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 0 -"norm"; this results in binary designs that characterize optimal sensor locations. We present numerical results for inference of the initial condition from spatio-temporal 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 0 -sparsified experimental designs obtained via a continuation method outperform 1 -sparsified designs.Key words. Optimal experimental design, A-optimal design, Bayesian inference, sensor placement, ill-posed inverse problems, low-rank approximation, randomized trace estimator, randomized SVD.
Abstract. We address the problem of optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs). The inverse problem seeks to infer an infinite-dimensional parameter from experimental data observed at a set of sensor locations and from the governing PDEs. The goal of the OED problem is to find an optimal placement of sensors so as to minimize the uncertainty in the inferred parameter field. Specifically, we seek an optimal subset of sensors from among a fixed set of candidate sensor locations. We formulate the OED objective function by generalizing the classical A-optimal experimental design criterion using the expected value of the trace of the posterior covariance. This expected value is computed through sample averaging over the set of likely experimental data. To cope with the infinite-dimensional character of the parameter field, we construct a Gaussian approximation to the posterior at the maximum a posteriori probability (MAP) point, and use the resulting covariance operator to define the OED objective function. We use randomized trace estimation to compute the trace of this covariance operator, which is defined only implicitly. The resulting OED problem includes as constraints the system of PDEs characterizing the MAP point, and the PDEs describing the action of the covariance (of the Gaussian approximation to the posterior) to vectors. We control the sparsity of the sensor configurations using sparsifying penalty functions. Variational adjoint methods are used to efficiently compute the gradient of the PDE-constrained OED objective function. We elaborate our OED method for the problem of determining the optimal sensor configuration to best infer the coefficient of an elliptic PDE. Furthermore, we provide numerical results for inference of the log permeability field in a porous medium flow problem. Numerical results show that the number of PDE solves required for the evaluation of the OED objective function and its gradient is essentially independent of both the parameter dimension and the sensor dimension (i.e., the number of candidate sensor locations). The number of quasi-Newton iterations for computing an OED also exhibits the same dimension invariance properties.Key words. Optimal experimental design, A-optimal design, Bayesian inference, sensor placement, nonlinear inverse problems, randomized trace estimator, sparsified designs.
Polynomial chaos (PC) expansions are used to propagate parametric uncertainties in ocean global circulation model. The computations focus on short-time, high-resolution simulations of the Gulf of Mexico, using the hybrid coordinate ocean model, with wind stresses corresponding to hurricane Ivan. A sparse quadrature approach is used to determine the PC coefficients which provides a detailed representation of the stochastic model response. The quality of the PC representation is first examined through a systematic refinement of the number of resolution levels. The PC representation of the stochastic model response is then utilized to compute distributions of quantities of interest (QoIs) and to analyze the local and global sensitivity of these QoIs to uncertain parameters. Conclusions are finally drawn regarding limitations of local perturbations and variancebased assessment and concerning potential application of the present methodology to inverse problems and to uncertainty management.
We present randomized algorithms for estimating the trace and determinant of Hermitian positive semi-definite matrices. The algorithms are based on subspace iteration, and access the matrix only through matrix vector products. We analyse the error due to randomization, for starting guesses whose elements are Gaussian or Rademacher random variables. The analysis is cleanly separated into a structural (deterministic) part followed by a probabilistic part. Our absolute bounds for the expectation and concentration of the estimators are non-asymptotic and informative even for matrices of low dimension. For the trace estimators, we also present asymptotic bounds on the number of samples (columns of the starting guess) required to achieve a user-specified relative error. Numerical experiments illustrate the performance of the estimators and the tightness of the bounds on low-dimensional matrices; and on a challenging application in uncertainty quantification arising from Bayesian optimal experimental design.
We consider Bayesian linear inverse problems in infinite-dimensional separable Hilbert spaces, with a Gaussian prior measure and additive Gaussian noise model, and provide an extension of the concept of Bayesian D-optimality to the infinite-dimensional case. To this end, we derive the infinite-dimensional version of the expression for the Kullback-Leibler divergence from the posterior measure to the prior measure, which is subsequently used to derive the expression for the expected information gain. We also study the notion of Bayesian Aoptimality in the infinite-dimensional setting, and extend the well known (in the finite-dimensional case) equivalence of the Bayes risk of the MAP estimator with the trace of the posterior covariance, for the Gaussian linear case, to the infinitedimensional Hilbert space case.
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