Download or read book Sparse Variational Bayesian Algorithms for Large scale Inverse Problems with Applications in Biomechanics written by Isabell Maria Franck and published by . This book was released on 2017 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Bayesian Inverse Problems written by Juan Chiachio-Ruano and published by CRC Press. This book was released on 2021-11-10 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to a special class of engineering problems called Bayesian inverse problems. These problems comprise not only the probabilistic Bayesian formulation of engineering problems, but also the associated stochastic simulation methods needed to solve them. Through this book, the reader will learn how this class of methods can be useful to rigorously address a range of engineering problems where empirical data and fundamental knowledge come into play. The book is written for a non-expert audience and it is contributed to by many of the most renowned academic experts in this field.
Download or read book Optimal Experimental Design for Large scale Bayesian Inverse Problems written by Keyi Wu (Ph. D.) and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bayesian optimal experimental design (BOED)—including active learning, Bayesian optimization, and sensor placement—provides a probabilistic framework to maximize the expected information gain (EIG) or mutual information (MI) for uncertain parameters or quantities of interest with limited experimental data. However, evaluating the EIG remains prohibitive for largescale complex models due to the need to compute double integrals with respect to both the parameter and data distributions. In this work, we develop a fast and scalable computational framework to solve Bayesian optimal experimental design (OED) problems governed by partial differential equations (PDEs) with application to optimal sensor placement by maximizing the EIG. We (1) exploit the low-rank structure of the Jacobian of the parameter-to-observable map to extract the intrinsic low-dimensional data-informed subspace, and (2) employ a series of approximations of the EIG that reduce the number of PDE solves while retaining a high correlation with the true EIG. This allows us to propose an efficient offline–online decomposition for the optimization problem, using a new swapping greedy algorithm for both OED problems and goal-oriented linear OED problems. The offline stage dominates the cost and entails precomputing all components requiring PDE solusion. The online stage optimizes sensor placement and does not require any PDE solves. We provide a detailed error analysis with an upper bound for the approximation error in evaluating the EIG for OED and goal-oriented OED linear cases. Finally, we evaluate the EIG with a derivative-informed projected neural network (DIPNet) surrogate for parameter-to-observable maps. With this surrogate, no further PDE solves are required to solve the optimization problem. We provided an analysis of the error propagated from the DIPNet approximation to the approximation of the normalization constant and the EIG under suitable assumptions. We demonstrate the efficiency and scalability of the proposed methods for both linear inverse problems, in which one seeks to infer the initial condition for an advection–diffusion equation, and nonlinear inverse problems, in which one seeks to infer coefficients for a Poisson problem, an acoustic Helmholtz problem and an advection–diffusion–reaction problem. This dissertation is based on the following articles: A fast and scalable computational framework for large-scale and high-dimensional Bayesian optimal experimental design by Keyi Wu, Peng Chen, and Omar Ghattas [88]; An efficient method for goal-oriented linear Bayesian optimal experimental design: Application to optimal sensor placement by Keyi Wu, Peng Chen, and Omar Ghattas [89]; and Derivative-informed projected neural network for large-scale Bayesian optimal experimental design by Keyi Wu, Thomas O’Leary-Roseberry, Peng Chen, and Omar Ghattas [90]. This material is based upon work partially funded by DOE ASCR DE-SC0019303 and DESC0021239, DOD MURI FA9550-21-1-0084, and NSF DMS-2012453
Download or read book A Computational Framework for the Solution of Infinite dimensional Bayesian Statistical Inverse Problems with Application to Global Seismic Inversion written by James Robert Martin (Ph. D.) and published by . This book was released on 2015 with total page 810 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantifying uncertainties in large-scale forward and inverse PDE simulations has emerged as a central challenge facing the field of computational science and engineering. The promise of modeling and simulation for prediction, design, and control cannot be fully realized unless uncertainties in models are rigorously quantified, since this uncertainty can potentially overwhelm the computed result. While statistical inverse problems can be solved today for smaller models with a handful of uncertain parameters, this task is computationally intractable using contemporary algorithms for complex systems characterized by large-scale simulations and high-dimensional parameter spaces. In this dissertation, I address issues regarding the theoretical formulation, numerical approximation, and algorithms for solution of infinite-dimensional Bayesian statistical inverse problems, and apply the entire framework to a problem in global seismic wave propagation. Classical (deterministic) approaches to solving inverse problems attempt to recover the "best-fit" parameters that match given observation data, as measured in a particular metric. In the statistical inverse problem, we go one step further to return not only a point estimate of the best medium properties, but also a complete statistical description of the uncertain parameters. The result is a posterior probability distribution that describes our state of knowledge after learning from the available data, and provides a complete description of parameter uncertainty. In this dissertation, a computational framework for such problems is described that wraps around the existing forward solvers, as long as they are appropriately equipped, for a given physical problem. Then a collection of tools, insights and numerical methods may be applied to solve the problem, and interrogate the resulting posterior distribution, which describes our final state of knowledge. We demonstrate the framework with numerical examples, including inference of a heterogeneous compressional wavespeed field for a problem in global seismic wave propagation with 106 parameters.
