Search NASA⌕ Search

SEARCH · Search NASA

Results for “Exponential Matrix”

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 OSTI2012

Using Laguerre polynomials to compute the matrix exponential in burnup calculations

An essential part of burnup analysis is to solve the burnup equations. The burnup equations can be regarded as a first-order linear system and solved by means of matrix exponential methods. Because of its large spectrum, it is difficult to compute the exponential of the burnup matrix. Conventional methods of computing the matrix exponential, such as the truncated Taylor expansion and the Pade approximation, are not applicable to burnup calculations. Recently the Chebyshev Rational Approximation Method (CRAM) has been applied to solve burnup matrix exponential and shown to be robust and accurate. However, the main defect of CRAM is that its coefficients are not easy to obtain. In this paper, an orthogonal polynomial expansion method, called Laguerre Polynomial Approximation Method (LPAM), is proposed to compute the matrix exponential in burnup calculations. The polynomial sequence of LPAM can be easily computed in any order and thus LPAM is quite convenient to be utilized into burnup codes. Two typical test cases with the decay and cross-section data taken from the standard ORIGEN 2.1 libraries are calculated for validation, against the reference results provided by CRAM of 14 order. Numerical results show that, LPAM is sufficiently accurate for burnup calculations. The influences of the parameters on the convergence of LPAM are also discussed. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

The Legendre Polynomial Axial Expansion Method

This work presents a new formulation of the axial expansion transport method explicitly using Legendre polynomials for arbitrarily high-order expansions. This new formulation also features an alternative method of axial leakage calculation to allow for nonextruded flat source region meshes. This alternative axial leakage is introduced alongside a balance equation requirement to ensure that neutron balance is preserved in the coarse mesh for a given axial leakage formulation, which allows for effective coarse mesh finite difference acceleration. A matrix exponential table method is derived to allow for fast computations of arbitrarily high-order matrix exponentials for this work and precludes the need for further research into matrix exponential calculations for this method. Numerical results are presented that demonstrate the stability of the axial expansion method in systems with voidlike regions, showcase the speedup from matrix exponential tables, and investigate the axial convergence of the method in terms of both expansion order and mesh size.

Herring, Nicholas↗
DOE OSTI2025

ORMATEX

The Oak Ridge Matrix Exponential (ORMATEX) software library contains methods to compute the matrix exponential and the action of the matrix exponential on a vector. Additionally, this package contains the related methods for the phi-functions which commonly appear in a wide class of exponential time integration methods. Krylov methods are provided to evaluate the matrix exponential-vector and phi-vector products for cases where the matrix is large and sparse. Utilizing these methods, ORMATEX implements performant exponential integrators for large systems of coupled ordinary differential equations (ODEs). The exponential time integration routines in ORMATEX are particularly suitable to large, stiff systems of equations. These routines may be utilized as a competitive alternative to classical implicit and explicit time integration schemes for certain classes of differential equations where the problem stiffness can be predominately explained by the linear terms.

Gurecky, William [Oak Ridge National Laboratory (O↗
DOE OSTI2022

Exponential time differencing scheme for mass transport and depletion in molten salt reactors

This work extends the capability previously shown for addressing the problem of computing depletion and mass transport calculations in molten salt reactors (MSRs) by calculating matrix exponentials. Additional algorithms are implemented to compute the matrix exponential and the action of the matrix exponential on a matrix. These algorithms include two methods based on the Pade approximation, a Taylor series method, and three methods based on Cauchy's integral formula. In addition to the added matrix exponential solvers, a variable-order total variation diminishing scheme is applied to the convective flux approximation to provide enhanced accuracy. Finally, a simplified MSR problem is shown for each of the exponential time differencing solvers along with classical backwards differencing integrators. The results show excellent convergence for exponential time differencing methods. Computation time is a key element for selecting the optimal solver in these problems, and this work shows that Pade and Cauchy-based solvers may provided the fastest and most accurate solutions. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2015

Auxiliary matrix formalism for interaction representation transformations, optimal control, and spin relaxation theories

