A mixed $$\ell_1$$ regularization approach for sparse simultaneous approximation of parameterized PDEs
Abstract
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.
- Authors:
-
[1];
[2];
- Univ. of Tennessee, Knoxville, TN (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Univ. of Tennessee, Knoxville, TN (United States); Simon Fraser Univ., Burnaby, BC (Canada)
- Publication Date:
- Research Org.:
- Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1564178
- Grant/Contract Number:
- AC05-00OR22725
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Mathematical Modelling and Numerical Analysis
- Additional Journal Information:
- Journal Volume: 53; Journal Issue: 6; Journal ID: ISSN 0764-583X
- Publisher:
- EDP Sciences
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING
Citation Formats
Webster, Clayton, Tran, Hoang, and Dexter, Nick. A mixed $\ell_1$ regularization approach for sparse simultaneous approximation of parameterized PDEs. United States: N. p., 2019.
Web. doi:10.1051/m2an/2019048.
Webster, Clayton, Tran, Hoang, & Dexter, Nick. A mixed $\ell_1$ regularization approach for sparse simultaneous approximation of parameterized PDEs. United States. https://doi.org/10.1051/m2an/2019048
Webster, Clayton, Tran, Hoang, and Dexter, Nick. Thu .
"A mixed $\ell_1$ regularization approach for sparse simultaneous approximation of parameterized PDEs". United States. https://doi.org/10.1051/m2an/2019048. https://www.osti.gov/servlets/purl/1564178.
@article{osti_1564178,
title = {A mixed $\ell_1$ regularization approach for sparse simultaneous approximation of parameterized PDEs},
author = {Webster, Clayton and Tran, Hoang and Dexter, Nick},
abstractNote = {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.},
doi = {10.1051/m2an/2019048},
journal = {Mathematical Modelling and Numerical Analysis},
number = 6,
volume = 53,
place = {United States},
year = {Thu Jun 27 00:00:00 EDT 2019},
month = {Thu Jun 27 00:00:00 EDT 2019}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 4 works
Citation information provided by
Web of Science
Web of Science
Figures / Tables:
Fig. 6.1: (left) Comparison of relative error statistics $ε^{rel−E}_{h,approx}$ from (6.1) for the SCS method (SCS-TD) and compressed sensing point-wise functional recovery (PCS-TD) methods, both computed with a total degree basis of order p = 2 in d = 100 dimensions having cardinality N = 5151. (center) Magnitudes of themore »
All figures and tables
(4 total)
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Signal Recovery by Proximal Forward-Backward Splitting
journal, January 2005
- Combettes, Patrick L.; Wajs, Valérie R.
- Multiscale Modeling & Simulation, Vol. 4, Issue 4
Sparse Legendre expansions via <mml:math altimg="si12.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>-minimization
journal, May 2012
- Rauhut, Holger; Ward, Rachel
- Journal of Approximation Theory, Vol. 164, Issue 5
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
journal, January 2004
- Daubechies, I.; Defrise, M.; De Mol, C.
- Communications on Pure and Applied Mathematics, Vol. 57, Issue 11
Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs
journal, November 2012
- Chkifa, Abdellah; Cohen, Albert; DeVore, Ronald
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 47, Issue 1
Error Forgetting of Bregman Iteration
journal, June 2012
- Yin, Wotao; Osher, Stanley
- Journal of Scientific Computing, Vol. 54, Issue 2-3
Polynomial approximation via compressed sensing of high-dimensional functions on lower sets
journal, September 2017
- Chkifa, Abdellah; Dexter, Nick; Tran, Hoang
- Mathematics of Computation, Vol. 87, Issue 311
On the Stability and Accuracy of Least Squares Approximations
journal, March 2013
- Cohen, Albert; Davenport, Mark A.; Leviatan, Dany
- Foundations of Computational Mathematics, Vol. 13, Issue 5
A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions
journal, January 2017
- Jakeman, John D.; Narayan, Akil; Zhou, Tao
- SIAM Journal on Scientific Computing, Vol. 39, Issue 3
Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing
journal, January 2008
- Yin, Wotao; Osher, Stanley; Goldfarb, Donald
- SIAM Journal on Imaging Sciences, Vol. 1, Issue 1
Robustness to Unknown Error in Sparse Regularization
journal, October 2018
- Brugiapaglia, Simone; Adcock, Ben
- IEEE Transactions on Information Theory, Vol. 64, Issue 10
A non-adapted sparse approximation of PDEs with stochastic inputs
journal, April 2011
- Doostan, Alireza; Owhadi, Houman
- Journal of Computational Physics, Vol. 230, Issue 8
Analysis of quasi-optimal polynomial approximations for parameterized PDEs with deterministic and stochastic coefficients
journal, March 2017
- Tran, Hoang; Webster, Clayton G.; Zhang, Guannan
- Numerische Mathematik, Vol. 137, Issue 2
Compressed sensing
journal, April 2006
- Donoho, D. L.