Download or read book Large Scale Variational Bayesian Inference with Applications to Image Deblurring written by Brian Jonathan Verbaken and published by . This book was released on 2011 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Data driven Reduction Strategies for Bayesian Inverse Problems written by Ellen Brooke Le and published by . This book was released on 2018 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: A persistent central challenge in computational science and engineering (CSE), with both national and global security implications, is the efficient solution of large-scale Bayesian inverse problems. These problems range from estimating material parameters in subsurface simulations to estimating phenomenological parameters in climate models. Despite recent progress, our ability to quantify uncertainties and solve large-scale inverse problems lags well behind our ability to develop the governing forward simulations. Inverse problems present unique computational challenges that are only magnified as we include larger observational data sets and demand higher-resolution parameter estimates. Even with the current state-of-the-art, solving deterministic large-scale inverse problems is prohibitively expensive. Large-scale uncertainty quantification (UQ), cast in the Bayesian inversion framework, is thus rendered intractable. To conquer these challenges, new methods that target the root causes of computational complexity are needed. In this dissertation, we propose data driven strategies for overcoming this "curse of di- mensionality." First, we address the computational complexity induced in large-scale inverse problems by high-dimensional observational data. We propose a randomized misfit approach (RMA), which uses random projections--quasi-orthogonal, information-preserving transformations--to map the high-dimensional data-misfit vector to a low dimensional space. We provide the first theoretical explanation for why randomized misfit methods are successful in practice with a small reduced data-misfit dimension (n = O(1)). Next, we develop the randomized geostatistical approach (RGA) for Bayesian sub- surface inverse problems with high-dimensional data. We show that the RGA is able to resolve transient groundwater inverse problems with noisy observed data dimensions up to 107, whereas a comparison method fails due to out-of-memory errors. Finally, we address the solution of Bayesian inverse problems with spatially localized data. The motivation is CSE applications that would gain from high-fidelity estimation over a smaller data-local domain, versus expensive and uncertain estimation over the full simulation domain. We propose several truncated domain inversion methods using domain decomposition theory to build model-informed artificial boundary conditions. Numerical investigations of MAP estimation and sampling demonstrate improved fidelity and fewer partial differential equation (PDE) solves with our truncated methods.
Download or read book Bayesian Non linear Statistical Inverse Problems written by Richard Nickl and published by European Mathematical Society. This book was released on 2023-06-15 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bayesian methods based on Gaussian process priors are frequently used in statistical inverse problems arising with partial differential equations (PDEs). They can be implemented by Markov chain Monte Carlo (MCMC) algorithms. The underlying statistical models are naturally high- or infinite-dimensional, and this book presents a rigorous mathematical analysis of the statistical performance, and algorithmic complexity, of such methods in a natural setting of non-linear random design regression. Due to the non-linearity present in many of these inverse problems, natural least squares functionals are non-convex, and the Bayesian paradigm presents an attractive alternative to optimization-based approaches. This book develops a general theory of Bayesian inference for non-linear forward maps and rigorously considers two PDE model examples arising with Darcy's problem and a Schrödinger equation. The focus is initially on statistical consistency of Gaussian process methods and then moves on to study local fluctuations and approximations of posterior distributions by Gaussian or log-concave measures whose curvature is described by PDE mapping properties of underlying “information operators”. Applications to the algorithmic runtime of gradient-based MCMC methods are discussed, as well as computation time lower bounds for worst case performance of some algorithms.