Auxiliary matrix exponential method is used to derive simple and numerically efficient general expressions for the following, historically rather cumbersome, and hard to compute, theoretical methods: (1) average Hamiltonian theory following interaction representation transformations; (2) Bloch-Redfield-Wangsness theory of nuclear and electron relaxation; (3) gradient ascent pulse engineering version of quantum optimal control theory. In the context of spin dynamics, the auxiliary matrix exponential method is more efficient than methods based on matrix factorizations and also exhibits more favourable complexity scaling with the dimension of the Hamiltonian matrix.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

VAN-DAMME: GPU-accelerated and symmetry-assisted quantum optimal control of multi-qubit systems

We present an open-source software package, VAN-DAMME (Versatile Approaches to Numerically Design, Accelerate, and Manipulate Magnetic Excitations), for massively-parallelized quantum optimal control (QOC) calculations of multi-qubit systems. To enable large QOC calculations, the VAN-DAMME software package utilizes symmetry-based techniques with custom GPU-enhanced algorithms. This combined approach allows for the simultaneous computation of hundreds of matrix exponential propagators that efficiently leverage the intra-GPU parallelism found in high-performance GPUs. In addition, to maximize the computational efficiency of the VAN-DAMME code, we carried out several extensive tests on data layout, computational complexity, memory requirements, and performance. These extensive analyses allowed us to develop computationally efficient approaches for evaluating complex-valued matrix exponential propagators based on Padé approximants. To assess the computational performance of our GPU-accelerated VAN-DAMME code, we carried out QOC calculations of systems containing 10 - 15 qubits, which showed that our GPU implementation is 18.4× faster than the corresponding CPU implementation. Our GPU-accelerated enhancements allow efficient calculations of multi-qubit systems, which can be used for the efficient implementation of QOC applications across multiple domains.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Solving the Bateman equations in CASMO5 using implicit ode numerical methods for stiff systems

The Bateman equations, which describe the transmutation of nuclides over time as a result of radioactive decay, absorption, and fission, are often numerically stiff. This is especially true if short-lived nuclides are included in the system. This paper describes the use of implicit numerical methods for o D Es applied to the stiff Bateman equations, specifically employing the Backward Differentiation Formulas (BDF) form of the linear multistep method. As is true in other domains, using an implicit method removes or lessens the (sometimes severe) step-length constraints by which explicit methods must abide. To gauge its accuracy and speed, the BDF method is compared to a variety of other solution methods, including Runge-Kutta explicit methods and matrix exponential methods such as the Chebyshev Rational Approximation Method (CRAM). A preliminary test case was chosen as representative of a PWR lattice depletion step and was solved with numerical libraries called from a Python front-end. The Figure of Merit (a combined measure of accuracy and efficiency) for the BDF method was nearly identical to that for CRAM, while explicit methods and other matrix exponential approximations trailed behind. The test case includes 319 nuclides, in which the shortest-lived nuclide is {sup 98}Nb with a half-life of 2.86 seconds. Finally, the BDF and CRAM methods were compared within CASMO5, where CRAM had a FOM about four times better than BDF, although the BDF implementation was not fully optimized. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2013

Accuracy considerations for Chebyshev rational approximation method (CRAM) in Burnup calculations

The burnup equations can in principle be solved by computing the exponential of the burnup matrix. However, due to the difficult numerical characteristics of burnup matrices, the problem is extremely stiff and the matrix exponential solution has previously been considered infeasible for an entire burnup system containing over a thousand nuclides. It was recently discovered by the author that the eigenvalues of burnup matrices are generally located near the negative real axis, which prompted introducing the Chebyshev rational approximation method (CRAM) for solving the burnup equations. CRAM can be characterized as the best rational approximation on the negative real axis and it has been shown to be capable of simultaneously solving an entire burnup system both accurately and efficiently. In this paper, the accuracy of CRAM is further studied in the context of burnup equations. The approximation error is analyzed based on the eigenvalue decomposition of the burnup matrix. It is deduced that the relative accuracy of CRAM may be compromised if a nuclide concentration diminishes significantly during the considered time step. Numerical results are presented for two test cases, the first one representing a small burnup system with 36 nuclides and the second one a full a decay system with 1531 nuclides. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of φ $-$functions of matrices via quadrature

