# Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification

## Abstract

Here, we analyze the convergence of probability density functions utilizing approximate models for both forward and inverse problems. We consider the standard forward uncertainty quantification problem where an assumed probability density on parameters is propagated through the approximate model to produce a probability density, often called a push-forward probability density, on a set of quantities of interest (QoI). The inverse problem considered in this paper seeks to update an initial probability density assumed on model input parameters such that the subsequent push-forward of this updated density through the parameter-to-QoI map matches a given probability density on the QoI. We prove that the densities obtained from solving the forward and inverse problems, using approximate models, converge to the true densities as the approximate models converge to the true models. Numerical results are presented to demonstrate convergence rates of densities for sparse grid approximations of parameter-to-QoI maps and standard spatial and temporal discretizations of PDEs and ODEs.

- Authors:

- Univ. of Colorado, Boulder, CO (United States)
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1479491

- Report Number(s):
- SAND-2018-6887J

Journal ID: ISSN 1064-8275; 664970

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- SIAM Journal on Scientific Computing

- Additional Journal Information:
- Journal Volume: 40; Journal Issue: 5; Journal ID: ISSN 1064-8275

- Publisher:
- SIAM

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; inverse problems; uncertainty quantification; density estimation; surrogate modeling; response surface approximations; discretization errors

### Citation Formats

```
Butler, Troy, Jakeman, John Davis, and Wildey, Timothy Michael.
```*Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification*. United States: N. p., 2018.
Web. doi:10.1137/18M1181675.

```
Butler, Troy, Jakeman, John Davis, & Wildey, Timothy Michael.
```*Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification*. United States. doi:10.1137/18M1181675.

```
Butler, Troy, Jakeman, John Davis, and Wildey, Timothy Michael. Thu .
"Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification". United States. doi:10.1137/18M1181675.
```

```
@article{osti_1479491,
```

title = {Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification},

author = {Butler, Troy and Jakeman, John Davis and Wildey, Timothy Michael},

abstractNote = {Here, we analyze the convergence of probability density functions utilizing approximate models for both forward and inverse problems. We consider the standard forward uncertainty quantification problem where an assumed probability density on parameters is propagated through the approximate model to produce a probability density, often called a push-forward probability density, on a set of quantities of interest (QoI). The inverse problem considered in this paper seeks to update an initial probability density assumed on model input parameters such that the subsequent push-forward of this updated density through the parameter-to-QoI map matches a given probability density on the QoI. We prove that the densities obtained from solving the forward and inverse problems, using approximate models, converge to the true densities as the approximate models converge to the true models. Numerical results are presented to demonstrate convergence rates of densities for sparse grid approximations of parameter-to-QoI maps and standard spatial and temporal discretizations of PDEs and ODEs.},

doi = {10.1137/18M1181675},

journal = {SIAM Journal on Scientific Computing},

issn = {1064-8275},

number = 5,

volume = 40,

place = {United States},

year = {2018},

month = {10}

}