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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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DOE OSTI2023

Enabling Hyper-Differential Sensitivity Analysis for Ill-Posed Inverse Problems

Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model that must be estimated. However, high dimensionality of the parameters and computational complexity of the PDE solves make such problems challenging. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to estimate the other parameters. In this article, hyper-differential sensitivity analysis (HDSA) is used to assess the sensitivity of the solution of the PDE-constrained optimization problem to changes in the auxiliary parameters. Foundational assumptions for HDSA require satisfaction of the optimality conditions which are not always practically feasible as a result of ill-posedness in the inverse problem. Here we introduce novel theoretical and computational approaches to justify and enable HDSA for ill-posed inverse problems by projecting the sensitivities on likelihood informed subspaces and defining a posteriori updates. Our proposed framework is demonstrated on a nonlinear multiphysics inverse problem motivated by estimation of spatially heterogeneous material properties in the presence of spatially distributed parametric modeling uncertainties.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A Physics-Driven Deep-Learning Network for Solving Nonlinear Inverse Problems

Geosteering inversion, which can be viewed as a nonlinear inverse problem, is an important technique used by directional drilling. Traditional methods that rely on iterative procedures and regularization are sensitive to the selection of initial values and can be slow due to convergence issues. In industrial applications, a lookup table is used to produce fast predictions. However, performance is not guaranteed by this approach due to the limitation of the hardware. In this paper, we propose a novel physics-driven deep-learning framework for providing a fast, accurate surrogate to solve the inverse problem. Particularly, leveraged by the forward physical model and 1D convolutional neural network (1D-CNN), the proposed method provides more reliable solutions to the inverse problem with improved performance. In addition, a new physics-driven loss function is introduced to accommodate both the model misfit and the data misfit. In conclusion, our experiments demonstrate the effectiveness of our method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems

Polynomial chaos expansions are used to reduce the computational cost in the Bayesian solutions of inverse problems by creating a surrogate posterior that can be evaluated inexpensively. We show, by analysis and example, that when the data contain significant information beyond what is assumed in the prior, the surrogate posterior can be very different from the posterior, and the resulting estimates become inaccurate. One can improve the accuracy by adaptively increasing the order of the polynomial chaos, but the cost may increase too fast for this to be cost effective compared to Monte Carlo sampling without a surrogate posterior.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

On the Ill-Posed Nature of Inverse Problems: An Exactly Solvable Paradigm Inverse Neutron Diffusion Problem Illustrating the Solution's Non-Computability - Paper 48

In general, a physical system and/or the result of an indirect experimental measurement are modeled mathematically in terms of: (a) A system of linear and/or nonlinear equations that relate the system's independent variables and parameters to the system's state (i.e., dependent) variables; (b) Probability distributions, moments thereof, inequality and/or equality constraints that define the range of variations of the system's parameters; (c) One or several quantities, customarily referred to as system responses (or objective functions, or indices of performance), which are computed using the mathematical model. The 'direct problem' solves the 'parameter-to-output' mapping that describes the 'cause-to-effect' relationship in the respective physical process. In practice, the model parameters and/or responses are experimentally measured quantities and are therefore subject to uncertainties. Many measurement problems are 'inverse' to the direct problem in that they seek to determine the properties of the medium (i.e., various cross sections), or the properties of the source, and/or the size of the medium on its boundaries from measurements of quantities that depend on the neutron flux (or from direct measurements of the neutron flux). In general, two problems are called inverses of one another if the formulation of each involves all or part of the solution of the other. Some authors further group such inverse problems into 'invasive', when the interior flux is accessible for measurements as opposed to 'non-invasive', in which only fluxes on the boundary of (or exterior to) the medium can be measured. Examples of inverse radiative transfer problems have been reviewed by McCormick, while examples of inverse source problems for neutron transport have been provided by Sanchez and McCormick. More recently, Bledsoe et al., have presented two methods (the Levenberg-Marquard, and 'differential evolution method') for addressing numerically inverse transport problems. However, although these works (as well as other works cited therein) mentioned the detrimental effects of 'small perturbations in the input data', none of the works published thus far on inverse problems for particle transport or diffusion actually showed explicitly the effects that uncertainties in the measured responses (e.g., detector measurements) would actually have on the 'inverse' deduction (from such measurements) of the underlying medium's properties, sources, etc. This work highlights explicitly the nature of the difficulties encountered in so-called 'inverse problems' by presenting a paradigm inverse neutron diffusion problem admitting an exact solution, in which the 'inverse problem' is to determine the medium's diffusion coefficient from flux measurements. It will be shown explicitly (analytically) that this is not possible to achieve in a unique manner, even in the absence of measurement uncertainties, unless additional error-free information is provided about the quantity to be determined (in this case, the medium's diffusion coefficient). It if further shown that, even if this additional error-free information about the diffusion coefficient to be determined is available, the unavoidable experimental measurements errors, albeit it vanishingly small, in the measured flux render the 'exact solution' useless for computational purposes, since the exact solution turns out to be a divergent series. As discussed in this paper's ENDNOTES section, the paradigm problem presented in this work also underscores the need for 'regularization', which is a qualifier indicating that the inverse problem is to be solved in some approximate manner. (authors)