Download or read book Finding Beauty in the Dissonance written by Bamdad Hosseini and published by . This book was released on 2017 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inverse problems - the process of recovering unknown parameters from indirect measurements - are encountered in various areas of science, technology and engineering including image processing, medical imaging, geosciences, astronomy, aeronautics engineering and machine learning. Statistical and probabilistic methods are promising approaches to solving such problems. Of these, the Bayesian methods provide a principled approach to incorporating our existing beliefs about the parameters (the prior model) and randomness in the data. These approaches are at the forefront of extensive current investigation. Overwhelmingly, Gaussian prior models are used in Bayesian inverse problems since they provide mathematically simple and computationally efficient formulations of important inverse problems. Unfortunately, these priors fail to capture a range of important properties including sparsity and natural constraints such as positivity, and so we are motivated to study non-Gaussian priors. In this thesis we provide a systematic study of the theory and applications of Bayesian approaches to inverse problems with non-Gaussian priors. We develop the theory of well-posedness of infinite-dimensional Bayesian inverse problems with convex, heavy-tailed or infinitely divisible prior measures. We also introduce new prior measures that aim to model compressible or sparse parameters. Next, we demonstrate the applications of Bayesian approaches to important inverse problems in industrial applications: the estimation of emission rates of particulate matter, and the estimation of acoustic aberrations in ultrasound treatment. We propose two Bayesian approaches for the problem of estimating the emission rates of particulate matter into the atmosphere from far field measurements of deposition. Next, we present a Bayesian method for estimation of acoustic aberrations in high intensity focused ultrasound treatment of tissue in the brain using magnetic resonance images. The final contribution of this thesis is a systematic construction and convergence analysis of regularizations of the Dirac delta distribution. Point sources arise naturally in many models and we discuss smooth regularizations of these.
Download or read book Bayesian Uncertainty Quantification for Large Scale Spatial Inverse Problems written by Anirban Mondal and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We considered a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a high dimension spatial field. The Bayesian approach contains a natural mechanism for regularization in the form of prior information, can incorporate information from heterogeneous sources and provides a quantitative assessment of uncertainty in the inverse solution. The Bayesian setting casts the inverse solution as a posterior probability distribution over the model parameters. Karhunen-Lo'eve expansion and Discrete Cosine transform were used for dimension reduction of the random spatial field. Furthermore, we used a hierarchical Bayes model to inject multiscale data in the modeling framework. In this Bayesian framework, we have shown that this inverse problem is well-posed by proving that the posterior measure is Lipschitz continuous with respect to the data in total variation norm. The need for multiple evaluations of the forward model on a high dimension spatial field (e.g. in the context of MCMC) together with the high dimensionality of the posterior, results in many computation challenges. We developed two-stage reversible jump MCMC method which has the ability to screen the bad proposals in the first inexpensive stage. Channelized spatial fields were represented by facies boundaries and variogram-based spatial fields within each facies. Using level-set based approach, the shape of the channel boundaries was updated with dynamic data using a Bayesian hierarchical model where the number of points representing the channel boundaries is assumed to be unknown. Statistical emulators on a large scale spatial field were introduced to avoid the expensive likelihood calculation, which contains the forward simulator, at each iteration of the MCMC step. To build the emulator, the original spatial field was represented by a low dimensional parameterization using Discrete Cosine Transform (DCT), then the Bayesian approach to multivariate adaptive regression spline (BMARS) was used to emulate the simulator. Various numerical results were presented by analyzing simulated as well as real data.
Download or read book Bayesian Inference for Inverse Problems written by Ali Mohammad-Djafari and published by . This book was released on 2018 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inverse problems arise everywhere we have indirect measurement. Regularization and Bayesian inference methods are two main approaches to handle inverse problems. Bayesian inference approach is more general and has much more tools for developing efficient methods for difficult problems. In this chapter, first, an overview of the Bayesian parameter estimation is presented, then we see the extension for inverse problems. The main difficulty is the great dimension of unknown quantity and the appropriate choice of the prior law. The second main difficulty is the computational aspects. Different approximate Bayesian computations and in particular the variational Bayesian approximation (VBA) methods are explained in details.
Download or read book Material Parameter Identification and Inverse Problems in Soft Tissue Biomechanics written by Stéphane Avril and published by Springer. This book was released on 2016-10-12 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: The articles in this book review hybrid experimental-computational methods applied to soft tissues which have been developed by worldwide specialists in the field. People developing computational models of soft tissues and organs will find solutions for calibrating the material parameters of their models; people performing tests on soft tissues will learn what to extract from the data and how to use these data for their models and people worried about the complexity of the biomechanical behavior of soft tissues will find relevant approaches to address this complexity.
Download or read book Bayesian Approach to Inverse Problems written by Jérôme Idier and published by John Wiley & Sons. This book was released on 2013-03-01 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many scientific, medical or engineering problems raise the issue of recovering some physical quantities from indirect measurements; for instance, detecting or quantifying flaws or cracks within a material from acoustic or electromagnetic measurements at its surface is an essential problem of non-destructive evaluation. The concept of inverse problems precisely originates from the idea of inverting the laws of physics to recover a quantity of interest from measurable data. Unfortunately, most inverse problems are ill-posed, which means that precise and stable solutions are not easy to devise. Regularization is the key concept to solve inverse problems. The goal of this book is to deal with inverse problems and regularized solutions using the Bayesian statistical tools, with a particular view to signal and image estimation. The first three chapters bring the theoretical notions that make it possible to cast inverse problems within a mathematical framework. The next three chapters address the fundamental inverse problem of deconvolution in a comprehensive manner. Chapters 7 and 8 deal with advanced statistical questions linked to image estimation. In the last five chapters, the main tools introduced in the previous chapters are put into a practical context in important applicative areas, such as astronomy or medical imaging.
