Search NASA⌕ Search

SEARCH · Search NASA

Results for “Convergence guarantee”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

A convex pseudolikelihood framework for high dimensional partial correlation estimation with convergence guarantees

Sparse high dimensional graphical model selection is a topic of much interest in modern day statistics. A popular approach is to apply l1-penalties to either parametric likelihoods, or regularized regression/pseudolikelihoods, with the latter having the distinct advantage that they do not explicitly assume Gaussianity. As none of the popular methods proposed for solving pseudolikelihood-based objective functions have provable convergence guarantees, it is not clear whether corresponding estimators exist or are even computable, or if they actually yield correct partial correlation graphs. We propose a new pseudolikelihood-based graphical model selection method that aims to overcome some of the shortcomings of current methods, but at the same time retain all their respective strengths. In particular, we introduce a novel framework that leads to a convex formulation of the partial covariance regression graph problem, resulting in an objective function comprised of quadratic forms. The objective is then optimized via a coordinate wise approach. The specific functional form of the objective function facilitates rigorous convergence analysis leading to convergence guarantees; an important property that cannot be established by using standard results, when the dimension is larger than the sample size, as is often the case in high dimensional applications. These convergence guarantees ensure that estimators are well defined under very general conditions and are always computable. In addition, the approach yields estimators that have good large sample properties and also respect symmetry. Furthermore, application to simulated and real data, timing comparisons and numerical convergence is demonstrated. We also present a novel unifying framework that places all graphical pseudolikelihood methods as special cases of a more general formulation, leading to important insights.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Scalable Plug-and-Play ADMM with Convergence Guarantees

Plug-and-play priors (PnP) is a broadly applicable methodology for solving inverse problems by exploiting statistical priors specified as denoisers. Recent work has reported the state-of-the-art performance of PnP algorithms using pre-trained deep neural nets as denoisers in a number of imaging applications. However, current PnP algorithms are impractical in large-scale settings due to their heavy computational and memory requirements. This work addresses this issue by proposing an incremental variant of the widely used PnP-ADMM algorithm, making it scalable to problems involving a large number measurements. Here, we theoretically analyze the convergence of the algorithm under a set of explicit assumptions, extending recent theoretical results in the area. Additionally, we show the effectiveness of our algorithm with nonsmooth data-fidelity terms and deep neural net priors, its fast convergence compared to existing PnP algorithms, and its scalability in terms of speed and memory.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Hierarchical Network Partitioning for Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations

The solution of potential-driven steady-state flow in large networks is a task which manifests in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and in general, there is no numerical algorithm that offers guaranteed convergence to the solution (assuming a solution exists). Some methods offer guarantees in cases where the network topology satisfies certain assumptions, but these methods fail for larger networks. On the other hand, the Newton-Raphson algorithm offers a convergence guarantee if the starting point lies close to the (unknown) solution. It would be advantageous to compute the solution of the large nonlinear system through the solution of smaller nonlinear sub-systems wherein the solution algorithms (Newton-Raphson or otherwise) are more likely to succeed. Here, this letter proposes and describes such a procedure, a hierarchical network partitioning algorithm that enables the solution of large nonlinear systems corresponding to potential-driven steady-state network flow equations.

42 ENGINEERING↗
DOE OSTI2019

A self-correcting variable-metric algorithm framework for nonsmooth optimization

Abstract An algorithm framework is proposed for minimizing nonsmooth functions. The framework is variable metric in that, in each iteration, a step is computed using a symmetric positive-definite matrix whose value is updated as in a quasi-Newton scheme. However, unlike previously proposed variable-metric algorithms for minimizing nonsmooth functions, the framework exploits self-correcting properties made possible through Broyden–Fletcher–Goldfarb–Shanno-type updating. In so doing, the framework does not overly restrict the manner in which the step computation matrices are updated, yet the scheme is controlled well enough that global convergence guarantees can be established. The results of numerical experiments for a few algorithms are presented to demonstrate the self-correcting behaviours that are guaranteed by the framework.

