6,617 research outputs found

    Approximate Computational Approaches for Bayesian Sensor Placement in High Dimensions

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    Since the cost of installing and maintaining sensors is usually high, sensor locations are always strategically selected. For those aiming at inferring certain quantities of interest (QoI), it is desirable to explore the dependency between sensor measurements and QoI. One of the most popular metric for the dependency is mutual information which naturally measures how much information about one variable can be obtained given the other. However, computing mutual information is always challenging, and the result is unreliable in high dimension. In this paper, we propose an approach to find an approximate lower bound of mutual information and compute it in a lower dimension. Then, sensors are placed where highest mutual information (lower bound) is achieved and QoI is inferred via Bayes rule given sensor measurements. In addition, Bayesian optimization is introduced to provide a continuous mutual information surface over the domain and thus reduce the number of evaluations. A chemical release accident is simulated where multiple sensors are placed to locate the source of the release. The result shows that the proposed approach is both effective and efficient in inferring QoI

    Optimal Experimental Design for Infinite-dimensional Bayesian Inverse Problems Governed by PDEs: A Review

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    We present a review of methods for optimal experimental design (OED) for Bayesian inverse problems governed by partial differential equations with infinite-dimensional parameters. The focus is on problems where one seeks to optimize the placement of measurement points, at which data are collected, such that the uncertainty in the estimated parameters is minimized. We present the mathematical foundations of OED in this context and survey the computational methods for the class of OED problems under study. We also outline some directions for future research in this area.Comment: 37 pages; minor revisions; added more references; article accepted for publication in Inverse Problem

    Efficient D-optimal design of experiments for infinite-dimensional Bayesian linear inverse problems

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    We develop a computational framework for D-optimal experimental design for PDE-based Bayesian linear inverse problems with infinite-dimensional parameters. We follow a formulation of the experimental design problem that remains valid in the infinite-dimensional limit. The optimal design is obtained by solving an optimization problem that involves repeated evaluation of the log-determinant of high-dimensional operators along with their derivatives. Forming and manipulating these operators is computationally prohibitive for large-scale problems. Our methods exploit the low-rank structure in the inverse problem in three different ways, yielding efficient algorithms. Our main approach is to use randomized estimators for computing the D-optimal criterion, its derivative, as well as the Kullback--Leibler divergence from posterior to prior. Two other alternatives are proposed based on a low-rank approximation of the prior-preconditioned data misfit Hessian, and a fixed low-rank approximation of the prior-preconditioned forward operator. Detailed error analysis is provided for each of the methods, and their effectiveness is demonstrated on a model sensor placement problem for initial state reconstruction in a time-dependent advection-diffusion equation in two space dimensions.Comment: 27 pages, 9 figure

    A Fast and Scalable Method for A-Optimal Design of Experiments for Infinite-dimensional Bayesian Nonlinear Inverse Problems

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    We address the problem of optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by PDEs. The goal is to find a placement of sensors, at which experimental data are collected, so as to minimize the uncertainty in the inferred parameter field. We formulate the OED objective function by generalizing the classical A-optimal experimental design criterion using the expected value of the trace of the posterior covariance. We seek a method that solves the OED problem at a cost (measured in the number of forward PDE solves) that is independent of both the parameter and sensor dimensions. To facilitate this, we construct a Gaussian approximation to the posterior at the maximum a posteriori probability (MAP) point, and use the resulting covariance operator to define the OED objective function. We use randomized trace estimation to compute the trace of this (implicitly defined) covariance operator. The resulting OED problem includes as constraints the PDEs characterizing the MAP point, and the PDEs describing the action of the covariance operator to vectors. The sparsity of the sensor configurations is controlled using sparsifying penalty functions. We elaborate our OED method for the problem of determining the sensor placement to best infer the coefficient of an elliptic PDE. Adjoint methods are used to compute the gradient of the PDE-constrained OED objective function. We provide numerical results for inference of the permeability field in a porous medium flow problem, and demonstrate that the number of PDE solves required for the evaluation of the OED objective function and its gradient is essentially independent of both the parameter and sensor dimensions. The number of quasi-Newton iterations for computing an OED also exhibits the same dimension invariance properties.Comment: 30 pages; minor revisions; accepted for publication in SIAM Journal on Scientific Computin

    Optimal Experimental Design Using A Consistent Bayesian Approach

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    We consider the utilization of a computational model to guide the optimal acquisition of experimental data to inform the stochastic description of model input parameters. Our formulation is based on the recently developed consistent Bayesian approach for solving stochastic inverse problems which seeks a posterior probability density that is consistent with the model and the data in the sense that the push-forward of the posterior (through the computational model) matches the observed density on the observations almost everywhere. Given a set a potential observations, our optimal experimental design (OED) seeks the observation, or set of observations, that maximizes the expected information gain from the prior probability density on the model parameters. We discuss the characterization of the space of observed densities and a computationally efficient approach for rescaling observed densities to satisfy the fundamental assumptions of the consistent Bayesian approach. Numerical results are presented to compare our approach with existing OED methodologies using the classical/statistical Bayesian approach and to demonstrate our OED on a set of representative PDE-based models

    A-optimal design of experiments for infinite-dimensional Bayesian linear inverse problems with regularized β„“0\ell_0-sparsification

