A greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters. Large-scale models are too costly to evaluate repeatedly, as is required in the statistical setting. Furthermore, these models often have high-dimensional parametric input spaces, which compounds the difficulty of effectively exploring the uncertainty space. We simultaneously address both challenges by constructing a projectionbased reduced model that accepts low-dimensional parameter inputs and whose model evaluations are inexpensive. The associated parameter and state bases are obtained through a greedy procedure that targets the governing equations, model outputs, and prior information. The methodology and results are presented for groundwater inverse problems in one and two dimensions.1. Introduction. Statistical inverse problems governed by partial differential equations (PDEs) with spatially distributed parameters pose a significant computational challenge for existing methods. While the cost of a repeated PDE solutions can be addressed by traditional model reduction techniques, the difficulty in sampling in high-dimensional parameter spaces remains. We present a model reduction algorithm that seeks low-dimensional representations of parameters and states while maintaining fidelity in outputs of interest. The resulting reduced model accelerates model evaluations and facilitates efficient sampling in the reduced parameter space. The result is a tractable procedure for the solution of statistical inverse problems involving PDEs with high-dimensional parametric input spaces.Given a parameterized mathematical model of a certain phenomenon, the forward problem is to compute output quantities of interest for specified parameter inputs. In many cases, the parameters are uncertain, but they can be inferred from observations by solving an inverse problem. Inference is often performed by solving an optimization problem to minimize the disparity between model-predicted outputs and observations. Many inverse problems of this form are ill-posed in the sense that there may be many values of the parameters whose model-predicted outputs reproduce the observations. The set of parameters consistent with the observations may be larger still if we also admit noise in the sensor instruments. In the deterministic setting, a regularization term is often included in the objective function to make the problem well-posed. The *

We compute u(t) = exp(−tA)ϕ using rational Krylov subspace reduction for 0 ≤ t < ∞, where u(t), ϕ ∈ R N and 0 < A = A * ∈ R N×N. A priori optimization of the rational Krylov subspaces for this problem was considered in [V. Druskin, L. Knizhnerman, and M. Zaslavsky, SIAM J. Sci. Comput., 31 (2009), pp. 3760-3780]. There was suggested an algorithm generating sequences of equidistributed shifts, which are asymptotically optimal for the cases with uniform spectral distributions. Here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum. The algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost. The effectiveness of the developed approach is demonstrated in an example of the three-dimensional diffusion problem for Maxwell's equation arising in geophysical exploration. We compare our approach with the one using the above-mentioned equidistributed sequences of shifts. Numerical examples show that our algorithm is able to adapt to the spectral density of operator A. For examples with near-uniform spectral distributions, both algorithms show the same convergence rates, but the new algorithm produces superior convergence for cases with nonuniform spectra.

Abstract. Inference of model parameters is one step in an engineering process often ending in predictions that support decision in the form of design or control. Incorporation of end goals into the inference process leads to more efficient goal-oriented algorithms that automatically target the most relevant parameters for prediction. In the linear setting the control-theoretic concepts underlying balanced truncation model reduction can be exploited in inference through a dimensionally optimal subspace regularizer. The inference-for-prediction method exactly replicates the prediction results of either truncated singular value decomposition, Tikhonov-regularized, or Gaussian statistical inverse problem formulations independent of data; it sacrifices accuracy in parameter estimate for online efficiency. The new method leads to low-dimensional parameterization of the inverse problem enabling solution on smartphones or laptops in the field. Many inverse problems are ill-posed; the data do not determine a unique solution. Inference approaches, therefore, rely on the injection of prior information. In deterministic formulations [9], this prior information is often manifested as a form of regularization. In Bayesian statistical formulations [17], the prior information is used to formulate a prior distribution reflecting the belief in probable parameter values. As a result, the distinction becomes blurred between inferred parameter modes informed by data and modes influenced largely or wholly by prior information. Without careful design of prior information, this injection of outside information can overshadow the information contained in the limited data that are obtained. Although ill-posedness will always be an issue to some extent in limited data settings, in this paper we show that it is possible to partially circumvent the deleterious effects of the use of regularizers or prior information by incorporating end goals.While in some cases estimation of unknown parameters is the end goal, there are many engineering processes where parameter estimation is one step in a multistep process ending with design. In such scenarios, engineers often define output quantities of interest to be optimized by the design. In consideration of this fact, we propose a goal-oriented approach to inference that accounts for the output quantities of interest. Generally, in an abstract sense, our experimental data are informative

SUMMARYHessian-based model reduction was previously proposed as an approach in deriving reduced models for the solution of large-scale linear inverse problems by targeting accuracy in observation outputs. A controltheoretic view of Hessian-based model reduction that hinges on the equality between the Hessian and the transient observability gramian of the underlying linear system is presented. The model reduction strategy is applied to a large-scale (O.10 6 / degrees of freedom) three-dimensional contaminant transport problem in an urban environment, an application that requires real-time computation. In addition to the inversion accuracy, the ability of reduced models of varying dimension to make predictions of the contaminant evolution beyond the time horizon of observations is studied. Results indicate that the reduced models have a factor O.1000/ speedup in computing time for the same level of accuracy.