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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Enhancing scalability and accuracy of quantum poisson solver

The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far either suffer from lack of accuracy and/or are limited to very small sizes of the problem and thus have no practical usage. In this regard, our previous work showed a proof-of-concept demonstration in advancing quantum Poisson solver algorithm and validated preliminary results for a simple case of 3 x 3 problem. In this work, we delve into comprehensive research details, presenting the results on up to 15 x 15 problems that include step-by-step improvements in Poisson equation solutions, scaling performance, and experimental exploration. In particular, we demonstrate the implementation of eigenvalue amplification by a factor of up to 2 8 , achieving a significant improvement in the accuracy of our quantum Poisson solver and comparing that to the exact solution. Additionally, we present success probability results, highlighting the reliability of our quantum Poisson solver. Moreover, we explore the scaling performance of our algorithm against the circuit depth and width, demonstrating how our approach scales with larger problem sizes and thus further solidifies the practicality of easy adaptation of this algorithm in real-world applications. We also discuss a multilevel strategy for how this algorithm might be further improved to explore much larger problems with greater performance. Finally, through our experiments on the IBM quantum hardware, we conclude that though overall results on the existing NISQ hardware are dominated by the error in the CNOT gates, this work opens a path to realizing a multidimensional Poisson solver on near-term quantum hardware.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Advanced Quantum Poisson Solver in the NISQ era

The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far, either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to the linear systems through the finite difference method, we adopt the Harrow-Hassidim-Lloyd (HHL) algorithm as the basic framework. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues through eigenvalue amplification as well as by increasing the accuracy of the controlled rotation angular coefficients, which are the critical factors in the HHL algorithm. We show that our algorithm not only increases the accuracy of the solutions, but also composes more practical and scalable circuits by dynamically controlling problem size in the NISQ devices. We present both simulated and experimental results, and discuss the sources of errors. Finally, we conclude that overall results on the quantum hardware are dominated by the error in the CNOT gates.

Robson, Walter↗
NASA NTRS1990

Multitasking domain decomposition fast Poisson solvers on the Cray Y-MP

The results of multitasking implementation of a domain decomposition fast Poisson solver on eight processors of the Cray Y-MP are presented. The object of this research is to study the performance of domain decomposition methods on a Cray supercomputer and to analyze the performance of different multitasking techniques using highly parallel algorithms. Two implementations of multitasking are considered: macrotasking (parallelism at the subroutine level) and microtasking (parallelism at the do-loop level). A conventional FFT-based fast Poisson solver is also multitasked. The results of different implementations are compared and analyzed. A speedup of over 7.4 on the Cray Y-MP running in a dedicated environment is achieved for all cases.

Chan, Tony F.↗
NASA NTRS1993

A mixed method Poisson solver for three-dimensional self-gravitating astrophysical fluid dynamical systems

A key ingredient in the simulation of self-gravitating astrophysical fluid dynamical systems is the gravitational potential and its gradient. This paper focuses on the development of a mixed method multigrid solver of the Poisson equation formulated so that both the potential and the Cartesian components of its gradient are self-consistently and accurately generated. The method achieves this goal by formulating the problem as a system of four equations for the gravitational potential and the three Cartesian components of the gradient and solves them using a distributed relaxation technique combined with conventional full multigrid V-cycles. The method is described, some tests are presented, and the accuracy of the method is assessed. We also describe how the method has been incorporated into our three-dimensional hydrodynamics code and give an example of an application to the collision of two stars. We end with some remarks about the future developments of the method and some of the applications in which it will be used in astrophysics.

Duncan, Comer↗
NASA NTRS1995

A fast Poisson solver for unsteady incompressible Navier-Stokes equations on the half-staggered grid

In this paper, a fast Poisson solver for unsteady, incompressible Navier-Stokes equations with finite difference methods on the non-uniform, half-staggered grid is presented. To achieve this, new algorithms for diagonalizing a semi-definite pair are developed. Our fast solver can also be extended to the three dimensional case. The motivation and related issues in using this second kind of staggered grid are also discussed. Numerical testing has indicated the effectiveness of this algorithm.

Golub, G. H.↗
NASA NTRS1984

Vectorized multigrid Poisson solver for the CDC CYBER 205

The full multigrid (FMG) method is applied to the two dimensional Poisson equation with Dirichlet boundary conditions. This has been chosen as a relatively simple test case for examining the efficiency of fully vectorizing of the multigrid method. Data structure and programming considerations and techniques are discussed, accompanied by performance details.

