Quantifying Uncertainties in Inverse Problems: Meaning and Usefulness of Error Bars on Information Extracted from Stockpile Stewardship Experiments
A presentation on quantifying uncertainties in inverse problems.
97 MATHEMATICS AND COMPUTING↗
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Solvability of Parametric Inverse Problem for Parabolic Equation
We establish the solvability of a parametric inverse problem for an n-dimensional parabolic equation. We obtain sufficient conditions for the existence and uniqueness of a solution.
97 MATHEMATICS AND COMPUTING↗
Solving inverse problems in stochastic models using deep neural networks and adversarial training
Inverse problems associated with stochastic models constitute a significant portion of scientific and engineering applications. In such cases the unknown quantities are distributions. The applicability of traditional methods is limited because of their demanding assumptions or prohibitive computational consumption; for example, maximum likelihood methods require closed-form density functions, and Markov Chain Monte Carlo needs a large number of simulations. We propose a new method that estimates the unknown distribution by matching the statistical properties between observed and simulated random processes. We leverage the expressive power of neural networks to approximate the unknown distribution and use a discriminative neural network for computing the statistical discrepancies between the observed and simulated random processes. Here we demonstrated numerically that the proposed methods can estimate both the model parameters and learn complicated unknown distributions.
42 ENGINEERING↗
Combining Push-Forward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems
We formulate, and present a numerical method for solving, an inverse problem for inferring parameters of a deterministic model from stochastic observational data on quantities of interest. The solution, given as a probability measure, is derived using a Bayesian updating approach for measurable maps that finds a posterior probability measure, that when propagated through the deterministic model produces a push-forward measure that exactly matches the observed probability measure on the data. Our approach for finding such posterior measures, which we call consistent Bayesian inference or push-forward based inference, is simple and only requires the computation of the push-forward probability measure induced by the combination of a prior probability measure and the deterministic model. We establish existence and uniqueness of observation-consistent posteriors and present both stability and error analyses. We also discuss the relationships between consistent Bayesian inference, classical/statistical Bayesian inference, and a recently developed measure-theoretic approach for inference. Finally, analytical and numerical results are presented to highlight certain properties of the consistent Bayesian approach and the differences between this approach and the two aforementioned alternatives for inference.
97 MATHEMATICS AND COMPUTING↗
Expectation propagation for nonlinear inverse problems – with an application to electrical impedance tomography
In this paper, we study a fast approximate inference method based on expectation propagation for exploring the posterior probability distribution arising from the Bayesian formulation of nonlinear inverse problems. It is capable of efficiently delivering reliable estimates of the posterior mean and covariance, thereby providing an inverse solution together with quantified uncertainties. Some theoretical properties of the iterative algorithm are discussed, and the efficient implementation for an important class of problems of projection type is described. The method is illustrated with one typical nonlinear inverse problem, electrical impedance tomography with complete electrode model, under sparsity constraints. Numerical results for real experimental data are presented, and compared with that by Markov chain Monte Carlo. The results indicate that the method is accurate and computationally very efficient.
97 MATHEMATICS AND COMPUTING↗
Data-Consistent Solutions to Stochastic Inverse Problems Using a Probabilistic Multi-Fidelity Method Based on Conditional Densities
In this work, we build upon a recently developed approach for solving stochastic inverse problems based on a combination of measure-theoretic principles and Bayes' rule. We propose a multi-fidelity method to reduce the computational burden of performing uncertainty quantification using high-fidelity models. This approach is based on a Monte Carlo framework for uncertainty quantification that combines information from solvers of various fidelities to obtain statistics on the quantities of interest of the problem. In particular, our goal is to generate samples from a high-fidelity push-forward density at a fraction of the costs of standard Monte Carlo methods, while maintaining flexibility in the number of random model input parameters. Key to this methodology is the construction of a regression model to represent the stochastic mapping between the low- and high-fidelity models, such that most of the computations can be leveraged to the low-fidelity model. To that end, we employ Gaussian process regression and present extensions to multi-level-type hierarchies as well as to the case of multiple quantities of interest. Finally, we demonstrate the feasibility of the framework in several numerical examples.
