arXiv subjects

Nick Dexter

Publications and source records attributed to Nick Dexter.

21 records · Page 2Linked to original sources

arXiv2018

A mixed $\ell_1$ regularization approach for sparse simultaneous approximation of parameterized PDEs

We present and analyze a novel sparse polynomial technique for the simultaneous approximation of parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our approach treats the numerical solution as a jointly sparse reconstruction problem through the reformulation of the standard basis pursuit denoising, where the set of jointly sparse vectors is infinite. To achieve global reconstruction of sparse solutions to parameterized elliptic PDEs over both physical and parametric domains, we combine the standard measurement scheme developed for compressed sensing in the context of bounded orthonormal systems with a novel mixed-norm based $\ell_1$ regularization method that exploits both energy and sparsity. In addition, we are able to prove that, with minimal sample complexity, error estimates comparable to the best $s$-term and quasi-optimal approximations are achievable, while requiring only a priori bounds on polynomial truncation error with respect to the energy norm. Finally, we perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.

math.NA↗

arXiv2016

Explicit cost bounds of stochastic Galerkin approximations for parameterized PDEs with random coefficients

This work analyzes the overall computational complexity of the stochastic Galerkin finite element method (SGFEM) for approximating the solution of parameterized elliptic partial differential equations with both affine and non-affine random coefficients. To compute the fully discrete solution, such approaches employ a Galerkin projection in both the deterministic and stochastic domains, produced here by a combination of finite elements and a global orthogonal basis, defined on an isotopic total degree index set, respectively. To account for the sparsity of the resulting system, we present a rigorous cost analysis that considers the total number of coupled finite element systems that must be simultaneously solved in the SGFEM. However, to maintain sparsity as the coefficient becomes increasingly nonlinear in the parameterization, it is necessary to also approximate the coefficient by an additional orthogonal expansion. In this case we prove a rigorous complexity estimate for the number of floating point operations (FLOPs) required per matrix-vector multiplication of the coupled system. Based on such complexity estimates we also develop explicit cost bounds in terms of FLOPs to solve the stochastic Galerkin (SG) systems to a prescribed tolerance, which are used to compare with the minimal complexity estimates of a stochastic collocation finite element method (SCFEM), shown in our previous work [16]. Finally, computational evidence complements the theoretical estimates and supports our conclusion that, in the case that the coefficient is affine, the coupled SG system can be solved more efficiently than the decoupled SC systems. However, as the coefficient becomes more nonlinear, it becomes prohibitively expensive to obtain an approximation with the SGFEM.

math.NA↗

arXiv2016

Polynomial approximation via compressed sensing of high-dimensional functions on lower sets

This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions. Of particular interest is the setting where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we present an innovative weighted $\ell_1$ minimization procedure with a precise choice of weights, and a new iterative hard thresholding method, for imposing the downward closed preference. Theoretical results reveal that our computational approaches possess a provably reduced sample complexity compared to existing compressed sensing techniques presented in the literature. In addition, the recovery of the corresponding best approximation using these methods is established through an improved bound for the restricted isometry property. Our analysis represents an extension of the approach for Hadamard matrices in [5] to the general case of continuous bounded orthonormal systems, quantifies the dependence of sample complexity on the successful recovery probability, and provides an estimate on the number of measurements with explicit constants. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the novel weighted $\ell_1$ minimization strategy.

math.NA↗