- IEEE Transactions on Information Theory, Vol. 52, Issue 4
Sparse Tensor Galerkin Discretization of Parametric and Random Parabolic PDEs---Analytic Regularity and Generalized Polynomial Chaos Approximation
journal, January 2013
- Hoang, Viet Ha; Schwab, Christoph
- SIAM Journal on Mathematical Analysis, Vol. 45, Issue 5
Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation
journal, January 2010
- Eldar, Yonina C.; Rauhut, Holger
- IEEE Transactions on Information Theory, Vol. 56, Issue 1
Sparse Adaptive Approximation of High Dimensional Parametric Initial Value Problems
journal, March 2013
- Hansen, Markus; Schwab, Christoph
- Vietnam Journal of Mathematics, Vol. 41, Issue 2
Regularity and Generalized Polynomial Chaos Approximation of Parametric and Random Second-Order Hyperbolic Partial Differential Equations
journal, July 2012
- Hoang, Viet Ha; Schwab, Christoph
- Analysis and Applications, Vol. 10, Issue 03
The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming
journal, January 1967
- Bregman, L. M.
- USSR Computational Mathematics and Mathematical Physics, Vol. 7, Issue 3
Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos
journal, September 2002
- Xiu, Dongbin; Em Karniadakis, George
- Computer Methods in Applied Mechanics and Engineering, Vol. 191, Issue 43
Stochastic finite element methods for partial differential equations with random input data
journal, May 2014
- Gunzburger, Max D.; Webster, Clayton G.; Zhang, Guannan
- Acta Numerica, Vol. 23
Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs
journal, December 2012
- Hansen, Markus; Schwab, Christoph
- Mathematische Nachrichten, Vol. 286, Issue 8-9
Sparse Tensor Discretization of Elliptic sPDEs
journal, January 2010
- Bieri, Marcel; Andreev, Roman; Schwab, Christoph
- SIAM Journal on Scientific Computing, Vol. 31, Issue 6
N-TERM WIENER CHAOS APPROXIMATION RATES FOR ELLIPTIC PDEs WITH LOGNORMAL GAUSSIAN RANDOM INPUTS
journal, January 2014
- Hoang, Viet Ha; Schwab, Christoph
- Mathematical Models and Methods in Applied Sciences, Vol. 24, Issue 04
Assessment of Collocation and Galerkin Approaches to Linear Diffusion Equations with Random data
journal, January 2011
- Tuminaro, Raymond S.; Phipps, Eric T.; Miller, Christopher W.
- International Journal for Uncertainty Quantification, Vol. 1, Issue 1
Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
journal, May 2016
- Rauhut, Holger; Schwab, Christoph
- Mathematics of Computation, Vol. 86, Issue 304
Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and Convergence
journal, January 2008
- Hale, Elaine T.; Yin, Wotao; Zhang, Yin
- SIAM Journal on Optimization, Vol. 19, Issue 3
Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors
journal, October 2008
- Mishali, M.; Eldar, Y. C.
- IEEE Transactions on Signal Processing, Vol. 56, Issue 10
Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs
journal, April 2015
- Chkifa, Abdellah; Cohen, Albert; Migliorati, Giovanni
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 49, Issue 3
On polynomial chaos expansion via gradient-enhanced ℓ1-minimization
journal, April 2016
- Peng, Ji; Hampton, Jerrad; Doostan, Alireza
- Journal of Computational Physics, Vol. 310
Atoms of All Channels, Unite! Average Case Analysis of Multi-Channel Sparse Recovery Using Greedy Algorithms
journal, October 2008
- Gribonval, Rémi; Rauhut, Holger; Schnass, Karin
- Journal of Fourier Analysis and Applications, Vol. 14, Issue 5-6
A Compressed Sensing Approach for Partial Differential Equations with Random Input Data
journal, October 2012
- Mathelin, L.; Gallivan, K. A.
