A MEASURE-THEORETIC COMPUTATIONAL METHOD FOR INVERSE SENSITIVITY PROBLEMS I: METHOD AND ANALYSIS

Novel and flexible parameter estimation methods for data-consistent inversion in mechanistic modelling

Predictions for physical systems often rely upon knowledge acquired from ensembles of entities, e.g. ensembles of cells in biological sciences. For qualitative and quantitative analysis, these ensembles are simulated with parametric families of mechanistic models (MMs). Two classes of methodologies, based on Bayesian inference and population of models, currently prevail in parameter estimation for physical systems. However, in Bayesian analysis, uninformative priors for MM parameters introduce undesirable bias. Here, we propose how to infer parameters within the framework of stochastic inverse problems (SIPs), also termed data-consistent inversion, wherein the prior targets only uncertainties that arise due to MM non-invertibility. To demonstrate, we introduce new methods to solve SIPs based on rejection sampling, Markov chain Monte Carlo, and generative adversarial networks (GANs). In addition, to overcome limitations of SIPs, we reformulate SIPs based on constrained optimization and present a novel GAN to solve the constrained optimization problem.

Definition and solution of a stochastic inverse problem for the Manning's n parameter field in hydrodynamic models

The uncertainty in spatially heterogeneous Manning's n fields is quantified using a novel formulation and numerical solution of stochastic inverse problems for physics-based models. The uncertainty is quantified in terms of a probability measure and the physics-based model considered here is the state-of-the-art ADCIRC model although the presented methodology applies to other hydrodynamic models. An accessible overview of the formulation and solution of the stochastic inverse problem in a mathematically rigorous framework based on measure theory is presented. Technical details that arise in practice by applying the framework to determine the Manning's n parameter field in a shallow water equation model used for coastal hydrodynamics are presented and an efficient computational algorithm and open source software package are developed. A new notion of "condition" for the stochastic inverse problem is defined and analyzed as it relates to the computation of probabilities. This notion of condition is investigated to determine effective output quantities of interest of maximum water elevations to use for the inverse problem for the Manning's n parameter and the effect on model predictions is analyzed.

A numerical method for solving a stochastic inverse problem for parameters

We review recent work (Briedt et al., 2011., 2012) on a new approach to the formulation and solution of the stochastic inverse parameter determination problem, i.e. determine the random variation of input parameters to a map that matches specified random variation in the output of the map, and then apply the various aspects of this method to the interesting Brusselator model. In this approach, the problem is formulated as an inverse problem for an integral equation using the Law of Total Probability. The solution method employs two steps: (1) we construct a systematic method for approximating set-valued inverse solutions and (2) we construct a computational approach to compute a measure-theoretic approximation of the probability measure on the input space imparted by the approximate set-valued inverse that solves the inverse problem. In addition to convergence analysis, we carry out an a posteriori error analysis on the computed probability distribution that takes into account all sources of stochastic and deterministic error.

A COMPUTATIONAL MEASURE THEORETIC APPROACH TO INVERSE SENSITIVITY PROBLEMS II: A POSTERIORI ERROR ANALYSIS

In part one of this paper [T. Butler and D. Estep, SIAM J. Numer. Anal., to appear], we develop and analyze a numerical method to solve a probabilistic inverse sensitivity analysis problem for a smooth deterministic map assuming that the map can be evaluated exactly. In this paper, we treat the situation in which the output of the map is determined implicitly and is difficult and/or expensive to evaluate, e.g., requiring the solution of a differential equation, and hence the output of the map is approximated numerically. The main goal is an a posteriori error estimate that can be used to evaluate the accuracy of the computed distribution solving the inverse problem, taking into account all sources of statistical and numerical deterministic errors. We present a general analysis for the method and then apply the analysis to the case of a map determined by the solution of an initial value problem.

A MEASURE-THEORETIC COMPUTATIONAL METHOD FOR INVERSE SENSITIVITY PROBLEMS I: METHOD AND ANALYSIS

We consider the inverse sensitivity analysis problem of quantifying the uncertainty of inputs to a deterministic map given specified uncertainty in a linear functional of the output of the map. This is a version of the model calibration or parameter estimation problem for a deterministic map. We assume that the uncertainty in the quantity of interest is represented by a random variable with a given distribution, and we use the law of total probability to express the inverse problem for the corresponding probability measure on the input space. Assuming that the map from the input space to the quantity of interest is smooth, we solve the generally ill-posed inverse problem by using the implicit function theorem to derive a method for approximating the set-valued inverse that provides an approximate quotient space representation of the input space. We then derive an efficient computational approach to compute a measure theoretic approximation of the probability measure on the input space imparted by the approximate set-valued inverse that solves the inverse problem.