Bayesian inference with optimal maps

We present a new approach to Bayesian inference that entirely avoids Markov chain simulation, by constructing a map that pushes forward the prior measure to the posterior measure. Existence and uniqueness of a suitable measure-preserving map is established by formulating the problem in the context of optimal transport theory. We discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem, exploiting gradient information from the forward model when possible. The resulting algorithm overcomes many of the computational bottlenecks associated with Markov chain Monte Carlo. Advantages of a map-based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves. The optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood. We demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension, involving the inference of parameters appearing in ordinary and partial differential equations.United States. Dept. of Energy. Office of Advanced Scientific Computing Research (Grant DE-SC0002517)United States. Dept. of Energy. Office of Advanced Scientific Computing Research (Grant DE-SC0003908

Calculation of Generalized Polynomial-Chaos Basis Functions and Gauss Quadrature Rules in Hierarchical Uncertainty Quantification

Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a set of orthonormal polynomials and a proper numerical quadrature rule. The former are used as the basis functions in a generalized polynomial chaos expansion. The latter is used to compute the integrals involved in stochastic spectral methods. Obtaining such information requires knowing the density function of the random input {\it a-priori}. However, individual system components are often described by surrogate models rather than density functions. In order to apply stochastic spectral methods in hierarchical uncertainty quantification, we first propose to construct physically consistent closed-form density functions by two monotone interpolation schemes. Then, by exploiting the special forms of the obtained density functions, we determine the generalized polynomial-chaos basis functions and the Gauss quadrature rules that are required by a stochastic spectral simulator. The effectiveness of our proposed algorithm is verified by both synthetic and practical circuit examples.Comment: Published by IEEE Trans CAD in May 201

Fluorescence Spectrometric Determination of Drugs Containing α-Methylene Sulfone/Sulfonamide Functional Groups Using N1-Methylnicotinamide Chloride as a Fluorogenic Agent

A simple spectrofluorometric method has been developed, adapted, and validated for the quantitative estimation of drugs containing α-methylene sulfone/sulfonamide functional groups using N1-methylnicotinamide chloride (NMNCl) as fluorogenic agent. The proposed method has been applied successfully to the determination of methyl sulfonyl methane (MSM) (1), tinidazole (2), rofecoxib (3), and nimesulide (4) in pure forms, laboratory-prepared mixtures, pharmaceutical dosage forms, spiked human plasma samples, and in volunteer's blood. The method showed linearity over concentration ranging from 1 to 150 μg/mL, 10 to 1000 ng/mL, 1 to 1800 ng/mL, and 30 to 2100 ng/mL for standard solutions of 1, 2, 3, and 4, respectively, and over concentration ranging from 5 to 150 μg/mL, 10 to 1000 ng/mL, 10 to 1700 ng/mL, and 30 to 2350 ng/mL in spiked human plasma samples of 1, 2, 3, and 4, respectively. The method showed good accuracy, specificity, and precision in both laboratory-prepared mixtures and in spiked human plasma samples. The proposed method is simple, does not need sophisticated instruments, and is suitable for quality control application, bioavailability, and bioequivalency studies. Besides, its detection limits are comparable to other sophisticated chromatographic methods

Inverse Problems in a Bayesian Setting

In a Bayesian setting, inverse problems and uncertainty quantification (UQ) --- the propagation of uncertainty through a computational (forward) model --- are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. We give a detailed account of this approach via conditional approximation, various approximations, and the construction of filters. Together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update in form of a filter is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and nonlinear Bayesian update in form of a filter on some examples.Comment: arXiv admin note: substantial text overlap with arXiv:1312.504

A hierarchical floating random walk algorithm for fabric-aware 3d capacitance extraction,” in Computer-Aided Design,

ABSTRACT With the adoption of ultra regular fabric paradigms for controlling design printability at the 22nm node and beyond, there is an emerging need for a layout-driven, pattern-based parasitic extraction of alternative fabric layouts. In this paper, we propose a hierarchical floating random walk (HFRW) algorithm for computing the 3D capacitances of a large number of topologically different layout configurations that are all composed of the same layout motifs. Our algorithm is not a standard hierarchical domain decomposition extension of the well established floating random walk technique, but rather a novel algorithm that employs Markov Transition Matrices. Specifically, unlike the fast-multipole boundary element method and hierarchical domain decomposition (which use a far-field approximation to gain computational efficiency), our proposed algorithm is exact and does not rely on any tradeoff between accuracy and computational efficiency. Instead, it relies on a tradeoff between memory and computational efficiency. Since floating random walk type of algorithms have generally minimal memory requirements, such a tradeoff does not result in any practical limitations. The main practical advantage of the proposed algorithm is its ability to handle a set of layout configurations in a complexity that is basically independent of the set size. For instance, in a large 3D layout example, the capacitance calculation of 120 different configurations made of similar motifs is accomplished in the time required to solve independently just 2 configurations, i.e. a 60× speedup