Sonjoy Das

Asymptotic Sampling Distribution for Polynomial Chaos Representation from Data: A Maximum Entropy and Fisher Information Approach

A procedure is presented for characterizing the asymptotic sampling distribution of the estimators of the polynomial chaos (PC) coefficients of physical process modeled as non-stationary, non-Gaussian second-order random process by using a collection of observations. These observations made over a denumerable subset of the indexing set of the process are considered to form a set of realizations of a random vector, y, representing a finite-dimensional model of the random process. The estimators of the PC coefficients of y are next deduced by relying on its reduced order representation obtained by employing Karhunen-Loeve decomposition and subsequent use of the maximum-entropy principle, Metropolis-Hastings Markov chain Monte Carlo algorithm and the Rosenblatt transformation. These estimators are found to be maximum likelihood estimators as well as consistent and asymptotically efficient estimators. The computation of the covariance matrix of the associated asymptotic normal distribution of the estimators of these PC coefficients requires evaluation of Fisher information matrix that is evaluated analytically and also estimated by using a sampling technique for the accompanied numerical illustration

A Bounded Random Matrix Approach for Stochastic Upscaling

A maximum entropy (MaxEnt) based probabilistic approach is developed to model mechanical systems characterized by symmetric positive-definite matrices bounded from below and above. These matrices are typically encountered in the constitutive modeling of heterogeneous materials, where the bounds are deduced by employing the principles of minimum complementary energy and minimum potential energy. Current random matrix approaches in the context of computational stochastic mechanics are adapted only to the Wishart or matrix-variate Gamma probability model supported over the entire space of symmetric positive-definite matrices and therefore unable to exploit additional information available through the lower and upper bounds when appropriate. Specifically, for a given material, the constitutive matrix is construed as a random matrix. A probability measure that reflects the constraints consistent with the energy-based bounds is constructed, and an associated sampling scheme is described to synthesize realizatio...

Polynomial chaos representation of spatio-temporal random fields from experimental measurements

Two numerical techniques are proposed to construct a polynomial chaos (PC) representation of an arbitrary second-order random vector. In the first approach, a PC representation is constructed by matching a target joint probability density function (pdf) based on sequential conditioning (a sequence of conditional probability relations) in conjunction with the Rosenblatt transformation. In the second approach, the PC representation is obtained by having recourse to the Rosenblatt transformation and simultaneously matching a set of target marginal pdfs and target Spearmans rank correlation coefficient (SRCC) matrix. Both techniques are applied to model an experimental spatio-temporal data set, exhibiting strong non-stationary and non-Gaussian features. The data consists of a set of oceanographic temperature records obtained from a shallow-water acoustics transmission experiment [1]. The measurement data, observed over a finite denumerable subset of the indexing set of the random process, is treated as a collection of observed samples of a second-order random vector that can be treated as a finite-dimensional approximation of the original random field. A set of properly ordered conditional pdfs, that uniquely characterizes the target joint pdf, in the first approach and a set of target marginal pdfs and a target SRCC matrix, in the second approach, are estimated from available experimental data. Digital realizations sampled from the constructed PC representations based on both schemes capture the observed statistical characteristics of the experimental data with sufficient accuracy. The relative advantages and disadvantages of the two proposed techniques are also highlighted.

Hybrid Representations of Coupled Nonparametric and Parametric Models for Dynamic Systems

Parametric modeling of stochastic systems has proven useful for systems with well-defined and well-structured sources of uncertainty. The suitability of such models is usually indicated by small levels of uncertainty associated with their parameters. The parametric model may not be efficiently employed to deal with problems associated with a high level of uncertainty, particularly due to the modeling uncertainty. The class of so-called nonparametric stochastic models has recently been introduced to address this specific issue and found to be useful to some extent. This paper presents a coupling technique adapted to the receptance frequency-response-function matrix that will be useful for analyzing a complex dynamic system, particularly when it consists of several stochastic subsystems, each of which is individually deemed to be suitable for either a parametric model or a nonparametric model. Such a complex dynamic system is otherwise difficult to analyze. The existing nonparametric approach was, to date, applied to the real-valued positive-definite/semidefinite random system matrix: for example, mass, damping, and stiffness matrices. In the present work, the nonparametric approach is also employed with the complex-valued symmetric receptance frequency-response-function matrix, now acting as the system matrix, by having recourse to Takagis factorization.

Asymptotic Sampling Distribution for Polynomial Chaos Representation of Data: A Maximum Entropy and Fisher information approach

A procedure is presented for characterizing the asymptotic sampling distribution of the estimators of the polynomial chaos (PC) coefficients of physical process modeled as non-stationary, non-Gaussian second-order random process by using a collection of observations. These observations made over a denumerable subset of the indexing set of the process are considered to form a set of realizations of a random vector, Y, representing a finite-dimensional model of the random process. The estimators of the PC coefficients of Y are next deduced by relying on its reduced order representation obtained by employing Karhunen-Loeve decomposition and subsequent use of the maximum-entropy principle, Metropolis-Hastings Markov chain Monte carlo algorithm and the Rosenblatt transformation. These estimators are found to be maximum likelihood estimators as well as consistent and asymptotically efficient estimators. The computation of the covariance matrix of the associated asymptotic normal distribution of the estimators of these PC coefficients requires evaluation of Fisher information matrix that is evaluated analytically and also estimated by using a sampling technique for the accompanied numerical illustration.

