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All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Algebraic Multigrid with Filtering: An Efficient Preconditioner for Interior Point Methods in Large-Scale Contact Mechanics Optimization

Large-scale contact mechanics simulations are crucial in many engineering fields such as structural design and manufacturing. In the frictionless case, contact can be modeled by minimizing an energy functional; however, these problems are often nonlinear, nonconvex, and increasingly difficult to solve as mesh resolution increases. In this work, we employ a Newton-based interior-point (IP) filter line-search method, an effective approach for large-scale constrained optimization. While this method converges rapidly, each iteration requires solving a large saddle-point linear system that becomes ill-conditioned as the optimization process converges, largely due to IP treatment of the contact constraints. Such ill-conditioning can hinder solver scalability and increase iteration counts with mesh refinement. Here, to address this, we introduce a novel preconditioner, algebraic multigrid with filtering (AMGF), tailored to the Schur complement of the saddle-point system. Building on the classical AMG solver, commonly used for elasticity, we augment it with a specialized subspace correction that filters near null space components introduced by contact interface constraints. Through theoretical analysis and numerical experiments on a range of linear and nonlinear contact problems, we demonstrate that the proposed solver achieves mesh independent convergence and maintains robustness against the ill-conditioning that notoriously plagues IP methods. These results indicate that AMGF makes contact mechanics simulations more tractable and broadens the applicability of Newton-based IP methods in challenging engineering scenarios. More broadly, AMGF is well suited for problems, optimization or otherwise, where solver performance is limited by a low-dimensional subspace, such as those arising from localized constraints, interface conditions, or model heterogeneities. This makes the method widely applicable beyond contact mechanics and constrained optimization.

Mathematics and Computing↗
DOE OSTI2022

An adaptive stochastic sequential quadratic programming with differentiable exact augmented lagrangians

In this study, we consider solving nonlinear optimization problems with a stochastic objective and deterministic equality constraints. We assume for the objective that its evaluation, gradient, and Hessian are inaccessible, while one can compute their stochastic estimates by, for example, subsampling. We propose a stochastic algorithm based on sequential quadratic programming (SQP) that uses a differentiable exact augmented Lagrangian as the merit function. To motivate our algorithm design, we first revisit and simplify an old SQP method Lucidi developed for solving deterministic problems, which serves as the skeleton of our stochastic algorithm. Based on the simplified deterministic algorithm, we then propose a non-adaptive SQP for dealing with stochastic objective, where the gradient and Hessian are replaced by stochastic estimates but the stepsizes are deterministic and prespecified. Finally, we incorporate a recent stochastic line search procedure Paquette and Scheinberg into the non-adaptive stochastic SQP to adaptively select the random stepsizes, which leads to an adaptive stochastic SQP. The global "almost sure" convergence for both non-adaptive and adaptive SQP methods is established. Numerical experiments on nonlinear problems in CUTEst test set demonstrate the superiority of the adaptive algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

gLaSDI: Parametric physics-informed greedy latent space dynamics identification

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. Here, an interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on the fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, a -nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 17 to 2,658× speed-up with 1 to 5% relative errors.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Taylor approximation variance reduction for approximation errors in PDE-constrained Bayesian inverse problems

In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e. the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e. before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.

Bayesian approximation error↗
DOE OSTI2024

Paired autoencoders for likelihood-free estimation in inverse problems

Abstract We consider the solution of nonlinear inverse problems where the forward problem is a discretization of a partial differential equation. Such problems are notoriously difficult to solve in practice and require minimizing a combination of a data-fit term and a regularization term. The main computational bottleneck of typical algorithms is the direct estimation of the data misfit. Therefore, likelihood-free approaches have become appealing alternatives. Nonetheless, difficulties in generalization and limitations in accuracy have hindered their broader utility and applicability. In this work, we use a paired autoencoder framework as a likelihood-free estimator (LFE) for inverse problems. We show that the use of such an architecture allows us to construct a solution efficiently and to overcome some known open problems when using LFEs. In particular, our framework can assess the quality of the solution and improve on it if needed. We demonstrate the viability of our approach using examples from full waveform inversion and inverse electromagnetic imaging.

Chung, Matthias (ORCID:0000000178224539)↗
DOE OSTI2022

Modeling design and control problems involving neural network surrogates

Here, we consider nonlinear optimization problems that involve surrogate models represented by neural networks. We demonstrate first how to directly embed neural network evaluation into optimization models, highlight a difficulty with this approach that can prevent convergence, and then characterize stationarity of such models. We then present two alternative formulations of these problems in the specific case of feedforward neural networks with ReLU activation: as a mixed-integer optimization problem and as a mathematical program with complementarity constraints. For the latter formulation we prove that stationarity at a point for this problem corresponds to stationarity of the embedded formulation. Each of these formulations may be solved with state-of-the-art optimization methods, and we show how to obtain good initial feasible solutions for these methods. We compare our formulations on three practical applications arising in the design and control of combustion engines, in the generation of adversarial attacks on classifier networks, and in the determination of optimal flows in an oil well network.

