Parameter and State Model Reduction for Large-Scale Statistical Inverse Problems

Abstract

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 *

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

“…Another example is large-scale statistical inverse problems for which Markov chain Monte Carlo methods are computationally intractable, either because they require excessive amounts of CPU time or because the parameter space is too large to be explored effectively by state-of-the-art sampling methods. In these cases, parametric model reduction over both state and parameter spaces can make tractable the solution of large-scale inverse problems that otherwise cannot be solved [80,102,156,220,72].…”

“…In [54], exponential convergence rates of greedy sampling are shown in a reduced basis framework, both for strong greedy (the optimal next sampling point is always found) as well as in computationally feasible variants. In [156], the optimization-based greedy sampling approach was extended to construct both a basis for the state and a basis for the parameter, leading to models that have both reduced state and reduced parameters. This approach was demonstrated on a subsurface model with a distributed parameter representing the hydraulic conductivity over the domain, and shown to scale up to the case of a discretized parameter vector of dimension d = 494.…”

A reduced computational and geometrical framework for inverse problems in hemodynamics

“…in order to compute sample statistics such as expectations, variances, and higher moments). Here we do not treat in detail statistical inverse problems and the various approaches that have been proposed in what is a vast field of applied statistics, but limit ourselves to mention that (i) reduction techniques prove to be mandatory also within a statistical approach (as detailed in [20,22,39]) and that (ii) the reduced basis framework is suitable also for uncertainty quantification [26] and more general probabilistic problems [8,46]. In this paper we identify two approaches to deal with uncertainty.…”

Computational Reduction for Parametrized PDEs: Strategies and Applications

“…Greedy algorithms have been applied in several contexts, involving also other reduction issues; some recent applications deal e.g. with a simultaneous parameter and state reduction, in problems requiring the exploration of high-dimensional parameter spaces [6,29].…”