Barkai, D.↗
NASA NTRS1994

An efficient three-dimensional Poisson solver for SIMD high-performance-computing architectures

We present an algorithm that solves the three-dimensional Poisson equation on a cylindrical grid. The technique uses a finite-difference scheme with operator splitting. This splitting maps the banded structure of the operator matrix into a two-dimensional set of tridiagonal matrices, which are then solved in parallel. Our algorithm couples FFT techniques with the well-known ADI (Alternating Direction Implicit) method for solving Elliptic PDE's, and the implementation is extremely well suited for a massively parallel environment like the SIMD architecture of the MasPar MP-1. Due to the highly recursive nature of our problem, we believe that our method is highly efficient, as it avoids excessive interprocessor communication.

Cohl, H.↗
DOE OSTI2024

A numerical Poisson solver with improved radial solutions for a self-consistent locally scaled self-interaction correction method

Abstract The universal applicability of density functional approximations is limited by self-interaction error made by these functionals. Recently, a novel one-electron self-interaction-correction (SIC) method that uses an iso-orbital indicator to apply the SIC at each point in space by scaling the exchange-correlation and Coulomb energy densities was proposed. The locally scaled SIC (LSIC) method is exact for the one-electron densities, and unlike the well-known Perdew–Zunger SIC (PZSIC) method recovers the uniform electron gas limit of the uncorrected density functional approximation, and reduces to PZSIC method as a special case when isoorbital indicator is set to the unity. Here, we present a numerical scheme that we have adopted to evaluate the Coulomb potential of the electron density scaled by the iso-orbital indicator required for the self-consistent LSIC calculations. After analyzing the behavior of the finite difference method (FDM) and the green function solution to the radial part of the Poisson equation, we adopt a hybrid approach that uses the FDM for the Coulomb potential due to the monopole and the GF for all higher-order terms. The performance of the resultant hybrid method is assessed using a variety of systems. The results show improved accuracy than earlier numerical schemes. We also find that, even with a generic set of radial grid parameters, accurate energy differences can be obtained using a numerical Coulomb solver in standard density functional studies.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

An Infinite Domain 3D Poisson Solver Based on the Barnes-Hut Algorithm

We present a domain decomposition method for the solution of the 3D Poisson equation with infinite domain boundary conditions. The method is based on an application of the Barnes-Hut tree particle scheme adapted to gridded data. Long range interactions are computed using the first two terms in the Cartesian multipole expansion of Green’s function convoluted with the charge while short range computations are performed using Hockney’s domain doubling algorithm. A standard domain decomposition strategy requires O(N 2 ) applications of Hockney’s algorithm, where N is the number of subdomains that intersect that charge support, while in the present approach only O(Nlog 2 N) such computations suffice. The discretization scheme employed is a sixth order Mehrstellen approximation of the 3D Laplace opera tor. The method exhibits satisfactory accuracy at a substantially reduced computational cost compared to the full domain decomposition Hockney’s algorithm.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1988

A vectorized Poisson solver over a spherical shell and its application to the quasi-geostrophic omega-equation

The quasi-geostrophic omega-equation in flux form is developed as an example of a Poisson problem over a spherical shell. Solutions of this equation are obtained by applying a two-parameter Chebyshev solver in vector layout for CDC 200 series computers. The performance of this vectorized algorithm greatly exceeds the performance of its scalar analog. The algorithm generates solutions of the omega-equation which are compared with the omega fields calculated with the aid of the mass continuity equation.

Mullenmeister, Paul↗
NASA NTRS1986

An exterior Poisson solver using fast direct methods and boundary integral equations with applications to nonlinear potential flow

A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson's equation on irregular exterior regions. The method requires O(N log N) operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.

Young, D. P.↗
NASA NTRS1990

Theory of multicolor lattice gas - A cellular automaton Poisson solver

The present class of models for cellular automata involving a quiescent hydrodynamic lattice gas with multiple-valued passive labels termed 'colors', the lattice collisions change individual particle colors while preserving net color. The rigorous proofs of the multicolor lattice gases' essential features are rendered more tractable by an equivalent subparticle representation in which the color is represented by underlying two-state 'spins'. Schemes for the introduction of Dirichlet and Neumann boundary conditions are described, and two illustrative numerical test cases are used to verify the theory. The lattice gas model is equivalent to a Poisson equation solution.