97 MATHEMATICS AND COMPUTING↗
A Nonlocal Inverse Problem for the Two-Dimensional Heat-Conduction Equation
We consider an inverse problem of determination of the time-dependent leading coefficient of a two-dimensional heat-conduction equation with nonlocal overdetermination condition. The existence and uniqueness conditions are established for the classical solution of the analyzed problem.
97 MATHEMATICS AND COMPUTING↗
Approaches to the Inverse Problem
In this talk, I describe some recent ideas relating to the spectral reconstruction inverse problem, which arises frequently in lattice QCD calculations of inclusive hadronic quantities, and provide some physical context for this work. Particular emphasis is given to a new method for rigorously bounding uncertainties using techniques from complex analysis.
Jay, William [Massachusetts Institute of Technolog↗
Impurity gas monitoring using ultrasonic sensing and neural networks: forward and inverse problems
Ultrasonic sensing is a non-invasive technique for monitoring impurity gas composition in various industrial applications where safety and regulatory compliance are crucial. In this study, ultrasonic sensing and neural networks were used to analyze impurity gases (i.e., air and argon) in helium. An experimental platform was established to acquire ultrasonic data. In the forward problem, an artificial neural network (ANN) model was used to forecast the response and time-of-flight (TOF) based on the excitation, and argon and air concentrations. The inverse problem was solved using a convolutional neural network (CNN) to predict the argon and air concentrations given the ultrasonic response and excitation. The results showed that the ANN accurately predicted the ultrasonic response and the change in TOF with concentration. As the air concentration was increased from 0 to 9.8%, the TOF sensitivity to detect argon decreased by 39.8% and 16.1% from ANN and sound speed theory, respectively. The CNN demonstrated high accuracy in predicting concentrations for inputs in the testing dataset. The application of the trained CNN indicated that it over-predicts air concentration while under-predicting the argon concentration. To improve accuracy, the predicted air and argon concentrations should be corrected by -0.992% and 1.027% bias, respectively.
47 OTHER INSTRUMENTATION↗
A Plug-and-Play Priors Approach for Solving Nonlinear Imaging Inverse Problems
In the past two decades, nonlinear image reconstruction methods have led to substantial improvements in the capabilities of numerous imaging systems. Such methods are traditionally formulated as optimization problems that are solved iteratively by simultaneously enforcing data consistency and incorporating prior models. Recently, the Plug-and-Play Priors (PPP) framework suggested that by using more sophisticated denoisers, not necessarily corresponding to an optimization objective, it is possible to improve the quality of reconstructed images. Here in this letter, we show that the PPP approach is applicable beyond linear inverse problems. In particular, we develop the fast iterative shrinkage/thresholding algorithm variant of PPP for model-based nonlinear inverse scattering. The key advantage of the proposed formulation over the original ADMM-based one is that it does not need to perform an inversion on the forward model. We show that the proposed method produces high quality images using both simulated and experimentally measured data.
97 MATHEMATICS AND COMPUTING↗
Incremental projection approach of regularization for inverse problems
This paper presents an alternative approach to the regularized least squares solution of ill-posed inverse problems. Instead of solving a minimization problem with an objective function composed of a data term and a regularization term, the regularization information is used to define a projection onto a convex subspace of regularized candidate solutions. The objective function is modified to include the projection of each iterate in the place of the regularization. Numerical experiments based on the problem of motion estimation for geophysical fluid images, show the improvement of the proposed method compared with regularization methods. For the presented test case, the incremental projection method uses 7 times less computation time than the regularization method, to reach the same error target. Moreover, at convergence, the incremental projection is two order of magnitude more accurate than the regularization method.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Algorithms for Forward and Inverse Problem Solutions in Phase Contrast Tomography
PhaseTorch is a python package that implements algorithms for forward and inverse problem solutions in X-ray phase contrast tomography (XPCT). XPCT is useful to reconstruct the 3D distribution of the refractive index decrement and absorption index of an imaged object. PhaseTorch implements algorithms for simulation, reconstruction, and analysis of XPCT data. PhaseTorch provides algorithms for simulation of XPCT data of an object with known refractive and absorption indices. It contains algorithms for reconstructing the projections of the absorption index and refractive index decrement.