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 OSTI2023

Linearization errors in discrete goal-oriented error estimation

This paper is concerned with goal-oriented a posteriori error estimation for nonlinear functionals in the context of nonlinear variational problems solved with continuous Galerkin finite element discretizations. A two-level, or discrete, adjoint-based approach for error estimation is considered. The traditional method to derive an error estimate in this context requires linearizing both the nonlinear variational form and the nonlinear functional of interest which introduces linearization errors into the error estimate. In this paper, we investigate these linearization errors. In particular, we develop a novel discrete goal-oriented error estimate that accounts for traditionally neglected nonlinear terms at the expense of greater computational cost. We demonstrate how this error estimate can be used to drive mesh adaptivity. Here, we show that accounting for linearization errors in the error estimate can improve its effectivity for several nonlinear model problems and quantities of interest. We also demonstrate that an adaptive strategy based on the newly proposed estimate can lead to more accurate approximations of the nonlinear functional with fewer degrees of freedom when compared to uniform refinement and traditional adjoint-based approaches.

42 ENGINEERING↗
DOE OSTI2022

An improved framework for the dynamic likelihood filtering approach to data assimilation

Here, we propose improvements to the Dynamic Likelihood Filter (DLF), a Bayesian data assimilation filtering approach, specifically tailored to wave problems. The DLF approach was developed to address the common challenge in the application of data assimilation to hyperbolic problems in the geosciences and in engineering, where observation systems are sparse in space and time. When these observations have low uncertainties, as compared to model uncertainties, the DLF exploits the inherent nature of information and uncertainties to propagate along characteristics to produce estimates that are phase aware as well as amplitude aware, as would be the case in the traditional data assimilation approach. Along characteristics, the stochastic partial differential equations underlying the linear or nonlinear stochastic dynamics are differential equations. This study focuses on developing the explicit challenges of relating dynamics and uncertainties in the Eulerian and Lagrangian frames via dynamic Gaussian processes. It also implements the approach using the ensemble Kalman filter (EnKF) and compares the DLF approach to the conventional one with respect to wave amplitude and phase estimates in linear and nonlinear wave problems. Numerical comparisons show that the DLF/EnKF outperforms the EnKF estimates, when applied to linear and nonlinear wave problems. This advantage is particularly noticeable when sparse, low uncertainty observations are used.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Asynchronous domain decomposition methods for nonlinear PDEs

One- and two-level parallel asynchronous methods for the numerical solution of nonlinear systems of equations, especially those arising from (nonlinear) partial differential equations, are studied. The proposed methods are based on domain decomposition techniques. Local convergence theorems are presented in several cases, with appropriate hypotheses. Computational results on a shared memory multiprocessor machine for various problems exhibiting nonlinearities are reported, illustrating the potential of these asynchronous methods, especially for heterogeneous clusters.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

On unifying randomized methods for inverse problems

This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems with Gaussian priors by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a sample average approximation (SAA). More importantly, we are able to prove a single theoretical result that guarantees the asymptotic convergence for a variety of randomized methods. Additionally, viewing randomized methods as an SAA enables us to prove, for the first time, a single non-asymptotic error result that holds for randomized methods under consideration. Another important consequence of our unified framework is that it allows us to discover new randomization methods. Here, we present various numerical results for linear, nonlinear, algebraic, and PDE-constrained inverse problems that verify the theoretical convergence results and provide a discussion on the apparently different convergence rates and the behavior for various randomized methods.

42 ENGINEERING↗
DOE OSTI2022

Protection settings optimizer

SAND2023-06672O The Protection Settings Optimizer (PSO) uses system and fault data as inputs to formulate the problem of calculating relay settings as a mixed integer, nonlinear optimization problem (MINLP). The MINLP is solved using a genetic algorithm-based optimizer that attempts to find settings to reduce the relay operating times. The PSO protects the power system by using the steady-state fault voltages and currents, which then calculates the optimal device setting to protect the power system. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Patel, Trupal↗
DOE OSTI2023

An Incremental Gradient Method for Optimization Problems With Variational Inequality Constraints

We consider minimizing a sum of agent-specific nondifferentiable merely convex functions over the solution set of a variational inequality (VI) problem in that each agent is associated with a local monotone mapping. This problem finds an application in computation of the best equilibrium in nonlinear complementarity problems arising in transportation networks. We develop an iteratively regularized incremental gradient method where at each iteration, agents communicate over a directed cycle graph to update their solution iterates using their local information about the objective and the mapping. The proposed method is single-timescale in the sense that it does not involve any excessive hard-to-project computation per iteration. We derive nonasymptotic agent-wise convergence rates for the suboptimality of the global objective function and infeasibility of the VI constraints measured by a suitably defined dual gap function. Finally, the proposed method appears to be the first fully iterative scheme equipped with iteration complexity that can address distributed optimization problems with VI constraints over cycle graphs.