- Communications in Computational Physics, Vol. 12, Issue 4
Compressed Sensing With Cross Validation
journal, December 2009
- Ward, Rachel
- IEEE Transactions on Information Theory, Vol. 55, Issue 12
Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs
journal, July 2010
- Cohen, Albert; DeVore, Ronald; Schwab, Christoph
- Foundations of Computational Mathematics, Vol. 10, Issue 6
Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies
journal, January 2015
- Hampton, Jerrad; Doostan, Alireza
- Journal of Computational Physics, Vol. 280
A dynamically adaptive sparse grids method for quasi-optimal interpolation of multidimensional functions
journal, June 2016
- Stoyanov, Miroslav K.; Webster, Clayton G.
- Computers & Mathematics with Applications, Vol. 71, Issue 11
An Iterative Regularization Method for Total Variation-Based Image Restoration
journal, January 2005
- Osher, Stanley; Burger, Martin; Goldfarb, Donald
- Multiscale Modeling & Simulation, Vol. 4, Issue 2
Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients
journal, April 2007
- Todor, Radu Alexandru; Schwab, Christoph
- IMA Journal of Numerical Analysis, Vol. 27, Issue 2
Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information
journal, February 2006
- Candes, E.J.; Romberg, J.; Tao, T.
- IEEE Transactions on Information Theory, Vol. 52, Issue 2, p. 489-509
Fixed-Point Continuation Applied to Compressed Sensing: Implementation and Numerical Experiments
journal, June 2010
- Zhang, Elaine T. Hale Wotao Yin and Yin
- Journal of Computational Mathematics, Vol. 28, Issue 2
Infinite-Dimensional Compressed Sensing and Function Interpolation
journal, April 2017
- Adcock, Ben
- Foundations of Computational Mathematics, Vol. 18, Issue 3
Explicit cost bounds of stochastic Galerkin approximations for parameterized PDEs with random coefficients
journal, June 2016
- Dexter, Nick C.; Webster, Clayton G.; Zhang, Guannan
- Computers & Mathematics with Applications, Vol. 71, Issue 11
A weighted -minimization approach for sparse polynomial chaos expansions
journal, June 2014
- Peng, Ji; Hampton, Jerrad; Doostan, Alireza
- Journal of Computational Physics, Vol. 267
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
journal, January 2008
- Nobile, F.; Tempone, R.; Webster, C. G.
- SIAM Journal on Numerical Analysis, Vol. 46, Issue 5
The Homogeneous Chaos
journal, October 1938
- Wiener, Norbert
- American Journal of Mathematics, Vol. 60, Issue 4
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
journal, January 2002
- Xiu, Dongbin; Karniadakis, George Em
- SIAM Journal on Scientific Computing, Vol. 24, Issue 2
Analytic Regularity and Polynomial Approximation of Parametric and Stochastic Elliptic Pde'S
journal, January 2011
- Cohen, Albert; Devore, Ronald; Schwab, Christoph
- Analysis and Applications, Vol. 09, Issue 01
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
journal, January 2008
- Nobile, F.; Tempone, R.; Webster, C. G.
- SIAM Journal on Numerical Analysis, Vol. 46, Issue 5
Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics
book, January 2010
- Le Maître, O. P.; Knio, Omar M.
- Scientific Computation
Reweighted minimization method for stochastic elliptic differential equations
journal, September 2013
- Yang, Xiu; Karniadakis, George Em
- Journal of Computational Physics, Vol. 248
Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs
journal, February 2015
- Chkifa, Abdellah; Cohen, Albert; Schwab, Christoph
- Journal de Mathématiques Pures et Appliquées, Vol. 103, Issue 2
Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients
journal, March 2014
- Beck, Joakim; Nobile, Fabio; Tamellini, Lorenzo
- Computers & Mathematics with Applications, Vol. 67, Issue 4
Infinite-Dimensional $$\ell ^1$$ ℓ 1 Minimization and Function Approximation from Pointwise Data
journal, February 2017
- Adcock, Ben
- Constructive Approximation, Vol. 45, Issue 3
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
journal, January 2007
- Babuška, Ivo; Nobile, Fabio; Tempone, Raúl
- SIAM Journal on Numerical Analysis, Vol. 45, Issue 3
STOCHASTIC COLLOCATION ALGORITHMS USING l1-MINIMIZATION
journal, January 2012
- Yan, Liang; Guo, Ling; Xiu, Dongbin
- International Journal for Uncertainty Quantification, Vol. 2, Issue 3
A gradient enhanced ℓ1-minimization for sparse approximation of polynomial chaos expansions
journal, August 2018
- Guo, Ling; Narayan, Akil; Zhou, Tao
- Journal of Computational Physics, Vol. 367
On the Optimal Polynomial Approximation of Stochastic pdes by Galerkin and Collocation Methods
journal, July 2012
- Beck, Joakim; Tempone, Raul; Nobile, Fabio
- Mathematical Models and Methods in Applied Sciences, Vol. 22, Issue 09
The null space property for sparse recovery from multiple measurement vectors
journal, May 2011
- Lai, Ming-Jun; Liu, Yang
- Applied and Computational Harmonic Analysis, Vol. 30, Issue 3
Theoretical and Empirical Results for Recovery From Multiple Measurements
journal, May 2010
- van den Berg, Ewout; Friedlander, Michael P.