Mathematics↗
DOE OSTI2025

Fair Concurrent Training of Multiple Models in Federated Learning

Federated learning (FL) enables collaborative learning across multiple clients. In most FL work, all clients train a single learning task. However, the recent proliferation of FL applications may increasingly require multiple FL tasks to be trained simultaneously, sharing clients’ computing resources, which we call Multiple-Model Federated Learning (MMFL). Current MMFL algorithms use naïve average-based client-task allocation schemes that often lead to unfair performance when FL tasks have heterogeneous difficulty levels, as the more difficult tasks may need more client participation to train effectively. Furthermore, in the MMFL setting, we face a further challenge that some clients may prefer training specific tasks to others, and may not even be willing to train other tasks, e.g., due to high computational costs, which may exacerbate unfairness in training outcomes across tasks. We address both challenges by firstly designing FedFairMMFL, a difficulty-aware algorithm that dynamically allocates clients to tasks in each training round, based on the tasks’ current performance levels. We provide guarantees on the resulting task fairness and FedFairMMFL’s convergence rate. We then propose novel auction designs that incentivizes clients to train multiple tasks, so as to fairly distribute clients’ training efforts across the tasks, and extend our convergence guarantees to this setting. Here, we finally evaluate our algorithm with multiple sets of learning tasks on real world datasets, showing that our algorithm improves fairness by improving the final model accuracy and convergence speed of the worst performing tasks, while maintaining the average accuracy across tasks.

Federated learning↗
DOE OSTI2021

Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm

The phase retrieval problem, where one aims to recover a complex-valued image from far-field intensity measurements, is a classic problem encountered in a range of imaging applications. Modern phase retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework. Calculating closed-form gradients for use in these methods is tedious work, and formulating second order derivatives is even more laborious. Additionally, second order techniques often require the storage and inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities. We use a reverse-mode automatic differentiation (AD) framework to implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a longstanding method that finds popular use in nonlinear least-square minimization problems but which has seen little use in phase retrieval. Furthermore, we extend the basic LM algorithm so that it can be applied for more general constrained optimization problems (including phase retrieval problems) beyond just the least-square applications. Since we use AD, we only need to specify the physics-based forward model for a specific imaging application; the first and second-order derivative terms are calculated automatically through matrix-vector products, without explicitly forming the large Jacobian or Gauss-Newton matrices typically required for the LM method. We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval problem and the constrained “blind” ptychographic object and probe retrieval problems, under the popular Gaussian noise model as well as the Poisson noise model. We compare this algorithm to state-of-the-art first order ptychographic reconstruction methods to demonstrate empirically that this method outperforms best-in-class first-order methods: it provides excellent convergence guarantees with (in many cases) a superlinear rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Nonlinear convergence in contact mechanics: Immersed boundary finite volume

In this report we present an immersed boundary finite volume (IBM) method for simulating quasistatic contact mechanics of linearly elastic domains at small strains. In IBM, all external boundaries and internal contacts of an object are represented by embedded surfaces inside a Cartesian mesh, which need not conform to the grid lines. The contact constraints consist of the non-penetrability condition and Coulomb’s friction law, which are discretized using special interpolation stencils and enforced via penalty parameters. The resulting nonlinear system depends on displacement unknowns only. To solve it, we use the Newton method but find that it diverges frequently. To understand the divergence pattern, we analyze a simplified 2-cell problem and show that the global convergence of Newton cannot be ensured for any choice of penalty parameters. We thus propose a modified Newton solver, which guarantees convergence for the 2-cell problem and is numerically verified to converge for all the challenging simulations considered herein. While both 1 st - and 2 nd -order variants of IBM, in displacement unknowns, are proposed, the modified Newton solver applies only to the 1 st -order variant.

42 ENGINEERING↗
DOE OSTI2026

Adaptive Model-Free Vehicle Path-Tracking via Fast-Converging Prescribed-Time Newton-Based Extremum-Seeking Control

Model-free control (MFC) offers a simple and effective approach to automated vehicle path-tracking without requiring an explicit plant model for control law design. However, gain tuning in MFC is typically carried out through trial-and-error, which can be time-consuming and may lead to suboptimal performance. To address this limitation, extremum-seeking-based adaptive MFC has shown promise by enabling real-time adaptation of control gains, without relying on a predefined vehicle model. Nonetheless, existing ESC approaches often suffer from slow convergence. This paper integrates MFC, employing longitudinal and lateral ultra-local models of a rear-wheel-drive vehicle, with a novel prescribed-time (PT) Newton-based extremum-seeking control (ESC) strategy that ensures rapid convergence of control gains within the prescribed time. Unlike conventional gradient-based ESC methods, the PT Newton-based ESC leverages artificial delays and time-periodic gains, not only to guarantee convergence within the specified time, but also to compensate for feedback delays. Simulation results demonstrate that the proposed approach significantly improves gain adaptation speed and tracking accuracy. This work advances adaptive model-free vehicle control by offering a high-performance, delay-resilient alternative to existing ESCMFC frameworks.

