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

## TL;DR

This paper introduces a new sparse polynomial method using mixed $$ regularization for efficiently approximating solutions to high-dimensional parameterized PDEs, combining compressed sensing with energy-based sparsity.

## Contribution

The paper develops a novel mixed-norm $$ regularization technique that improves sparse approximation of parameterized PDE solutions, with theoretical guarantees and extensive numerical validation.

## Key findings

- Achieves quasi-optimal approximation with minimal samples.
- Demonstrates superior recovery in high-dimensional PDE models.
- Provides theoretical error bounds comparable to best $s$-term approximations.

## Abstract

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.