### Abstract

We present an efficient method for computing A-optimal experimental designs for infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs). Specifically, we address the problem of optimizing the location of sensors (at which observational data are collected) to minimize the uncertainty in the parameters estimated by solving the inverse problem, where the uncertainty is expressed by the trace of the posterior covariance. Computing optimal experimental designs (OEDs) is particularly challenging for inverse problems governed by computationally expensive PDE models with infinite-dimensional (or, after discretization, high-dimensional) parameters. To alleviate the computational cost, we exploit the problem structure and build a low-rank approximation of the parameter-to-observable map, preconditioned with the square root of the prior covariance operator. The availability of this low-rank surrogate, relieves our method from expensive PDE solves when evaluating the optimal experimental design objective function, i.e., the trace of the posterior covariance, and its derivatives. Moreover, we employ a randomized trace estimator for efficient evaluation of the OED objective function. We control the sparsity of the sensor configuration by employing a sequence of penalty functions that successively approximate the l0-"norm"; this results in binary designs that characterize optimal sensor locations. We present numerical results for inference of the initial condition from spatiotemporal observations in a time-dependent advection-diffusion problem in two and three space dimensions. We find that an optimal design can be computed at a cost, measured in number of forward PDE solves, that is independent of the parameter and sensor dimensions. Moreover, the numerical optimization problem for finding the optimal design can be solved in a number of interior-point quasi-Newton iterations that is insensitive to the parameter and sensor dimensions. We demonstrate numerically that l0-sparsified experimental designs obtained via a continuation method outperform l1-sparsified designs.

Original language | English (US) |
---|---|

Pages (from-to) | A2122-A2148 |

Journal | SIAM Journal on Scientific Computing |

Volume | 36 |

Issue number | 5 |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

- A-optimal design
- Bayesian inference
- Ill-posed inverse problems
- Low-rank approximation
- Optimal experimental design
- Randomized SVD
- Randomized trace estimator
- Sensor placement

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*36*(5), A2122-A2148. https://doi.org/10.1137/130933381

**A-Optimal design of experiments for infinite-dimensional Bayesian linear inverse problems with regularized l0-Sparsification.** / Alexanderian, Alen; Petra, Noemi; Stadler, Georg; Ghattas, Omar.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 36, no. 5, pp. A2122-A2148. https://doi.org/10.1137/130933381

}

TY - JOUR

T1 - A-Optimal design of experiments for infinite-dimensional Bayesian linear inverse problems with regularized l0-Sparsification

AU - Alexanderian, Alen

AU - Petra, Noemi

AU - Stadler, Georg

AU - Ghattas, Omar

PY - 2014

Y1 - 2014

N2 - We present an efficient method for computing A-optimal experimental designs for infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs). Specifically, we address the problem of optimizing the location of sensors (at which observational data are collected) to minimize the uncertainty in the parameters estimated by solving the inverse problem, where the uncertainty is expressed by the trace of the posterior covariance. Computing optimal experimental designs (OEDs) is particularly challenging for inverse problems governed by computationally expensive PDE models with infinite-dimensional (or, after discretization, high-dimensional) parameters. To alleviate the computational cost, we exploit the problem structure and build a low-rank approximation of the parameter-to-observable map, preconditioned with the square root of the prior covariance operator. The availability of this low-rank surrogate, relieves our method from expensive PDE solves when evaluating the optimal experimental design objective function, i.e., the trace of the posterior covariance, and its derivatives. Moreover, we employ a randomized trace estimator for efficient evaluation of the OED objective function. We control the sparsity of the sensor configuration by employing a sequence of penalty functions that successively approximate the l0-"norm"; this results in binary designs that characterize optimal sensor locations. We present numerical results for inference of the initial condition from spatiotemporal observations in a time-dependent advection-diffusion problem in two and three space dimensions. We find that an optimal design can be computed at a cost, measured in number of forward PDE solves, that is independent of the parameter and sensor dimensions. Moreover, the numerical optimization problem for finding the optimal design can be solved in a number of interior-point quasi-Newton iterations that is insensitive to the parameter and sensor dimensions. We demonstrate numerically that l0-sparsified experimental designs obtained via a continuation method outperform l1-sparsified designs.

AB - We present an efficient method for computing A-optimal experimental designs for infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs). Specifically, we address the problem of optimizing the location of sensors (at which observational data are collected) to minimize the uncertainty in the parameters estimated by solving the inverse problem, where the uncertainty is expressed by the trace of the posterior covariance. Computing optimal experimental designs (OEDs) is particularly challenging for inverse problems governed by computationally expensive PDE models with infinite-dimensional (or, after discretization, high-dimensional) parameters. To alleviate the computational cost, we exploit the problem structure and build a low-rank approximation of the parameter-to-observable map, preconditioned with the square root of the prior covariance operator. The availability of this low-rank surrogate, relieves our method from expensive PDE solves when evaluating the optimal experimental design objective function, i.e., the trace of the posterior covariance, and its derivatives. Moreover, we employ a randomized trace estimator for efficient evaluation of the OED objective function. We control the sparsity of the sensor configuration by employing a sequence of penalty functions that successively approximate the l0-"norm"; this results in binary designs that characterize optimal sensor locations. We present numerical results for inference of the initial condition from spatiotemporal observations in a time-dependent advection-diffusion problem in two and three space dimensions. We find that an optimal design can be computed at a cost, measured in number of forward PDE solves, that is independent of the parameter and sensor dimensions. Moreover, the numerical optimization problem for finding the optimal design can be solved in a number of interior-point quasi-Newton iterations that is insensitive to the parameter and sensor dimensions. We demonstrate numerically that l0-sparsified experimental designs obtained via a continuation method outperform l1-sparsified designs.

KW - A-optimal design

KW - Bayesian inference

KW - Ill-posed inverse problems

KW - Low-rank approximation

KW - Optimal experimental design

KW - Randomized SVD

KW - Randomized trace estimator

KW - Sensor placement

UR - http://www.scopus.com/inward/record.url?scp=84911378439&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84911378439&partnerID=8YFLogxK

U2 - 10.1137/130933381

DO - 10.1137/130933381

M3 - Article

AN - SCOPUS:84911378439

VL - 36

SP - A2122-A2148

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 5

ER -