Here, in this article, we propose an algorithm for approximating the action of φ $-$ functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the φ $-$ functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Memristive linear algebra

The advent of memristive devices offers a promising avenue for efficient and scalable analog computing, particularly for linear algebra operations essential in various scientific and engineering applications. This paper investigates the potential of memristive crossbars in implementing matrix inversion algorithms. We explore both static and dynamic approaches, emphasizing the advantages of analog and in-memory computing for matrix operations beyond multiplication. In particular, we demonstrate that the electrical properties of memristive crossbars uniquely suit them for the evolution of a family of matrix exponentials, which can be exploited for the efficient computation of matrix inverses and online solutions for linear problems. Our results demonstrate that memristive arrays can reduce computational complexity. We also study power consumption and show a tradeoff between precision and energy. Furthermore, we address the challenges of device variability, precision, and scalability, providing insights into the practical implementation of these algorithms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Efficient Integration of Algebraic Constraint for Exponential Time Integration

This work is a continuation of a previous project where the high efficiency of exponential integration methods for DRMHD systems was demonstrated. Often algebraic constraints must also be enforced for DRMHD systems of interest. Straightforward application of exponential methods to DRMHD equations with constraints leads to prohibitively computationally expensive methods. In this work, we propose new exponential schemes that allow the constraints to be removed from evaluation of exponential matrix functions which drastically reduces computational complexity by eliminating the need to perform computations enforcing constraints in exponential calculations while still preserving a high order of accuracy and allowing for a large time step even when the problem is stiff. This idea is similar to the W-methods and is achieved by carefully designing a method with the desired order of accuracy, even with an incomplete Jacobian matrix used as an argument of exponential-like functions. The constraints are accounted for by including them in the evaluation of the right-hand-side forcing function of the spatially discretized system. We study performance of the new methods on test problems and outline future research directions that this work opens.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Exponential time differencing for the tracer equations appearing in primitive equation ocean models

The tracer equations are part of the primitive equations used in ocean modeling and describe the transport of tracers, such as temperature, salinity or chemicals, in the ocean. Depending on the number of tracers considered, several equations may be added to and coupled to the dynamics system. In many relevant situations, the time-step requirements of explicit methods imposed by the transport and mixing in the vertical direction are more restrictive than those for the horizontal, and this may cause the need to use very small time steps if a fully explicit method is employed. To overcome this issue, we propose an exponential time differencing (ETD) solver where the vertical terms (transport and diffusion) are treated with a matrix exponential, whereas the horizontal terms are dealt with in an explicit way. In this work, we investigate numerically the computational speed-ups that can be obtained over other semi-implicit methods, and we analyze the advantages of the method in the case of multiple tracers.

42 ENGINEERING↗
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 OSTI2020

Localized Exponential Time Differencing Method for Shallow Water Equations: Algorithms and Numerical Study

Here, we explore the performance of the exponential time differencing (ETD) method applied to the rotating shallow water equations. Comparing with explicit time stepping of the same order accuracy in time, the ETD algorithms could reduce the computational time in many cases by allowing the use of large time step sizes while still maintaining numerical stability. To accelerate the ETD simulations, we propose a localized approach that synthesizes the ETD method and overlapping domain decomposition. By dividing the original problem into many subdomain problems of smaller sizes and solving them locally, the proposed approach could speed up the calculation of matrix exponential vector products. Several standard test cases for shallow water equations of one or multiple layers are considered. The results show great potential of the localized ETD method for high-performance computing because each subdomain problem can be naturally solved in parallel at every time step.

