# Sparse-grid, reduced-basis Bayesian inversion: Nonaffine-parametric nonlinear equations

## Abstract

We extend the reduced basis (RB) accelerated Bayesian inversion methods for affine-parametric, linear operator equations which are considered in [16,17] to non-affine, nonlinear parametric operator equations. We generalize the analysis of sparsity of parametric forward solution maps in [20] and of Bayesian inversion in [48,49] to the fully discrete setting, including Petrov–Galerkin high-fidelity (“HiFi”) discretization of the forward maps. We develop adaptive, stochastic collocation based reduction methods for the efficient computation of reduced bases on the parametric solution manifold. The nonaffinity and nonlinearity with respect to (w.r.t.) the distributed, uncertain parameters and the unknown solution is collocated; specifically, by the so-called Empirical Interpolation Method (EIM). For the corresponding Bayesian inversion problems, computational efficiency is enhanced in two ways: first, expectations w.r.t. the posterior are computed by adaptive quadratures with dimension-independent convergence rates proposed in [49]; the present work generalizes [49] to account for the impact of the PG discretization in the forward maps on the convergence rates of the Quantities of Interest (QoI for short). Second, we propose to perform the Bayesian estimation only w.r.t. a parsimonious, RB approximation of the posterior density. Based on the approximation results in [49], the infinite-dimensional parametric, deterministic forward map and operator admit N-termmore »

- Authors:

- The Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th Street, Stop C0200, Austin, TX 78712-1229 (United States)
- Seminar für Angewandte Mathematik, Eidgenössische Technische Hochschule, Römistrasse 101, CH-8092 Zürich (Switzerland)

- Publication Date:

- OSTI Identifier:
- 22572326

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 316; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; CONVERGENCE; CURRENTS; EFFICIENCY; EQUATIONS; ERRORS; EXPERIMENT DESIGN; INTERPOLATION; NONLINEAR PROBLEMS; QUADRATURES; STOCHASTIC PROCESSES

### Citation Formats

```
Chen, Peng, E-mail: peng@ices.utexas.edu, and Schwab, Christoph, E-mail: christoph.schwab@sam.math.ethz.ch.
```*Sparse-grid, reduced-basis Bayesian inversion: Nonaffine-parametric nonlinear equations*. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.02.055.

```
Chen, Peng, E-mail: peng@ices.utexas.edu, & Schwab, Christoph, E-mail: christoph.schwab@sam.math.ethz.ch.
```*Sparse-grid, reduced-basis Bayesian inversion: Nonaffine-parametric nonlinear equations*. United States. doi:10.1016/J.JCP.2016.02.055.

```
Chen, Peng, E-mail: peng@ices.utexas.edu, and Schwab, Christoph, E-mail: christoph.schwab@sam.math.ethz.ch. Fri .
"Sparse-grid, reduced-basis Bayesian inversion: Nonaffine-parametric nonlinear equations". United States.
doi:10.1016/J.JCP.2016.02.055.
```

```
@article{osti_22572326,
```

title = {Sparse-grid, reduced-basis Bayesian inversion: Nonaffine-parametric nonlinear equations},

author = {Chen, Peng, E-mail: peng@ices.utexas.edu and Schwab, Christoph, E-mail: christoph.schwab@sam.math.ethz.ch},

abstractNote = {We extend the reduced basis (RB) accelerated Bayesian inversion methods for affine-parametric, linear operator equations which are considered in [16,17] to non-affine, nonlinear parametric operator equations. We generalize the analysis of sparsity of parametric forward solution maps in [20] and of Bayesian inversion in [48,49] to the fully discrete setting, including Petrov–Galerkin high-fidelity (“HiFi”) discretization of the forward maps. We develop adaptive, stochastic collocation based reduction methods for the efficient computation of reduced bases on the parametric solution manifold. The nonaffinity and nonlinearity with respect to (w.r.t.) the distributed, uncertain parameters and the unknown solution is collocated; specifically, by the so-called Empirical Interpolation Method (EIM). For the corresponding Bayesian inversion problems, computational efficiency is enhanced in two ways: first, expectations w.r.t. the posterior are computed by adaptive quadratures with dimension-independent convergence rates proposed in [49]; the present work generalizes [49] to account for the impact of the PG discretization in the forward maps on the convergence rates of the Quantities of Interest (QoI for short). Second, we propose to perform the Bayesian estimation only w.r.t. a parsimonious, RB approximation of the posterior density. Based on the approximation results in [49], the infinite-dimensional parametric, deterministic forward map and operator admit N-term RB and EIM approximations which converge at rates which depend only on the sparsity of the parametric forward map. In several numerical experiments, the proposed algorithms exhibit dimension-independent convergence rates which equal, at least, the currently known rate estimates for N-term approximation. We propose to accelerate Bayesian estimation by first offline construction of reduced basis surrogates of the Bayesian posterior density. The parsimonious surrogates can then be employed for online data assimilation and for Bayesian estimation. They also open a perspective for optimal experimental design.},

doi = {10.1016/J.JCP.2016.02.055},

journal = {Journal of Computational Physics},

number = ,

volume = 316,

place = {United States},

year = {Fri Jul 01 00:00:00 EDT 2016},

month = {Fri Jul 01 00:00:00 EDT 2016}

}