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    We present an efficient method for computing A-optimal experimental designs for infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs). Specifically, we address the problem of optimizing the location of sensors (at which observational data are collected) to minimize the uncertainty in the parameters estimated by solving the inverse problem, where the uncertainty is expressed by the trace of the posterior covariance. Computing optimal experimental designs (OEDs) is particularly challenging for inverse problems governed by computationally expensive PDE models with infinite-dimensional (or, after discretization, high-dimensional) parameters. To alleviate the computational cost, we exploit the problem structure and build a low-rank approximation of the parameter-to-observable map, preconditioned with the square root of the prior covariance operator. This relieves our method from expensive PDE solves when evaluating the optimal experimental design objective function and its derivatives. Moreover, we employ a randomized trace estimator for efficient evaluation of the OED objective function. We control the sparsity of the sensor configuration by employing a sequence of penalty functions that successively approximate the β„“0\ell_0-"norm"; this results in binary designs that characterize optimal sensor locations. We present numerical results for inference of the initial condition from spatio-temporal observations in a time-dependent advection-diffusion problem in two and three space dimensions. We find that an optimal design can be computed at a cost, measured in number of forward PDE solves, that is independent of the parameter and sensor dimensions. We demonstrate numerically that β„“0\ell_0-sparsified experimental designs obtained via a continuation method outperform β„“1\ell_1-sparsified designs.Comment: 27 pages, accepted for publication in SIAM Journal on Scientific Computin

    Bayesian Optimal Design of Experiments For Inferring The Statistical Expectation Of A Black-Box Function

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    Bayesian optimal design of experiments (BODE) has been successful in acquiring information about a quantity of interest (QoI) which depends on a black-box function. BODE is characterized by sequentially querying the function at specific designs selected by an infill-sampling criterion. However, most current BODE methods operate in specific contexts like optimization, or learning a universal representation of the black-box function. The objective of this paper is to design a BODE for estimating the statistical expectation of a physical response surface. This QoI is omnipresent in uncertainty propagation and design under uncertainty problems. Our hypothesis is that an optimal BODE should be maximizing the expected information gain in the QoI. We represent the information gain from a hypothetical experiment as the Kullback-Liebler (KL) divergence between the prior and the posterior probability distributions of the QoI. The prior distribution of the QoI is conditioned on the observed data and the posterior distribution of the QoI is conditioned on the observed data and a hypothetical experiment. The main contribution of this paper is the derivation of a semi-analytic mathematical formula for the expected information gain about the statistical expectation of a physical response. The developed BODE is validated on synthetic functions with varying number of input-dimensions. We demonstrate the performance of the methodology on a steel wire manufacturing problem.Comment: 27 pages, 19 figure

    A fast and scalable computational framework for large-scale and high-dimensional Bayesian optimal experimental design

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    We develop a fast and scalable computational framework to solve large-scale and high-dimensional Bayesian optimal experimental design problems. In particular, we consider the problem of optimal observation sensor placement for Bayesian inference of high-dimensional parameters governed by partial differential equations (PDEs), which is formulated as an optimization problem that seeks to maximize an expected information gain (EIG). Such optimization problems are particularly challenging due to the curse of dimensionality for high-dimensional parameters and the expensive solution of large-scale PDEs. To address these challenges, we exploit two essential properties of such problems: the low-rank structure of the Jacobian of the parameter-to-observable map to extract the intrinsically low-dimensional data-informed subspace, and the high correlation of the approximate EIGs by a series of approximations to reduce the number of PDE solves. We propose an efficient offline-online decomposition for the optimization problem: an offline stage of computing all the quantities that require a limited number of PDE solves independent of parameter and data dimensions, and an online stage of optimizing sensor placement that does not require any PDE solve. For the online optimization, we propose a swapping greedy algorithm that first construct an initial set of sensors using leverage scores and then swap the chosen sensors with other candidates until certain convergence criteria are met. We demonstrate the efficiency and scalability of the proposed computational framework by a linear inverse problem of inferring the initial condition for an advection-diffusion equation, and a nonlinear inverse problem of inferring the diffusion coefficient of a log-normal diffusion equation, with both the parameter and data dimensions ranging from a few tens to a few thousands

    State observation and sensor selection for nonlinear networks

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    A large variety of dynamical systems, such as chemical and biomolecular systems, can be seen as networks of nonlinear entities. Prediction, control, and identification of such nonlinear networks require knowledge of the state of the system. However, network states are usually unknown, and only a fraction of the state variables are directly measurable. The observability problem concerns reconstructing the network state from this limited information. Here, we propose a general optimization-based approach for observing the states of nonlinear networks and for optimally selecting the observed variables. Our results reveal several fundamental limitations in network observability, such as the trade-off between the fraction of observed variables and the observation length on one side, and the estimation error on the other side. We also show that owing to the crucial role played by the dynamics, purely graph- theoretic observability approaches cannot provide conclusions about one's practical ability to estimate the states. We demonstrate the effectiveness of our methods by finding the key components in biological and combustion reaction networks from which we determine the full system state. Our results can lead to the design of novel sensing principles that can greatly advance prediction and control of the dynamics of such networks.Comment: Matches publication version to appear in IEEE Transactions on Control of Network Systems. 28 pages and 13 figure

    Optimal Sensor Positioning (OSP); A Probability Perspective Study

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    We propose a method to optimally position a sensor system, which consists of multiple sensors, each has limited range and viewing angle, and they may fail with a certain failure rate. The goal is to find the optimal locations as well as the viewing directions of all the sensors and achieve the maximal surveillance of the known environment. We setup the problem using the level set framework. Both the environment and the viewing range of the sensors are represented by level set functions. Then we solve a system of ordinary differential equations (ODEs) to find the optimal viewing directions and locations of all sensors together. Furthermore, we use the intermittent diffusion, which converts the ODEs into stochastic differential equations (SDEs), to find the global maximum of the total surveillance area. The numerical examples include various failure rates of sensors, different rate of importance of surveillance region, and 3-D setups. They show the effectiveness of the proposed method
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