Waleed khan, Muhammad [The University of Texas at ↗
DOE OSTI2017

Spectral partitioning with blends of eigenvectors

We report many common methods for data analysis rely on linear algebra. We provide new results connecting data analysis error to numerical accuracy in the context of spectral graph partitioning. We provide pointwise convergence guarantees so that spectral blends (linear combinations of eigenvectors) can be employed to solve data analysis problems with confidence in their accuracy. We apply this theory to an accessible model problem, the ring of cliques, by deriving the relevant eigenpairs and finding necessary and sufficient solver tolerances. Analysis of the ring of cliques provides an upper bound on eigensolver tolerances for graph partitioning problems. These results bridge the gap between linear algebra based data analysis methods and the convergence theory of iterative approximation methods. These results explain how the combinatorial structure of a problem can be recovered much faster than numerically accurate solutions to the associated linear algebra problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Integral Kernel Methods for Nonlinear Parabolic-Elliptic Systems

Nonlinear parabolic-elliptic systems arise in many physical, biological, and chemical phenomena such as chemotaxis, ion transport, self-gravitating particles, and Brownian vortices. Existing methods struggle with the strong coupling and high nonlinearity and nonlocality of some of these systems, especially the ill-conditioned, convection-dominated problems. To overcome numerical difficulties, current approaches rely on initial guesses, preconditioning, or iterative techniques with no convergence guarantees. They might suffer from poor scalability, large memory usage, and difficulty to parallelize. Inspired by the connection of parabolic-elliptic systems to stochastic processes, we introduce a novel meshless, monolithic, and fully explicit method that naturally encapsulates the elliptic and parabolic operators into a single step which updates each node deterministically with global information. By being fully quadrature-based, it avoids solving systems of discretized equations and does not utilize initial guesses or preconditioning, while requiring little memory and being easy to parallelize. We first derive the method in an integral kernel formulation with quadratic complexity in the number of integration nodes and then leverage kernel-independent fast multipole methods (FMM) to present a scalable algorithm with linear complexity. We provide numerical examples for the Poisson-Nernst-Planck equations in one, two, and three dimensions, together with the derivation of the integral kernel for each case. Furthermore, the examples demonstrate the fast convergence and scalability of the FMM-accelerated algorithm, as well as its suitability for convection-dominated problems, making it competitive against traditional PDE solvers.

PDE systems↗
DOE OSTI2021

Contingency Analysis Based on Partitioned and Parallel Holomorphic Embedding

In the steady-state contingency analysis, the traditional Newton-Raphson method suffers from non-convergence issues when solving post-outage power flow problems, which hinders the integrity and accuracy of security assessment. In this paper, we propose a novel robust contingency analysis approach based on holomorphic embedding (HE). Here, the HE-based simulator provides theoretical convergence guarantee, which is desirable because it avoids the influence of numerical issues and provides a credible security assessment conclusion. In addition, based on the multi-area characteristics of real-world power systems, a partitioned HE (PHE) method is proposed with an interfacebased partitioning of HE formulation. The PHE method does not undermine the numerical robustness of HE and significantly reduces the computation burden in large-scale contingency analysis. The PHE method is further enhanced by parallel or distributed computation to become parallel PHE (P2HE). Tests on a 458-bus system, a synthetic 419-bus system and a large-scale 21447-bus system demonstrate the advantages of the proposed methods in robustness and efficiency.

42 ENGINEERING↗
DOE OSTI2018

A fast and accurate method for perturbative resummation of transverse momentum-dependent observables

Here, we propose a novel strategy for the perturbative resummation of transverse momentum-dependent (TMD) observables, using the q T spectra of gauge bosons (γ*, Higgs) in pp collisions in the regime of low (but perturbative) transverse momentum q T as a specific example. First we introduce a scheme to choose the factorization scale for virtuality in momentum space instead of in impact parameter space, allowing us to avoid integrating over (or cutting off) a Landau pole in the inverse Fourier transform of the latter to the former. The factorization scale for rapidity is still chosen as a function of impact parameter b, but in such a way designed to obtain a Gaussian form (in ln b) for the exponentiated rapidity evolution kernel, guaranteeing convergence of the b integral. We then apply this scheme to obtain the q T spectra for Drell-Yan and Higgs production at NNLL accuracy. In addition, using this scheme we are able to obtain a fast semi-analytic formula for the perturbative resummed cross sections in momentum space: analytic in its dependence on all physical variables at each order of logarithmic accuracy, up to a numerical expansion for the pure mathematical Bessel function in the inverse Fourier transform that needs to be performed just once for all observables and kinematics, to any desired accuracy.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2018

A fully distributed asynchronous approach for multi-area coordinated network-constrained unit commitment

