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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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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations

In this work, we develop implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac–Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) (Ceruti et al. in BIT Numer Math 62(4):1149–1174, 2022) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using an implicit solves for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. Here, we prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Model-based and Model-free Designs for an Extended Continuous-time LQR with Exogenous Inputs

We present an extended linear quadratic regulator (LQR) design for continuous-time linear time-invariant (LTI) systems in the presence of exogenous inputs. We first propose a model-based solution with cost minimization guarantees for states and inputs using dynamic programming (DP). The control law consists of a combination of the optimal state feedback and an additional optimal term dependent on the exogenous inputs. The control gains for the two components are obtained by solving a set of matrix differential equations. We provide these solutions for both finite horizons and steady-state cases. In the second part of the paper, we formulate a reinforcement learning (RL) based algorithm which does not need any model information except the input matrix, and can compute an approximate steady-state LQR gain using measurements of the states, the control inputs, and the exogenous inputs. Both model-based and data-driven optimal control algorithms are tested with a numerical example under different exogenous inputs showcasing the effectiveness of the designs.

Mukherjee, Sayak↗
DOE OSTI2026

Laplace Transform–Based Quantum Eigenvalue Transformation via Linear Combination of Hamiltonian Simulation

Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for nonnormal matrices. Here, we propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation method [D. An, J.-P. Liu, and L. Lin, Phys. Rev. Lett., 131 (2023), 150603; D. An, A. M. Childs, and L. Lin, Commun. Math. Phys. 407, 19 (2026)] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, 𝐴 −𝑘 , and the exponential of the matrix inverse, 𝑒 −𝐴 −1 . The latter can be interpreted as the solution of a mass-matrix differential equation of the form form 𝐴⁢𝑢′⁡⁡(𝑡) =−𝑢⁡(𝑡). We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting 𝐴, thereby reducing the computational complexity.

Laplace transform↗
DOE OSTI2019

Integrable nonlinear evolution of the qubit

Highlights: •We study the nonlinear generalization of the von Neumann equation. •We discuss the nonlinear von Neumann equation for the qubit states. •We analyze solutions to the Riccati system corresponding to evolution of a qubit. -- Abstract: The nonlinear generalization of the von Neumann equation is studied. The global version of this equation preserves convexity of the space of states. The particular case of the evolution of pure states refers to the nonlinear Schrödinger equation discovered by Gisin. The concrete realization of the nonlinear von Neumann equation is investigated in detail for the density matrix representing the qubit states. Such equation reduces to the integrable classical Riccati system of nonlinear ordinary differential equations. An interesting property of the nonlinear dynamics described by this system is the global asymptotical stability of stationary solutions related to evolution from mixed states to pure states.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Fast and Scalable FFT-Based GPU-Accelerated Algorithms for Block-Triangular Toeplitz Matrices with Application to Linear Inverse Problems Governed by Autonomous Dynamical Systems

In this work, we present an efficient and scalable algorithm for performing matrix-vector multiplications (matvecs) for block Toeplitz matrices. Such matrices, which are shift-invariant with respect to their blocks, arise in the context of solving inverse problems governed by autonomous systems, and time-invariant systems in particular. In this article, we consider inverse problems that infer unknown parameters from observational data of a linear time-invariant dynamical system given in the form of partial differential equations (PDEs). Matrix-free Newton-conjugate-gradient methods are often the gold standard for solving these inverse problems, but they require numerous actions of the Hessian on a vector. Matrix-free adjoint-based Hessian matvecs require solution of a pair of linearized forward/adjoint PDE solves per Hessian action, which may be prohibitive for large-scale inverse problems. Time invariance of the forward PDE problem leads to a block Toeplitz structure of the discretized parameter-to-observable (p2o) map defining the mapping from inputs (parameters) to outputs (observables) of the PDEs. This block Toeplitz structure enables us to exploit two key properties: (1) compact storage of the p2o map and its adjoint, and (2) efficient fast Fourier transform–based Hessian matvecs. The proposed algorithm is mapped onto large multi-GPU clusters and achieves more than 80% of peak bandwidth on NVIDIA A100 GPUs. Excellent weak scaling is shown for up to 48 A100 GPUs. For the targeted problems, the implementation executes Hessian matvecs within fractions of a second, which is orders of magnitude faster than can be achieved by conventional matrix-free Hessian matvecs via forward/adjoint PDE solves.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Asymptotics of the General Solution of a Linear Singularly Perturbed System of Higher-Order Differential Equations with Degenerations