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2024

Analysis of a Computational Framework for Bayesian Inverse Problems: Ensemble Kalman Updates and MAP Estimators under Mesh Refinement

This paper analyzes a popular computational framework to solve infinite-dimensional Bayesian inverse problems, discretizing the prior and the forward model in a finite-dimensional weighted inner product space. We demonstrate the benefit of working on a weighted space by establishing operator-norm bounds for finite element and graph-based discretizations of Matérn-type priors and deconvolution forward models. For linear-Gaussian inverse problems, we develop a general theory to characterize the error in the approximation to the posterior. We also embed the computational framework into ensemble Kalman methods and MAP estimators for nonlinear inverse problems. Furthermore, our operator-norm bounds for prior discretizations guarantee the scalability and accuracy of these algorithms under mesh refinement.

Bayesian inverse problem↗
DOE OSTI2024

Sequential Kalman tuning of the t -preconditioned Crank-Nicolson algorithm: efficient, adaptive and gradient-free inference for Bayesian inverse problems

Ensemble Kalman Inversion (EKI) has been proposed as an efficient method for the approximate solution of Bayesian inverse problems with expensive forward models. However, when applied to the Bayesian inverse problem EKI is only exact in the regime of Gaussian target measures and linear forward models. Here, in this work we propose embedding EKI and Flow Annealed Kalman Inversion, its normalizing flow (NF) preconditioned variant, within a Bayesian annealing scheme as part of an adaptive implementation of the t-preconditioned Crank-Nicolson (tpCN) sampler. The tpCN sampler differs from standard pCN in that its proposal is reversible with respect to the multivariate t-distribution. The more flexible tail behaviour allows for better adaptation to sampling from non-Gaussian targets. Within our Sequential Kalman Tuning (SKT) adaptation scheme, EKI is used to initialize and precondition the tpCN sampler for each annealed target. The subsequent tpCN iterations ensure particles are correctly distributed according to each annealed target, avoiding the accumulation of errors that would otherwise impact EKI. We demonstrate the performance of SKT for tpCN on three challenging numerical benchmarks, showing significant improvements in the rate of convergence compared to adaptation within standard SMC with importance weighted resampling at each temperature level, and compared to similar adaptive implementations of standard pCN. The SKT scheme applied to tpCN offers an efficient, practical solution for solving the Bayesian inverse problem when gradients of the forward model are not available. Code implementing the SKT schemes for tpCN is available at https://github.com/RichardGrumitt/KalmanMC.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Aspects of the inverse problem for the Toda chain

We generalize Babelon's approach to equations in dual variables so as to be able to treat new types of operators which we build out of the sub-constituents of the model's monodromy matrix. Further, we also apply Sklyanin's recent monodromy matrix identities so as to obtain equations in dual variables for yet other operators. The schemes discussed in this paper appear to be universal and thus, in principle, applicable to many models solvable through the quantum separation of variables.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2019

Transient and Steady-State Inverse Problems in Sierra/Aria

Inverse problems arise in a wide range of applications, whenever unknown model parameters cannot be measured directly. Instead, the unknown parameters are estimated using experimental data and forward simulations. Thermal inverse problems, such as material characterization problems, are often large-scale and transient. Therefore, they require intrusive adjoint-based gradient implementations in order to be solved efficiently. The capability to solve large-scale transient thermal inverse problems using an adjoint-based approach was recently implemented in SNL Sierra Mechanics, a massively parallel capable multiphysics code suite. This report outlines the theory, optimization formulation, and path taken to implement thermal inverse capabilities in Sierra within a unit test framework. The capability utilizes Sierra/Aria and Sierra/Fuego data structures, the Rapid Optimization Library, and an interface to the Sierra/InverseOpt library. The existing Sierra/Aria time integrator is leveraged to implement a time-dependent adjoint solver.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems

Deep learning (DL), in particular deep neural networks, by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are often not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates for missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, hence promoting better generalization. This paper develops a model-constrained DL approach and its variant TNet—a Tikhonov neural network—which are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solutions with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for one-dimensional (1D) deconvolution, 5 for an inverse 2D heat conductivity problem, 100 for inverse initial conditions for a time-dependent 2D Burgers’s equation, and 50 for inverse initial conditions for 2D Navier–Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. Furthermore, this is possible owing to the model-constrained term, replications, and randomization.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Bi-fidelity Gradient-Based Approach for Nonlinear Well Logging Inverse Problems

Solving a non-linear inverse problem is challenging in computational science and engineering. Sampling based methods require a large number of model valuations; gradientbased methods require fewer model evaluations but only find the local minima. Multifidelity optimization combines the low fidelity model and the high-fidelity model to achieve both high accuracy and high efficiency. In this paper, we present a bi-fidelity approach to solve non-linear inverse problems. In the bi-fidelity inversion method, the low-fidelity model is used to acquire a good initial guess, and the high-fidelity model is used to locate the global minimum. Combined with a multi-start optimization scheme, the proposed approach significantly increases the possibility of finding the global minimum for nonlinear inverse problems with many local minima. The method is tested with two toy problems and then applied to an electromagnetic well logging inverse problem, which is difficult to solve using traditional gradient-based methods. The bi-fidelity method provides promising inversion results and can be easily applied to traditional gradient-based methods.