Download or read book Nonlinear Programming written by Lorenz T. Biegler and published by SIAM. This book was released on 2010-01-01 with total page 411 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book addresses modern nonlinear programming (NLP) concepts and algorithms, especially as they apply to challenging applications in chemical process engineering. The author provides a firm grounding in fundamental NLP properties and algorithms, and relates them to real-world problem classes in process optimization, thus making the material understandable and useful to chemical engineers and experts in mathematical optimization.
Download or read book Handbook of Approximate Bayesian Computation written by Scott A. Sisson and published by CRC Press. This book was released on 2018-09-03 with total page 679 pages. Available in PDF, EPUB and Kindle. Book excerpt: As the world becomes increasingly complex, so do the statistical models required to analyse the challenging problems ahead. For the very first time in a single volume, the Handbook of Approximate Bayesian Computation (ABC) presents an extensive overview of the theory, practice and application of ABC methods. These simple, but powerful statistical techniques, take Bayesian statistics beyond the need to specify overly simplified models, to the setting where the model is defined only as a process that generates data. This process can be arbitrarily complex, to the point where standard Bayesian techniques based on working with tractable likelihood functions would not be viable. ABC methods finesse the problem of model complexity within the Bayesian framework by exploiting modern computational power, thereby permitting approximate Bayesian analyses of models that would otherwise be impossible to implement. The Handbook of ABC provides illuminating insight into the world of Bayesian modelling for intractable models for both experts and newcomers alike. It is an essential reference book for anyone interested in learning about and implementing ABC techniques to analyse complex models in the modern world.
Download or read book Active Inference written by Thomas Parr and published by MIT Press. This book was released on 2022-03-29 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first comprehensive treatment of active inference, an integrative perspective on brain, cognition, and behavior used across multiple disciplines. Active inference is a way of understanding sentient behavior—a theory that characterizes perception, planning, and action in terms of probabilistic inference. Developed by theoretical neuroscientist Karl Friston over years of groundbreaking research, active inference provides an integrated perspective on brain, cognition, and behavior that is increasingly used across multiple disciplines including neuroscience, psychology, and philosophy. Active inference puts the action into perception. This book offers the first comprehensive treatment of active inference, covering theory, applications, and cognitive domains. Active inference is a “first principles” approach to understanding behavior and the brain, framed in terms of a single imperative to minimize free energy. The book emphasizes the implications of the free energy principle for understanding how the brain works. It first introduces active inference both conceptually and formally, contextualizing it within current theories of cognition. It then provides specific examples of computational models that use active inference to explain such cognitive phenomena as perception, attention, memory, and planning.
Download or read book FastSLAM written by Michael Montemerlo and published by Springer. This book was released on 2007-04-27 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph describes a new family of algorithms for the simultaneous localization and mapping (SLAM) problem in robotics, called FastSLAM. The FastSLAM-type algorithms have enabled robots to acquire maps of unprecedented size and accuracy, in a number of robot application domains and have been successfully applied in different dynamic environments, including a solution to the problem of people tracking.
Download or read book The Mathematics and Mechanics of Biological Growth written by Alain Goriely and published by Springer. This book was released on 2017-05-29 with total page 651 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents a general mathematical theory for biological growth. It provides both a conceptual and a technical foundation for the understanding and analysis of problems arising in biology and physiology. The theory and methods are illustrated on a wide range of examples and applications. A process of extreme complexity, growth plays a fundamental role in many biological processes and is considered to be the hallmark of life itself. Its description has been one of the fundamental problems of life sciences, but until recently, it has not attracted much attention from mathematicians, physicists, and engineers. The author herein presents the first major technical monograph on the problem of growth since D’Arcy Wentworth Thompson’s 1917 book On Growth and Form. The emphasis of the book is on the proper mathematical formulation of growth kinematics and mechanics. Accordingly, the discussion proceeds in order of complexity and the book is divided into five parts. First, a general introduction on the problem of growth from a historical perspective is given. Then, basic concepts are introduced within the context of growth in filamentary structures. These ideas are then generalized to surfaces and membranes and eventually to the general case of volumetric growth. The book concludes with a discussion of open problems and outstanding challenges. Thoughtfully written and richly illustrated to be accessible to readers of varying interests and background, the text will appeal to life scientists, biophysicists, biomedical engineers, and applied mathematicians alike.