By using the theory of polynomial matrix pencils, we construct the asymptotics of linearly independent solutions of homogeneous singularly perturbed system of linear differential equations of any order m with matrices at higher derivatives degenerating as a small parameter approaches zero. The general case is analyzed. In this case, the limit matrix pencil has several finite and infinite elementary divisors of both the same and different multiplicities. The corresponding asymptotic estimates are presented.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Dual Representations and H ∞ -Optimal Control of Partial Differential Equations

We consider H ∞ -optimal state-feedback control of the class of linear Partial Differential Equations (PDEs) which admit a Partial Integral Equation (PIE) representation. While linear matrix inequalities are commonly used for optimal control of Ordinary Differential Equations (ODEs), the absence of a universal state-space representation and suitable dual form prevents such methods from being applied to optimal control of PDEs. Specifically, for ODEs, the controller synthesis problem is defined in state-space, and duality is used to resolve the bilinearity of that synthesis problem. Recently, the PIE representation was proposed as a universal state-space representation for linear PDE systems. In this paper, we show that any PDE system represented by a PIE admits a dual PIE with identical stability and I/O properties. This result allows us to reformulate the stabilizing and optimal state-feedback control problems as convex optimization over the cone of positive Partial Integral (PI) operators. Operator inversion formulae then allow us to construct feedback gains for the original PDE system. The results are verified through application to several canonical problems in optimal control of PDEs and indicate the resulting bounds on H ∞ norm are not conservative.

42 ENGINEERING↗
DOE OSTI2025

Fluctuations in Hill’s equation parameters and application to cosmic reheating

Cosmic inflation provides a compelling framework for explaining several observed features of our Universe, but its viability depends on an efficient reheating phase that converts the inflaton’s energy into Standard Model particles. This conversion often proceeds through nonperturbative mechanisms such as parametric resonance, which is described by Hill’s equation. In this work, we investigate how stochastic fluctuations in the parameters of Hill’s equation can influence particle production during reheating. We show that such fluctuations can arise from couplings to light scalar fields and can significantly alter the stability bands in the resonance structure, thereby enhancing the growth of fluctuations and broadening the region of efficient energy transfer. Using random matrix theory and stochastic differential equations, we decompose the particle growth rate into deterministic and noise-induced components and demonstrate analytically and numerically that even modest noise leads to substantial particle production in otherwise stable regimes. Furthermore, these results suggest that stochastic effects can robustly enhance the efficacy of reheating across a wide swath of parameter space, with implications for early Universe cosmology, UV completions involving multiple scalar fields, and the resolution of the cosmological moduli problem.

Cosmology↗
DOE OSTI2020

Parallel-in-Time Simulation of Lindblad's Equation

Constructing fast quantum logic gates is critical to building a scalable quantum computer. We consider a qudit, a quantum version of a bit that can take an arbitrary number of states, coupled with a cavity. In this project, we wish to force the qudit to reach the 0-state, for any possible initial state. The coupled system changes in time according to Lindblad’s equation, an ordinary differential equation on the density matrix of the quantum system. Lindblad’s equation contains some parameters that we can control, so-called control functions. We seek control functions which force the qudit to the 0-state within 2 microseconds, which is much faster than what is currently done in practice. The search method is gradient descent, a numerical optimization method that uses gradient information to iteratively improve the control parameters. My contribution to this project is an attempt to speed up the computation of the gradient. It currently takes about 40 seconds to compute the gradient which involves solving a set of ODEs sequentially. Current supercomputers have thousands of cores, but sequential computations can only make use of 1 core at a time. We wish to divide up the work better, so that we can use many more cores at once. To this end, we have implemented the Multigrid Reduction in Time (MGRIT) algorithm. We perform a systematic parameter search on how to best apply this algorithm. Results indicate a 25 percent speed up for solving Lindblad’s equation and determining how close the final state is the 0-state.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Composing preconditioners for multiphysics PDE systems with applications to Generalized MHD

