A mixed $\ell_1$ regularization approach for sparse simultaneous approximation of parameterized PDEs
This work provides a novel theoretical and practical framework for efficient approximation of parameterized PDEs, benefiting computational science and engineering applications.
The paper introduces a sparse polynomial technique for simultaneously approximating parameterized PDEs, treating the solution as a jointly sparse reconstruction problem. The method achieves error estimates comparable to best s-term approximations with minimal sample complexity, as demonstrated on high-dimensional elliptic PDE models.
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.