Chen, H.↗
DOE OSTI2021

Gravitational Self-force Errors of Poisson Solvers on Adaptively Refined Meshes

An error in the gravitational force that the source of gravity induces on itself (a self-force error) violates both the conservation of linear momentum and the conservation of energy. If such errors are present in a self-gravitating system and are not sufficiently random to average out, the obtained numerical solution will become progressively more unphysical with time: the system will acquire or lose momentum and energy due to numerical effects. In this paper, we demonstrate how self-force errors can arise in the case where self-gravity is solved on an adaptively refined mesh when the refinement is nonuniform. Here, we provide the analytical expression for the self-force error and numerical examples that demonstrate such self-force errors in idealized settings. We also show how these errors can be corrected to an arbitrary order by straightforward addition of correction terms at the refinement boundaries.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
NASA NTRS1986

Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. 1: One-step method

An algorithm for solving a large class of two- and three-dimensional nonseparable elliptic partial differential equations (PDE's) is developed and tested. It uses a modified D'Yakanov-Gunn iterative procedure in which the relaxation factor is grid-point dependent. It is easy to implement and applicable to a variety of boundary conditions. It is also computationally efficient, as indicated by the results of numerical comparisons with other established methods. Furthermore, the current algorithm has the advantage of possessing two important properties which the traditional iterative methods lack; that is: (1) the convergence rate is relatively insensitive to grid-cell size and aspect ratio, and (2) the convergence rate can be easily estimated by using the coefficient of the PDE being solved.

Chang, S. C.↗
NASA NTRS1986

Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. 2: Two-step method

A two-step semidirect procedure is developed to accelerate the one-step procedure described in NASA TP-2529. For a set of constant coefficient model problems, the acceleration factor increases from 1 to 2 as the one-step procedure convergence rate decreases from + infinity to 0. It is also shown numerically that the two-step procedure can substantially accelerate the convergence of the numerical solution of many partial differential equations (PDE's) with variable coefficients.

Chang, S. C.↗
NASA NTRS1986

Solution of elliptic PDEs by fast Poisson solvers using a local relaxation factor

A large class of two- and three-dimensional, nonseparable elliptic partial differential equations (PDEs) is presently solved by means of novel one-step (D'Yakanov-Gunn) and two-step (accelerated one-step) iterative procedures, using a local, discrete Fourier analysis. In addition to being easily implemented and applicable to a variety of boundary conditions, these procedures are found to be computationally efficient on the basis of the results of numerical comparison with other established methods, which lack the present one's: (1) insensitivity to grid cell size and aspect ratio, and (2) ease of convergence rate estimation by means of the coefficient of the PDE being solved. The two-step procedure is numerically demonstrated to outperform the one-step procedure in the case of PDEs with variable coefficients.

Chang, Sin-Chung↗
NASA NTRS1994

Nonlinear hydrodynamics of cosmological sheets. 1: Numerical techniques and tests

We present the numerical techniques and tests used to construct and validate a computer code designed to study the multidimensional nonlinear hydrodynamics of large-scale sheet structures in the universe, especially the fragmentation of such structures under various instabilities. This code is composed of two codes, the hydrodynamical code ZEUS-2D and a particle-mesh code. The ZEUS-2D code solves the hydrodynamical equations in two dimensions using explicit Eulerian finite-difference techniques, with modifications made to incorporate the expansion of the universe and the gas cooling due to Compton scattering, bremsstrahlung, and hydrogen and helium cooling. The particle-mesh code solves the equation of motion for the collisionless dark matter. The code uses two-dimensional Cartesian coordinates with a nonuniform grid in one direction to provide high resolution for the sheet structures. A series of one-dimensional and two-dimensional linear perturbation tests are presented which are designed to test the hydro solver and the Poisson solver with and without the expansion of the universe. We also present a radiative shock wave test which is designed to ensure the code's capability to handle radiative cooling properly. And finally a series of one-dimensional Zel'dovich pancake tests used to test the dark matter code and the hydro solver in the nonlinear regime are discussed and compared with the results of Bond et al. (1984) and Shapiro & Struck-Marcell (1985). Overall, the code is shown to produce accurate and stable results, which provide us a powerful tool to further our studies.

Anninos, Wenbo Y.↗