- IEEE Transactions on Information Theory, Vol. 56, Issue 5
Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
journal, December 2016
- Bachmayr, Markus; Cohen, Albert; DeVore, Ronald
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 1
Interpolation via weighted ℓ1 minimization
journal, March 2016
- Rauhut, Holger; Ward, Rachel
- Applied and Computational Harmonic Analysis, Vol. 40, Issue 2
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
journal, January 2010
- Babuška, Ivo; Nobile, Fabio; Tempone, Raúl
- SIAM Review, Vol. 52, Issue 2
Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs
text, January 2013
- Albert, Cohen,; Abdellah, Chkifa,; Christoph, Schwab,
- ETH Zurich
Sparse adaptive approximation of high dimensional parametric initial value problems
text, January 2012
- Hansen, M.; Schwab, Christoph
- ETH Zurich
Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs
text, January 2011
- Abdellah, Chkifa,; Albert, Cohen,; Ronald, DeVore,
- ETH Zurich
N-term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs
text, January 2013
- Hoang, Viet Ha; Schwab, Christoph
- ETH Zurich
Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs
text, January 2009
- Cohen, Albert; DeVore, Ronald; Schwab, Christoph
- ETH Zurich
Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients
text, January 2007
- Tudor, Radu A.; Schwab, Christoph
- ETH Zurich
Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
text, January 2014
- Rauhut, Holger; Schwab, Christoph
- ETH Zurich
Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs
text, January 2011
- Hansen, Markus; Schwab, Christoph
- ETH Zurich
Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors
text, January 2008
- Mishali, Moshe; Eldar, Yonina C.
- arXiv
Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation
preprint, January 2009
- Eldar, Yonina C.; Rauhut, Holger
- arXiv
A non-adapted sparse approximation of PDEs with stochastic inputs
text, January 2010
- Doostan, Alireza; Owhadi, Houman
- arXiv
On the stability and accuracy of least squares approximations
preprint, January 2011
- Cohen, Albert; Davenport, Mark A.; Leviatan, Dany
- arXiv
Explicit cost bounds of stochastic Galerkin approximations for parameterized PDEs with random coefficients
text, January 2015
- Dexter, Nick; Webster, Clayton; Zhang, Guannan
- arXiv
Infinite-dimensional compressed sensing and function interpolation
preprint, January 2015
- Adcock, Ben
- arXiv
A gradient enhanced $\ell_1$-minimization for sparse approximation of polynomial chaos expansions
text, January 2018
- Guo, Ling; Narayan, Akil; Zhou, Tao
- arXiv
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
preprint, January 2003
- Daubechies, Ingrid; Defrise, Michel; De Mol, Christine
- arXiv
Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information
preprint, January 2004
- Candes, Emmanuel; Romberg, Justin; Tao, Terence
- arXiv
Atoms of All Channels, Unite! Average Case Analysis of Multi-Channel Sparse Recovery Using Greedy Algorithms
journal, October 2008
- Gribonval, Rémi; Rauhut, Holger; Schnass, Karin
- Journal of Fourier Analysis and Applications, Vol. 14, Issue 5-6
Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression
journal, June 2015
- Hampton, Jerrad; Doostan, Alireza
- Computer Methods in Applied Mechanics and Engineering, Vol. 290
Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs
journal, April 2015
- Chkifa, Abdellah; Cohen, Albert; Migliorati, Giovanni
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 49, Issue 3
Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
journal, May 2016
- Rauhut, Holger; Schwab, Christoph
- Mathematics of Computation, Vol. 86, Issue 304
Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors
text, January 2008
- Mishali, Moshe; Eldar, Yonina C.
- arXiv
A non-adapted sparse approximation of PDEs with stochastic inputs
text, January 2010
- Doostan, Alireza; Owhadi, Houman
- arXiv
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
preprint, January 2003
- Daubechies, Ingrid; Defrise, Michel; De Mol, Christine
- arXiv
Figures / Tables found in this record:
Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.