42 ENGINEERING↗
DOE OSTI2025

Robust A-Optimal Experimental Design for Sensor Placement in Bayesian Linear Inverse Problems

Optimal design of experiments for Bayesian inverse problems has recently gained wide popularity and attracted much attention, especially in the computational science and Bayesian inversion communities. An optimal design maximizes a predefined utility function that is formulated in terms of the elements of an inverse problem, an example being optimal sensor placement for parameter identification. The state-of-the-art algorithmic approaches following this simple formulation generally overlook misspecification of the elements of the inverse problem, such as the prior or the measurement uncertainties. This work presents an efficient algorithmic approach for designing optimal experimental design schemes for Bayesian linear inverse problems such that the optimal design is robust to misspecification of elements of the inverse problem. Specifically, we consider a worst-case scenario approach for the uncertain or misspecified parameters, formulate robust objectives, and propose an algorithmic approach for optimizing such objectives. Furthermore, both relaxation and stochastic solution approaches are discussed with detailed analysis and insight into the interpretation of the problem and the proposed algorithmic approach. Extensive numerical experiments to validate and analyze the proposed approach are carried out for sensor placement in a parameter identification problem.

Bayesian inverse problems↗
DOE OSTI2018

A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs

Joint inversion refers to the simultaneous inference of multiple parameter fields from observations of systems governed by single or multiple forward models. In many cases these parameter fields reflect different attributes of a single medium and are thus spatially correlated or structurally similar. By imposing prior information on their spatial correlations via a joint regularization term, we seek to improve the reconstruction of the parameter fields relative to inversion for each field independently. One of the main challenges is to devise a joint regularization functional that conveys the spatial correlations or structural similarity between the fields while at the same time permitting scalable and efficient solvers for the joint inverse problem. We describe several joint regularizations that are motivated by these goals: a cross-gradient and a normalized cross-gradient structural similarity term, the vectorial total variation, and a joint regularization based on the nuclear norm of the gradients. Based on numerical results from three classes of inverse problems with piecewise-homogeneous parameter fields, we conclude that the vectorial total variation functional is preferable to the other methods considered. Besides resulting in good reconstructions in all experiments, it allows for scalable, efficient solvers for joint inverse problems governed by PDE forward models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

SAGIPS: a physics-inspired scalable asynchronous generative inverse-problem solver

Abstract Solving large-scale inverse problems using deep-learning algorithms have become an essential part of modern research and industrial applications. The complexity of the underlying inverse problem may require the utilization of high performance computing systems which poses a challenge on the algorithmic design of the inverse problem solver. Most deep learning algorithms require, due to their design, custom parallelization techniques in order to be resource efficient while showing a reasonable convergence. In this paper we introduce a S calable A synchronous G enerative I nverse P roblem S olver (SAGIPS) on high-performance computing systems. We present a workflow that utilizes an asynchronous ring-allreduce algorithm to transfer the gradients of the generator network across multiple GPUs. Experiments with a scientific proxy application demonstrate that SAGIPS shows near linear weak scaling, together with a convergence quality that is comparable to traditional methods. The approach presented here allows leveraging Generative Adverserial Network across multiple GPUs, promising advancements in solving complex inverse problems at scale.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Numerical methods for the inverse problem of density functional theory

Here, the inverse problem of Kohn–Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to one-dimensional finite and periodic model systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Robust Optimal Experimental Design of Infinite-Dimensional Bayesian Nonlinear Inverse Problems

Abstract. We consider robust optimal experimental design (ROED) for nonlinear Bayesian inverse problems governed by partial differential equations (PDEs). An optimal design is one that maximizes some utility quantifying the quality of the solution of an inverse problem. However, the optimal design is dependent on elements of the inverse problem such as the simulation model, the prior, or the measurement error model. ROED aims to produce an optimal design that is aware of the additional uncertainties encoded in the inverse problem and remains optimal even after variations in them. We follow a worst-case scenario approach to develop a new framework for robust optimal design of nonlinear Bayesian inverse problems. The proposed framework (a) is scalable and designed for infinite-dimensional Bayesian nonlinear inverse problems constrained by PDEs; (b) develops efficient approximations of the utility, namely the expected information gain; (c) employs eigenvalue sensitivity techniques to develop analytical forms and efficient evaluation methods of the gradient of the utility with respect to the uncertainties against which we wish to be robust; and (d) employs a probabilistic optimization paradigm that properly defines and efficiently solves the resulting combinatorial max-min optimization problem. The effectiveness of the proposed approach is illustrated for optimal sensor placement problem in an inverse problem governed by an elliptic PDE.

Chowdhary, Abhijit↗