New patch smoothers or relaxation techniques are developed for solving linear matrix equations coming from systems of discretized partial differential equations (PDEs). One key linear solver challenge for many PDE systems arises when the resulting discretization matrix has a near null space that has a large dimension, which can occur in generalized magnetohydrodynamic (GMHD) systems. Patch-based relaxation is highly effective for problems when the null space can be spanned by a basis of locally supported vectors. The patch-based relaxation methods that we develop can be used either within an algebraic multigrid (AMG) hierarchy or as stand-alone preconditioners. These patch-based relaxation techniques are a form of well-known overlapping Schwarz methods where the computational domain is covered with a series of overlapping sub-domains (or patches). Patch relaxation then corresponds to solving a set of independent linear systems associated with each patch. In the context of GMHD, we also reformulate the underlying discrete representation used to generate a suitable set of matrix equations. In general, deriving a discretization that accurately approximates the curl operator and the Hall term while also producing linear systems with physically meaningful near null space properties can be challenging. Unfortunately, many natural discretization choices lead to a near null space that includes non-physical oscillatory modes and where it is not possible to span the near null space with a minimal set of locally supported basis vectors. Further discretization research is needed to understand the resulting trade-offs between accuracy, stability, and ease in solving the associated linear systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A model for discrete fracture-clay rock interaction incorporating electrostatic effects on transport

Abstract A model based on the code CrunchClay is presented for a fracture-clay matrix system that takes electrostatic effects on transport into account. The electrostatic effects on transport include those associated with the development of a diffusion potential as captured by the Nernst-Planck equation, and the formation of a diffuse layer bordering negatively charged clay particles within which partial anion exclusion occurs. The model is based on a dual continuum formulation that accounts for diffuse layer and bulk water pore space, providing a more flexible framework than is found in the classical mean electrostatic potential models. The diffuse layer model is obtained by volume averaging ion concentrations in the Poisson-Boltzmann equation, but also includes the treatment of longitudinal transport within this continuum. The calculation of transport within the bulk and diffuse layer porosity is based on a new formulation for the Nernst-Planck equation that considers averaging of diffusion coefficients and accumulation factors at grid cell interfaces. Equations for function residuals and the associated Jacobian matrix are presented such that the system of nonlinear differential-algebraic equations can be solved with Newton’s method. As an example, we consider a 2D system with a single discrete fracture within which flow and advective transport occurs that is coupled to diffusion in the clay-rich matrix. The simulation results demonstrate the lack of retardation for anions (e.g., 36 Cl − ) of the contaminant plume within the fracture flow system because they are largely excluded from the charged clay rock, while the migration of cations (e.g., 90 Sr ++ ) is more strongly attenuated. The diffusive loss of divalent cations in particular from the fracture is accentuated by their accumulation in the diffuse layer within the clay-rich matrix.

58 GEOSCIENCES↗
DOE OSTI2023

Domain Aware Deep-learning Algorithms Integrated with Scientific-computing Technologies (DADAIST)

This technical report summarized the contribution of the DADAIST project funded by the Data Model Convergence Initiative via the Laboratory Directed Research and Development (LDRD) investments at Pacific Northwest National Laboratory (PNNL). Specifically, we report the development of the NeuroMANCER (Neural Modules with Adaptive Nonlinear Constraints and Efficient Regularizations), a new open-source Scientific Machine Learning library for formulating and solving parametric constrained optimization problems, physics-informed system identification, and parametric optimal control problems. NeuroMANCER is using differentiable programming to combine modern data-driven models and optimization modeling language into a coherent algorithmic and software framework. NeuroMANCER is a Pytorch-based framework and adopts much of its philosophy focused on research and development, rapid prototyping, and streamlined deployment. Strong emphasis is given to extensibility, interoperability with the PyTorch ecosystem, and quick adaptability to custom domain problems. Neuromancer repository contains a comprehensive library of differentiable modules, including custom activation functions, matrix factorizations, deep learning architectures, neural differential equations, differential equation solvers, implicit layers such as iterative solvers, high-level API for symbolic expressions, API for modeling and control of dynamical systems, and extensive set of tutorial code examples in the form of python scripts and jupyter notebooks.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Improved quantum algorithms for linear and nonlinear differential equations

