Engineering topics

Dexter, Nick

Publications and source records attributed to Dexter, Nick.

3 records

DOE OSTI2021

On the Strong Convergence of Forward-Backward Splitting in Reconstructing Jointly Sparse Signals

We consider the problem of reconstructing an infinite set of sparse, finite-dimensional vectors, that share a common sparsity pattern, from incomplete measurements. This is in contrast to the work (Daubechies et al., Pure Appl. Math. 57(11), 1413–1457, 2004), where the single vector signal can be infinite-dimensional, and (Fornasier and Rauhut, SIAM J. Numer. Anal. 46(2), 577613, 2008), which extends the aforementioned work to the joint sparse recovery of finite number of infinite-dimensional vectors. In our case, to take account of the joint sparsity and promote the coupling of nonvanishing components, we employ a convex relaxation approach with mixed norm penalty ℓ 2,1 . This paper discusses the computation of the solutions of linear inverse problems with such relaxation by a forward-backward splitting algorithm. However, since the solution matrix possesses infinitely many columns, the arguments of Daubechies et al. (Pure Appl. Math. 57(11), 1413–1457, 2004) no longer apply. As such, we establish new strong convergence results for the algorithm, in particular when the set of jointly sparse vectors is infinite.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Reconstructing high-dimensional Hilbert-valued functions via compressed sensing

We present and analyze a novel sparse polynomial technique for approximating high-dimensional Hilbert-valued functions, with application to parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our theoretical framework treats the function approximation problem as a joint sparse recovery problem, where the set of jointly sparse vectors is possibly infinite. To achieve the simultaneous reconstruction of Hilbert-valued functions in both parametric domain and Hilbert space, we propose a novel mixed-norm based `1 regularization method that exploits both energy and sparsity. Our approach requires extensions of concepts such as the restricted isometry and null space properties, allowing us to prove recovery guarantees for sparse Hilbert-valued function reconstruction. We complement the enclosed theory with an algorithm for Hilbert-valued recovery, based on standard forward-backward algorithm, meanwhile establishing its strong convergence in the considered infinite-dimensional setting. Finally, we demonstrate the minimal sample complexity requirements of our approach, relative to other popular methods, with numerical experiments approximating the solutions of high-dimensional parameterized elliptic PDEs.

Dexter, Nick↗

DOE OSTI2019

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

We introduce and assess 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. Moreover, 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 {\em 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.

97 MATHEMATICS AND COMPUTING↗