Journal ArticleDOI
Monte Carlo analysis of inverse problems
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TLDR
In this paper, Monte Carlo sampling is used for nonlinear inverse problems where no analytical expression for the forward relation between data and model parameters is available, and where linearization is unsuccessful.Abstract:
Monte Carlo methods have become important in analysis of nonlinear inverse problems where no analytical expression for the forward relation between data and model parameters is available, and where linearization is unsuccessful. In such cases a direct mathematical treatment is impossible, but the forward relation materializes itself as an algorithm allowing data to be calculated for any given model. Monte Carlo methods can be divided into two categories: the sampling methods and the optimization methods. Monte Carlo sampling is useful when the space of feasible solutions is to be explored, and measures of resolution and uncertainty of solution are needed. The Metropolis algorithm and the Gibbs sampler are the most widely used Monte Carlo samplers for this purpose, but these methods can be refined and supplemented in various ways of which the neighbourhood algorithm is a notable example. Monte Carlo optimization methods are powerful tools when searching for globally optimal solutions amongst numerous local optima. Simulated annealing and genetic algorithms have shown their strength in this respect, but they suffer from the same fundamental problem as the Monte Carlo sampling methods: no provably optimal strategy for tuning these methods to a given problem has been found, only a number of approximate methods.read more
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Book
Parameter estimation and inverse problems
TL;DR: "Parameter Estimation and Inverse Problems, 2/e" introduces readers to both Classical and Bayesian approaches to linear and nonlinear problems with particular attention paid to computational, mathematical, and statistical issues related to their application to geophysical problems.
Journal ArticleDOI
Recent advances in diffuse optical imaging.
TL;DR: The current state-of-the-art of diffuse optical imaging is reviewed, which is an emerging technique for functional imaging of biological tissue and recent work on in vivo applications including imaging the breast and brain is reviewed.
Journal ArticleDOI
Monte carlo methods in geophysical inverse problems
TL;DR: The development and application of Monte Carlo methods for inverse problems in the Earth sciences and in particular geophysics are traced from the earliest work of the Russian school and the pioneering studies in the west by Press [1968] to modern importance sampling and ensemble inference methods.
Journal ArticleDOI
Stochastic spectral methods for efficient Bayesian solution of inverse problems
TL;DR: This work presents a reformulation of the Bayesian approach to inverse problems, that seeks to accelerate Bayesian inference by using polynomial chaos expansions to represent random variables, and evaluates the utility of this technique on a transient diffusion problem arising in contaminant source inversion.
Journal ArticleDOI
Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification
Rohit Tripathy,Ilias Bilionis +1 more
TL;DR: Deep neural networks (DNN) are used to construct surrogate models for numerical simulators in a manner that lends the DNN surrogate the interpretation of recovering a low-dimensional nonlinear manifold.
References
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Book
Genetic algorithms in search, optimization, and machine learning
TL;DR: In this article, the authors present the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields, including computer programming and mathematics.
Journal ArticleDOI
Optimization by Simulated Annealing
TL;DR: There is a deep and useful connection between statistical mechanics and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters), and a detailed analogy with annealing in solids provides a framework for optimization of very large and complex systems.
Journal ArticleDOI
Equation of state calculations by fast computing machines
TL;DR: In this article, a modified Monte Carlo integration over configuration space is used to investigate the properties of a two-dimensional rigid-sphere system with a set of interacting individual molecules, and the results are compared to free volume equations of state and a four-term virial coefficient expansion.
Book
Adaptation in natural and artificial systems
TL;DR: Names of founding work in the area of Adaptation and modiication, which aims to mimic biological optimization, and some (Non-GA) branches of AI.
Journal ArticleDOI
Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images
Stuart Geman,Donald Geman +1 more
TL;DR: The analogy between images and statistical mechanics systems is made and the analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations, creating a highly parallel ``relaxation'' algorithm for MAP estimation.