Sridhar, Venkatesh↗
On Approximation of Coefficient Inverse Problems for Differential Equations in Functional Spaces
This paper is devoted to the theory of approximation of coefficient inverse problems for differential equations of parabolic, elliptic, and hyperbolic types in functional spaces. We present general statements of problems and their approximations and review results obtained earlier in the literature.
97 MATHEMATICS AND COMPUTING↗
InverseBench: Inverse design benchmark suite that contains inverse problems from science and engineering (InverseBench) v0.0.1
A software package that contains three inverse design blackbox problems to investigate the efficiency and accuracy of inverse design machine learning models. The software contains highly accurate forward machine learning models that can be used to assess the inverse predictions. The package also contains separate test data for each problem. The inverse design problems that are in the package are: airfoil inverse design, scalar boundary reconstruction and photonic surfaces inverse design.
Grbcic, Luka [Lawrence Berkeley National Laborator↗
Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems
We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators: correcting the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction–diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.
97 MATHEMATICS AND COMPUTING↗
Adaptive Interface-PINNs (AdaI-PINNs) for inverse problems: Determining material properties for heterogeneous systems
Here, we determine spatially varying discontinuous material properties using a domain-decomposition based physics-informed neural networks (PINNs) framework named the Adaptive Interface-PINNs or AdaI-PINNs (Roy et al., 2024). We propose the use of distinct neural networks for the field variables and material properties within each material, utilizing adaptive activation functions. While the neural networks across different materials share the same weights and biases, their activation functions are uniquely tailored using a hyperparameter that influences the slope of the activation function. The proposed framework is tested on several one-dimensional and two-dimensional benchmark examples, and its performance is compared with conventional PINNs and existing domain-decomposition PINNs frameworks, namely, the Multi-domain physics-informed neural network (M-PINN), and the eXtended physics-informed neural networks (XPINNs). The results demonstrate that the proposed approach can determine randomly distributed discontinuous material properties with an L 2 error of $\mathscr{O}$ (10 -3 ) for the material property and the root-mean-square error of $\mathscr{O}$ (10 -3 ) for the primary variable while the other approaches yield errors that are approximately two orders of magnitude larger (that is, $\mathscr{O}$ (10 -1 )). Moreover, the spatial distribution of material properties obtained using the proposed framework is in close agreement with the true distribution, whereas the other approaches fare much worse. Additionally, the proposed approach is approximately 40% faster than its competitors, indicating its potential as a robust alternative for solving inverse problems in heterogeneous materials.
36 MATERIALS SCIENCE↗
Inverse problem for extragalactic transport of ultra-high energy cosmic rays
The energy spectra and composition of ultra-high energy cosmic rays are changing in a course of propagation in the expanding Universe filled with background radiation. We developed a numerical code for solution of inverse problem for cosmic-ray transport equations that allows the determination of average source spectra of different nuclei from the cosmic ray spectra observed at the Earth. Employing this approach, the injection spectra of protons and Iron nuclei in extragalactic sources are found assuming that only these species are accelerated at the source. The data from the Auger experiment and the combined data from the Telescope Array + HiRes experiments are used to illustrate the method.
79 ASTRONOMY AND ASTROPHYSICS↗
Bound Constrained Partial DifferentialEquation Inverse Problem Solution by theSemi-Smooth Newton Method
We present the mathematical derivation, software implementation details, and computational results for a semi-smooth Newton method applied to two inverse problems governed by partial differential equations with bound constraints. The two problems share mathematical structural similarities to density-based topology optimization problems. The semi-smooth Newton method provides a mesh independent solution computation for the two test problems. A key step is that the complementarity part of the necessary optimality conditions are reformulated with the use of a complementarity functionφsuch that the complementarity conditions are satisfied if and only if a zero of a nonsmooth function has been obtained. The modular finite element package MFEM is utilized for the software implementation. In addition we constructed a matrix-free Operator to enable the use of efficient Krylov subspace IterativeSolver of MFEM for the solution of our two target problems.
97 MATHEMATICS AND COMPUTING↗