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 ℓ 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.
Original language | English |
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Pages (from-to) | 2025-2045 |
Number of pages | 21 |
Journal | ESAIM: Mathematical Modelling and Numerical Analysis |
Volume | 53 |
Issue number | 6 |
DOIs | |
State | Published - Nov 1 2019 |
Funding
Acknowledgements. This material is based upon work supported in part by: the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contracts and awards ERKJ314, ERKJ331, ERKJ345; and Scientific Discovery through Advanced Computing (SciDAC) program through the FASTMath Institute under Contract No. DE-AC02-05CH11231; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC., for the U.S. Department of Energy under contract DE-AC05-00OR22725.
Keywords
- Basis pursuit denoising
- Best approximation
- Bounded orthonormal systems
- Compressed sensing
- Convex regularization
- High-dimensional
- Hilbert-valued signals
- Joint sparsity
- Parameterized PDEs
- Polynomial expansions
- Quasi-optimal
- Sparse recovery