convergence↗
DOE OSTI2024

DEPLOYING FAST CHARGING INFRASTRUCTURE FOR ELECTRIC VEHICLES IN URBAN NETWORKS: AN ACTIVITY-BASED APPROACH

This paper explores an important problem under the domain of network modeling, the optimal configuration of charging infrastructure for electric vehicles (EVs) in urban networks considering EV users' daily activities and charging behavior. This study proposes a charging behavior simulation model considering different initial state of charge (SOC), travel distance, availability of home chargers, and the daily schedule of trips for each traveler. The proposed charging behavior simulation model examines the complete chain of trips for EV users as well as the interdependency of trips traveled by each driver. The problem of finding the optimum charging configuration is then formulated as a mixed-integer nonlinear programming problem that considers the dynamics of travel time and travel distance, the interdependency of trips made by each driver, limited range of EVs, remaining battery capacity for recharging, waiting time in queue, and detour to access a charging station. This problem is solved using a metaheuristic approach for a large-scale case network. A series of examples are presented to demonstrate the model efficacy and explore the impact of energy consumption on the final SOC and the optimum charging infrastructure.

Chain of Trips↗
DOE OSTI2023

Subspace recursive Fermi-operator expansion strategies for large-scale DFT eigenvalue problems on HPC architectures

Quantum mechanical calculations for material modeling using Kohn–Sham density functional theory (DFT) involve the solution of a nonlinear eigenvalue problem for N smallest eigenvector-eigenvalue pairs, with N proportional to the number of electrons in the material system. Here, these calculations are computationally demanding and have asymptotic cubic scaling complexity with the number of electrons. Large-scale matrix eigenvalue problems arising from the discretization of the Kohn–Sham DFT equations employing a systematically convergent basis traditionally rely on iterative orthogonal projection methods, which are shown to be computationally efficient and scalable on massively parallel computing architectures. However, as the size of the material system increases, these methods are known to incur dominant computational costs through the Rayleigh–Ritz projection step of the discretized Kohn–Sham Hamiltonian matrix and the subsequent subspace diagonalization of the projected matrix. This work explores the potential of polynomial expansion approaches based on recursive Fermi-operator expansion as an alternative to the subspace diagonalization of the projected Hamiltonian matrix to reduce the computational cost. Subsequently, we perform a detailed comparison of various recursive polynomial expansion approaches to the traditional approach of explicit diagonalization on both multi-node central processing unit and graphics processing unit architectures and assess their relative performance in terms of accuracy, computational efficiency, scaling behavior, and energy efficiency.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Multilevel Parareal Algorithm with Averaging for Oscillatory Problems

The present study is an extension of the work done by Peddle, Haut, and Wingate and Haut and Wingate, where a two-level Parareal method with mapping and averaging is examined. The method proposed in this paper is a multilevel Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for nonlinear multiscale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The method is applied to nonlinear differential equations. The nonlinearities can generate a range of frequencies in the problem. The computational cost of the new method is investigated and studied on several examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

An Adaptive Newton-Based Free-Boundary Grad–Shafranov Solver

Equilibria in magnetic confinement devices result from force balancing between the Lorentz force and the plasma pressure gradient. In an axisymmetric configuration like a tokamak, such an equilibrium is described by an elliptic equation for the poloidal magnetic flux, commonly known as the Grad–Shafranov equation. It is challenging to develop a scalable and accurate free-boundary Grad–Shafranov solver, since it is a fully nonlinear optimization problem that simultaneously solves for the magnetic field coil current outside the plasma to control the plasma shape. In this work, we develop a Newton-based free-boundary Grad–Shafranov solver using adaptive finite elements and preconditioning strategies. The free-boundary interaction leads to the evaluation of a domain-dependent nonlinear form of which its contribution to the Jacobian matrix is achieved through shape calculus. The optimization problem aims to minimize the distance between the plasma boundary and specified control points while satisfying two nontrivial constraints, which correspond to the nonlinear finite element discretization of the Grad–Shafranov equation and a constraint on the total plasma current involving a nonlocal coupling term. The linear system is solved by a block factorization, and AMG is called for subblock elliptic operators. The unique contributions of this work include the treatment of a global constraint, preconditioning strategies, nonlocal reformulation, and the implementation of adaptive finite elements. Furthermore, it is found that the resulting Newton solver is robust, successfully reducing the nonlinear residual to 1e-6 and lower in a small handful of iterations while addressing the challenging case to find a Taylor state equilibrium where conventional Picard-based solvers fail to converge.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