58 GEOSCIENCES↗
DOE OSTI2020

Depletion capabilities in the OpenMC Monte Carlo particle transport code

A depletion solver has been implemented in OpenMC and is described herein. The depletion solver is implemented in Python and interfaces with OpenMC’s transport solver through a C++ application programming interface, which enables an in-memory transport-depletion coupling. Multiple integration methods for advancing in time have been implemented and exhibit tradeoffs in cost, accuracy, and memory use. For all time integration methods, evaluation of the matrix exponential is performed by using the incomplete partial fraction form of the Chebyshev rational approximation method. Simulations of a pressurized water reactor (PWR) pincell and a sodium-cooled fast reactor (SFR) assembly were carried out with OpenMC and Serpent. For both problems, the use of a high-fidelity depletion chain results in predictions of k eff that agree within 20–30 pcm between OpenMC and Serpent. Predicted actinide concentrations were found to agree to a fraction of a percent, and most fission product concentrations were found to agree within 1%. Here, the few cases where larger differences were observed can be attributed either to differences in how the energy dependence of fission product yields is handled or deficiencies in the nuclear data used. OpenMC simulations of the PWR and SFR problems using a simplified 228-nuclide depletion chain demonstrate that it achieves accuracy close to that of the full, high-fidelity depletion chain with respect to the studied responses.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Magnus method for electronic structure calculations at extreme conditions

We present the application of Magnus based methods to the solution of first order coupled ordinary differential equations in High Energy Density (HED) physics applications. Our focus is on the application to quantum mechanical methods, specifically on the solution of the radial Dirac equation for real and complex energies. HED applications require accurate solutions across a wide range of spatial and energy domains, including regimes where the solutions exhibit pronounced oscillatory behavior. Such cases pose significant computational challenges. We demonstrate that Magnus-based integrators can efficiently and accurately address these challenges. We discuss the implementation of the Magnus method for the solution of the radial Dirac equation, including practical considerations such as the evaluation of matrix exponentials, numerical integration, error estimation, and adaptive step size control. We also discuss the application of these methods to complex energy Green’s function techniques and the efficient approximation of integrals of the solutions relevant to HED electronic structure calculations. Here, we demonstrate the accuracy and robustness of the resulting method in applications to the free-particle case, for which analytic solutions are available for comparison, as well as the challenging case of gold at HED conditions.

general physics↗
DOE OSTI2019

Reaction Dynamics of Cyanohydrins with Hydrosulfide in Water

We studied the reaction dynamics of a proposed prebiotic reaction theoretically. The chemical process involves acetone cyanohydrin or formalcyanohydrin reacting with hydrosulfide in an aqueous environment. Rate constants and populations of reactant and product bimolecular geometric orientations for the reactions were obtained by using density functional theory for the energies, transition-state theory for the rates, and matrix exponentiation as well as the hybrid tau-leaping algorithm for the population dynamics. The role of including the solvent explicitly versus implicitly was also investigated. We found that adding explicit water or hydrogen sulfide molecules lowers the activation energy barrier and leads to a more efficient reaction pathway. In particular, hydrogen sulfide was a better proton donor. Lastly, we studied the role of including more than one reactant and product bimolecular orientation geometry in the dynamics. Including all initial pathways, reactant to reactant, product to product, reactant to product, and product to reactant led to a larger reaction rate constant compared to the single minimum energy pathway approach. Overall, we found that most reactions which involve formalcyanohydrin occur more rapidly or at the same speed as reactions which involve acetone cyanohydrin at room temperature.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Modified Newton-Raphson GRAPE methods for optimal control of spin systems

Quadratic convergence throughout the active space is achieved for the gradient ascent pulse engineering (GRAPE) family of quantum optimal control algorithms. We demonstrate in this communication that the Hessian of the GRAPE fidelity functional is unusually cheap, having the same asymptotic complexity scaling as the functional itself. This leads to the possibility of using very efficient numerical optimization techniques. In particular, the Newton-Raphson method with a rational function optimization (RFO) regularized Hessian is shown in this work to require fewer system trajectory evaluations than any other algorithm in the GRAPE family. This communication describes algebraic and numerical implementation aspects (matrix exponential recycling, Hessian regularization, etc.) for the RFO Newton-Raphson version of GRAPE and reports benchmarks for common spin state control problems in magnetic resonance spectroscopy.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