An efficient calculation of Fisher information matrix: Monte Carlo approach using prior information

The Fisher information matrix (FIM) is a critical quantity in several aspects of system identification, including input selection and confidence region calculation. Analytical determination of the FIM in a general system identification setting may be difficult or almost impossible due to intractable modeling requirements and/or high-dimensional integration. A Monte Carlo (MC) simulation-based technique was introduced by the second author to address these difficulties (Spall, 2005). This paper proposes an extension of the MC algorithm in order to enhance the statistical qualities of the estimator of the FIM. This modified MC algorithm is particularly useful in those cases where the FIM has a structure with some elements being analytically known from prior information and the others being unknown. The estimator of the FIM, obtained by using the proposed MC algorithm, simultaneously preserves the analytically known elements and reduces the variances of the estimators of the unknown elements by capitalizing on the information contained in the known elements.

Stochastic Upscaling for Inelastic Material Behavior from Limited Experimental Data

A stochastic upscaling approach based on random matrix theory has recently been proposed to characterize a coarse scale (continuum) constitutive elasticity matrix of heterogeneous materials (Das, “Model, identification & analysis of complex stochastic systems: Applications in stochastic partial differential equations and multiscale mechanics”, PhD thesis, University of Southern California, Los Angeles, USA, 2008; Das and Ghanem, “SIAM Multiscale Modeling and Simulation”, 8(1):296–325, 2009). The approach, originally developed for linear elastic behavior, is adapted in the present work to characterize nonlinear elastic and strain-hardening plastic behavior. This is achieved by assuming that the heterogeneous material is locally elastic at any given point on the nonlinear constitutive stress–strain curve. The associated constitutive tangential elasticity matrix or tangential elastoplastic matrix is treated as a random matrix that evolves with the effective strain which depends on the current strain state of the material. The uncertainty characterized by such constitutive tangential matrix can be construed as a reflection, on the coarse scales, of fluctuations of the fine scale features from which the constitutive matrices are constructed. Under certain conditions, such constitutive matrix can be shown to be symmetric, positive-definite and bounded from below and above, in the positive-definite sense, by two symmetric and positive-definite matrices. The condition of boundedness follows from the applications of the principles of minimum complementary energy and minimum potential energy. The lower and upper matrix-valued bounds can be obtained fairly accurately from a limited amount of fine scale experimental data. They are typically computed by carrying out micromechanical analyses on a small volume element of the heterogeneous material, the size of which depends on the particular application of interest. A probability measure of the random matrix that reflects the constraints consistent with these energy-based bounds is constructed, and a sampling scheme is developed to synthesize realizations of the random matrix.

Efficient Monte Carlo computation of Fisher information matrix using prior information

The Fisher information matrix (FIM) is a critical quantity in several aspects of mathematical modeling, including input selection, model selection, and confidence region calculation. For example, the determinant of the FIM is the main performance metric for choosing input values in a scientific experiment with the aims of achieving the most accurate resulting parameter estimates in a mathematical model. However, analytical determination of the FIM in a general setting, especially in nonlinear models, may be difficult or almost impossible due to intractable modeling requirements and/or intractable high-dimensional integration.n To circumvent these difficulties, a Monte Carlo (MC) simulation-based technique, resampling algorithm, based on the values of log-likelihood function or its exact stochastic gradient computed by using a set of pseudo data vectors, is usually recommended. This paper proposes an extension of the current algorithm in order to enhance the statistical characteristics of the estimator of the FIM. This modified algorithm is particularly useful in those cases where the FIM has a structure with some elements being analytically known from prior information and the others being unknown. The estimator of the FIM, obtained by using the proposed algorithm, simultaneously preserves the analytically known elements and reduces the variances of the estimators of the unknown elements by capitalizing on the information contained in the known elements.

Sensor configuration and optimization for detection of micro-anomalies in structural materials

The current work presents a sensor placement strategy to detect microanomalies (microcracks), that are not discernible by naked eyes, in structural materials. Such microlevel defects can be potentially dangerous for structural and mechanical systems due to fatigue cyclic loading that results in initiation of fatigue cracks. Analysis of this precursory state of internal damage evolution, before a macrocrack visibly appears, is beyond the scope of the conventional crack propagation analysis, e.g., fracture mechanics. The present work utilizes a stochastic upscaling scheme that has recently been developed to characterize continuum constitutive material properties which have the capability to capture signatures of microcracks at 30–70 μm level for aluminum (other materials can be similarly characterized). Relying on this novel feature, the present work proposes an optimal sensor placement scheme that, in a certain sense, maximizes the probability of detection of microcracks. The developed algorithm is also likely to be useful in other areas of threat detection.