We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes the run time of quantum algorithms for linear ODEs opening the door to an application to a wider class of linear and nonlinear ODEs. In [1], a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices including singular matrices. The algorithm here is also exponentially faster than the bounds derived in [1] for certain classes of diagonalizable matrices. Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in [2]). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by [3], but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas [2] and [3] additionally require normality.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Parametric matrix models

We present a general class of machine learning algorithms called parametric matrix models. In contrast with most existing machine learning models that imitate the biology of neurons, parametric matrix models use matrix equations that emulate physical systems. Similar to how physics problems are usually solved, parametric matrix models learn the governing equations that lead to the desired outputs. Parametric matrix models can be efficiently trained from empirical data, and the equations may use algebraic, differential, or integral relations. While originally designed for scientific computing, we prove that parametric matrix models are universal function approximators that can be applied to general machine learning problems. After introducing the underlying theory, we apply parametric matrix models to a series of different challenges that show their performance for a wide range of problems. For all the challenges tested here, parametric matrix models produce accurate results within an efficient and interpretable computational framework that allows for input feature extrapolation.

Computational science↗
DOE OSTI2021

Geometry optimization speedup through a geodesic approach to internal coordinates

We present a new geodesic-based method for geometry optimization in a basis set of redundant internal coordinates. Overall, our method updates the molecular geometry by following the geodesic generated by a displacement vector on the internal coordinate manifold, which dramatically reduces the number of steps required to converge to a minimum. Our method can be implemented in any existing optimization code, requiring only implementation of derivatives of the Wilson B-matrix and the ability to numerically solve an ordinary differential equation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Greedy emulators for nuclear two-body scattering

Applications of reduced basis method emulators are increasing in low-energy nuclear physics because they enable fast and accurate sampling of high-fidelity calculations, enabling robust uncertainty quantification. Here, in this paper, we develop, implement, and test two model-driven emulators based on the (Petrov-)Galerkin projection using the prototypical test case of two-body scattering with the Minnesota potential and a more realistic local chiral potential. The high-fidelity scattering equations are solved with the matrix Numerov method, a reformulation of the popular Numerov recurrence relation for solving special second-order differential equations as a linear system of coupled equations. A novel error estimator based on reduced-space residuals is applied to an active learning approach (a greedy algorithm) to choosing training samples (“snapshots”) for the emulator and contrasted with a proper orthogonal decomposition (POD) approach. Both approaches allow for computationally efficient offline-online decompositions, but the greedy approach requires many fewer snapshot calculations. These developments set the groundwork for emulating scattering observables based on chiral nucleon-nucleon and three-nucleon interactions and optical models, where computational speed-ups are necessary for Bayesian uncertainty quantification. Our emulators and error estimators are widely applicable to linear systems.

Bayesian methods↗
DOE OSTI2023

Binary operations on neuromorphic hardware with application to linear algebraic operations and stochastic equations

Abstract Non-von Neumann computational hardware, based on neuron-inspired, non-linear elements connected via linear, weighted synapses—so-called neuromorphic systems—is a viable computational substrate. Since neuromorphic systems have been shown to use less power than CPUs for many applications, they are of potential use in autonomous systems such as robots, drones, and satellites, for which power resources are at a premium. The power used by neuromorphic systems is approximately proportional to the number of spiking events produced by neurons on-chip. However, typical information encoding on these chips is in the form of firing rates that unarily encode information. That is, the number of spikes generated by a neuron is meant to be proportional to an encoded value used in a computation or algorithm. Unary encoding is less efficient (produces more spikes) than binary encoding. For this reason, here we present neuromorphic computational mechanisms for implementing binary two’s complement operations. We use the mechanisms to construct a neuromorphic, binary matrix multiplication algorithm that may be used as a primitive for linear differential equation integration, deep networks, and other standard calculations. We also construct a random walk circuit and apply it in Brownian motion simulations. We study how both algorithms scale in circuit size and iteration time.

97 MATHEMATICS AND COMPUTING↗
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↗