators. For instance, the popular Matérn kernel with smoothness ν in dimension d is the Green's function of an (Whittle, 1954(Whittle, , 1963Lindgren et al, 2011;Hou and Zhang, 2017;Fasshauer, 2012). The Galerkin (e.g.…”

“…Hence, they contain a discretisation of the gamblets ψ (k) i . As mentioned in Owhadi and Scovel (2017) and exploited for low rank compression of operators in Hou and Zhang (2017), the gamblets provide an approximation of the principal components of the operator G. In particular, in Hou and Zhang (2017) it was conjectured that gamblets can also be computed directly from the covariance operator. Algorithm 2, with computation restricted to the near-sparsity patterns, provides a method for achieving this computation in near linear time based on and following the initial basis transformation.…”

“…In Owhadi and Scovel (2017), the results of Owhadi (2017) were generalised to the abstract setting of bounded operators on Banach spaces (including arbitrary continuous linear bijections from H s 0 (Ω) to H −s (Ω)). Although the proof of exponential decay of gamblets provided in Owhadi and Scovel (2017) allows for very general measurement functions φ (k) i (including masses of Diracs, indicator functions and higher order polynomials, see also Hou and Zhang (2017) for strongly elliptic PDEs with higher order polynomials as measurement functions), the proof Owhadi and Scovel (2017) that the matrices B (k) have uniformly bounded condition numbers rely on vanishing moments of the underlying multiresolution basis. That is, they rely on the property that…”

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“…Let V be a basis of R n . For any subset S ⊂ V, we denote P S as the orthogonal projection onto S. Following the notations in [10], we also denote A S , A S and A S as the restricted, interior and closed energy of S with respect to A and E. coarse space Φ that is locally computable and has good interpolation property; (iii) construct the modified coarse space Ψ = A −1 (Φ) of R n as proposed in [11,14,19]. If an appropriate partitioning is given, we have the following error estimate for operator compression.…”

“…The exponential decaying property of these localized basis functions is also proved independently using the idea of gamblets. Hou and Zhang in [11] further extended these works and constructed localized basis functions for higher order strongly elliptic operators. To further promote the operator compression for situations where the physical domain is unknown or is embedded in some nontrivial high dimensional manifolds, Hou et.…”