This paper discusses a consensus-based alternating direction method of multipliers (ADMM) approach to solve the multi-area coordinated network-constrained unit commitment (NCUC) problem in a distributed manner. Due to political and technical difficulties, it is neither practical nor feasible to solve the multi-area coordination problem in a centralized fashion, which requires full access to all data of individual areas. In comparison, in the proposed fully-distributed approach, local NCUC problems of individual areas can be solved independently, and only limited information is exchanged among adjacent areas for facilitating multi-area coordination. Furthermore, as traditional ADMM can only guarantee convergence for convex problems, this paper discusses several strategies to mitigate oscillations, enhance convergence performance, and derive good-enough feasible solutions, including: (i) A tie-line power flow based area coordination strategy is designed to reduce the number of global consensus variables; (ii) Different penalty parameters \r{ho} are assigned to individual consensus variables and are updated via certain rules during the iterative procedure, which would reduce the impact of initial values of \r{ho} on convergence performance; (iii) Heuristic rules are adopted to fix certain unit commitment variables for avoiding oscillations during the iterative procedure; and (iv) An asynchronous distributed strategy is studied, which solves NCUC subproblems of small areas multiple times and exchanges information with adjacent areas more frequently within one complete run of slower NCUC subproblems of large areas. Numerical cases illustrate effectiveness of the proposed asynchronous fully-distributed NCUC approach, and investigate key factors that would affect its convergence performance.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2025

A randomized sketching trust-region secant method for low-memory dynamic optimization

The numerical solution of dynamic optimization problems is often limited by the memory required to store the state trajectory, which is used to evaluate the objective function and its derivatives. Recently, [R. Muthukumar et al., SIAM Journal on Optimization 31(2), pp. 1242–1275 (2021)] introduced a trust-region method for dynamic optimization that employs randomized sketching to compress the state trajectory, resulting in inexact derivative computations. By adaptively learning the sketch rank, the trust-region algorithm achieves rigorous convergence guarantees. Here, we extend this approach to use secant Hessian approximations. Due to the randomness introduced by the sketch, the traditional secant update formulae can produce poor Hessian approximations. In particular, the difference of two gradients, computed from two different sketches, may be inconsistent. To overcome this, we employ a sketched approximation of the Hessian application, in lieu of computing the gradient difference. We numerically demonstrate the improved stability of this approach on an example from PDE-constrained optimization.

dynamic optimization↗
DOE OSTI2015

Representing ductile damage with the dual domain material point method

In this study, we incorporate a ductile damage material model into a computational framework based on the Dual Domain Material Point (DDMP) method. As an example, simulations of a flyer plate experiment involving ductile void growth and material failure are performed. The results are compared with experiments performed on high purity tantalum. We also compare the numerical results obtained from the DDMP method with those obtained from the traditional Material Point Method (MPM). Effects of an overstress model, artificial viscosity, and physical viscosity are investigated. Our results show that a physical bulk viscosity and overstress model are important in this impact and failure problem, while physical shear viscosity and artificial shock viscosity have negligible effects. A simple numerical procedure with guaranteed convergence is introduced to solve for the equilibrium plastic state from the ductile damage model.

36 MATERIALS SCIENCE↗
DOE OSTI2023

An inexact semismooth Newton method with application to adaptive randomized sketching for dynamic optimization

In many applications, one can only access the inexact gradients and inexact hessian times vector products. Thus it is essential to consider algorithms that can handle such inexact quantities with a guaranteed convergence to solution. An inexact adaptive and provably convergent semismooth Newton method is considered to solve constrained optimization problems. In particular, dynamic optimization problems, which are known to be highly expensive, are the focus. A memory efficient semismooth Newton algorithm is introduced for these problems. The source of efficiency and inexactness is the randomized matrix sketching. Further, applications to optimization problems constrained by partial differential equations are also considered.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Generative AI models for learning flow maps of stochastic dynamical systems in bounded domains

Simulating stochastic differential equations (SDEs) in bounded domains, presents significant computational challenges due to particle exit phenomena, which requires accurate modeling of interior stochastic dynamics and boundary interactions. Despite the success of machine learning-based methods in learning SDEs, existing learning methods are not applicable to SDEs in bounded domains because they cannot accurately capture the particle exit dynamics. We present a unified hybrid data-driven approach that combines a conditional diffusion model with an exit prediction neural network to capture both interior stochastic dynamics and boundary exit phenomena. Our ML model consists of two major components: a neural network that learns exit probabilities using binary cross-entropy loss with rigorous convergence guarantees, and a training-free diffusion model that generates state transitions for non-exiting particles using closed-form score functions. The two components are integrated through a probabilistic sampling algorithm that determines particle exit at each time step and generates appropriate state transitions. Here, the performance of the proposed approach is demonstrated via three test cases: a one-dimensional simplified problem for theoretical verification, a two-dimensional advection-diffusion problem in a bounded domain, and a three-dimensional problem of interest to magnetically confined fusion plasmas.

Bounded domains↗