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We present an efficient numerical strategy for the Bayesian solution of inverse problems. Stochastic collocation methods, based on generalized polynomial chaos (gPC), are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. This approximation then defines a surrogate posterior probability densityâ€¦ (More)

Bayesian inference provides a probabilistic framework for combining prior knowledge with mathematical models and observational data. Characterizing a Bayesian posterior probability distribution can be a computationally challenging undertaking, however, particularly when evaluations of the posterior density are expensive and when the posterior has complexâ€¦ (More)

Solution of statistical inverse problems via the frequentist or Bayesian approaches described in earlier chapters can be a computationally intensive endeavor, particularly when faced with large-scale forward models characteristic of many engineering and science applications. High computational cost arises in several ways. First, thousands or millions ofâ€¦ (More)

- Benjamin Peherstorfera, Tiangang Cui, Youssef Marzouk, Karen Willcox
- 2015

Estimating statistics of model outputs with the Monte Carlo method often requires a large number of model evaluations. This leads to long runtimes if the model is expensive to evaluate. Importance sampling is one approach that can lead to a reduction in the number of model evaluations. Importance sampling uses a biasing distribution to sample the model moreâ€¦ (More)

- Tsui-Wei Weng, Zheng Zhang, Zhan Su, Youssef Marzouk, Andrea Melloni, Luca Daniel
- Optics express
- 2015

Process variations can significantly degrade device performance and chip yield in silicon photonics. In order to reduce the design and production costs, it is highly desirable to predict the statistical behavior of a device before the final fabrication. Monte Carlo is the mainstream computational technique used to estimate the uncertainties caused byâ€¦ (More)

The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intensive forward model to produce the equivalent of oneâ€¦ (More)

Cities today are strained by the exponential growth in population where they are homes to the majority of world's population. Understanding the complexities underlying the emerging behaviors of human travel patterns on the city level is essential toward making informed decision-making pertaining to urban transportation infrastructures This thesis includesâ€¦ (More)

We present an algorithm to identify sparse dependence structure in continuous and non-Gaussian probability distributions, given a corresponding set of data. The conditional independence structure of an arbitrary distribution can be represented as an undirected graph (or Markov random field), but most algorithms for learning this structure are restricted toâ€¦ (More)

We propose optimal dimensionality reduction techniques for the solution of goalâ€“oriented linearâ€“Gaussian inverse problems, where the quantity of interest (QoI) is a function of the inversion parameters. These approximations are suitable for large-scale applications. In particular, we study the approximation of the posterior covariance of the QoI as aâ€¦ (More)

Design of complex engineering systems requires coupled analyses of the multiple disciplines affecting system performance. The coupling among disciplines typically contributes significantly to the computational cost of analyzing the system, and can become particularly burdensome when coupled analyses are embedded within a design or optimization loop. In manyâ€¦ (More)