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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

HyKKT

HyKKT (pronounced as "hiked") is a package for solving systems of linear equations of Karush-Kuhn-Tucker (KKT) form, which typically arise in optimization problems, such as optimal power flow analysis. HyKKT uses Cholesky instead of LDL^T factorization and solves the general KKT system to a desired numerical precision via block reduction and conjugate gradient on the Schur complement. Such implementation is more suitable for implementation on graphic processing units (GPUs).

Regev, Shaked↗
DOE OSTI2021

Scalable Techniques for Stochastic Power Flow Problems (Final Report)

The proposed research focuses on developing scalable algorithms for two-stage security-constrained OPF problems with AC power flow constraints, a class of problems complicated by (i) scale arising from a scenario representation; and (ii) the presence of nonlinearity, nonconvexity, and possibly second-stage discreteness or complementarity. Unfortunately, most existing solvers cannot contend with both challenges simultaneously; accordingly, the proposed research focuses on developing solution techniques that can both scale with the number of scenarios and contend with nonconvexity and second-stage complementarity. We consider three avenues for addressing such problems: (i) Variable sample-size SQP (VS-SQP) methods that combine sparse Quasi-Newton updates with a scalable variance-reduced stochastic gradient scheme for stochastic QP subproblems, allowing for contending with second-stage complementarity via regularization; (ii) Variable sample-size stochastic Interior-point (VS-sIP) schemes that propose a sampling-based regularized (to allow for contending with complementarity) interior-point schemes in which a Schur-complement technique is employed for decomposing the Newton direction computation step; (iii) Variable sample-size tractable ADMM (VS-tADMM) schemes combine variable sample-sizes with carefully designed techniques for resolving each of the nonconvex updates (by leveraging the QCQP structures). We intend to compare the three schemes using performance profiles in terms of solution quality, scalability, etc. and then select one scheme which will then be developed and further refined in Python for purposes of the GO competition.

42 ENGINEERING↗
NASA NTRS1991

A natural partitioning scheme for parallel simulation of multibody systems

A parallel partitioning scheme based on physical-coordinate variables is presented to systematically eliminate system constraint forces and yield the equations of motion of multibody dynamics systems in terms of their independent coordinates. Key features of the present scheme include an explicit determination of the independent coordinates, a parallel construction of the null space matrix of the constraint Jacobian matrix, an easy incorporation of the previously developed two-stage staggered solution procedure, and Schur complement based parallel preconditioned conjugate gradient numerical algorithm.

Chiou, J. C.↗
NASA NTRS1993

A natural partitioning scheme for parallel simulation of multibody systems

A parallel partitioning scheme based on physical-co-ordinate variables is presented to systematically eliminate system constraint forces and yield the equations of motion of multibody dynamics systems in terms of their independent coordinates. Key features of the present scheme include an explicit determination of the independent coordinates, a parallel construction of the null space matrix of the constraint Jacobian matrix, an easy incorporation of the previously developed two-stage staggered solution procedure and a Schur complement based parallel preconditioned conjugate gradient numerical algorithm.

Chiou, J. C.↗
NASA NTRS1993

Parallel O(log n) algorithms for open- and closed-chain rigid multibody systems based on a new mass matrix factorization technique

In this paper, parallel O(log n) algorithms for computation of rigid multibody dynamics are developed. These parallel algorithms are derived by parallelization of new O(n) algorithms for the problem. The underlying feature of these O(n) algorithms is a drastically different strategy for decomposition of interbody force which leads to a new factorization of the mass matrix (M). Specifically, it is shown that a factorization of the inverse of the mass matrix in the form of the Schur Complement is derived as M(exp -1) = C - B(exp *)A(exp -1)B, wherein matrices C, A, and B are block tridiagonal matrices. The new O(n) algorithm is then derived as a recursive implementation of this factorization of M(exp -1). For the closed-chain systems, similar factorizations and O(n) algorithms for computation of Operational Space Mass Matrix lambda and its inverse lambda(exp -1) are also derived. It is shown that these O(n) algorithms are strictly parallel, that is, they are less efficient than other algorithms for serial computation of the problem. But, to our knowledge, they are the only known algorithms that can be parallelized and that lead to both time- and processor-optimal parallel algorithms for the problem, i.e., parallel O(log n) algorithms with O(n) processors. The developed parallel algorithms, in addition to their theoretical significance, are also practical from an implementation point of view due to their simple architectural requirements.

Fijany, Amir↗
NASA NTRS1996

A Note on Substructuring Preconditioning for Nonconforming Finite Element Approximations of Second Order Elliptic Problems

In this paper an algebraic substructuring preconditioner is considered for nonconforming finite element approximations of second order elliptic problems in 3D domains with a piecewise constant diffusion coefficient. Using a substructuring idea and a block Gauss elimination, part of the unknowns is eliminated and the Schur complement obtained is preconditioned by a spectrally equivalent very sparse matrix. In the case of quasiuniform tetrahedral mesh an appropriate algebraic multigrid solver can be used to solve the problem with this matrix. Explicit estimates of condition numbers and implementation algorithms are established for the constructed preconditioner. It is shown that the condition number of the preconditioned matrix does not depend on either the mesh step size or the jump of the coefficient. Finally, numerical experiments are presented to illustrate the theory being developed.

Maliassov, Serguei↗
NASA NTRS1996

A New Coarsening Operator for the Optimal Preconditioning of the Dual and Primal Domain Decomposition Methods: Application to Problems with Severe Coefficient Jumps

We present an optimal preconditioning algorithm that is equally applicable to the dual (FETI) and primal (Balancing) Schur complement domain decomposition methods, and which successfully addresses the problems of subdomain heterogeneities including the effects of large jumps of coefficients. The proposed preconditioner is derived from energy principles and embeds a new coarsening operator that propagates the error globally and accelerates convergence. The resulting iterative solver is illustrated with the solution of highly heterogeneous elasticity problems.

Farhat, Charbel↗
NASA NTRS1995

Parallel CFD Algorithms for Aerodynamical Flow Solvers on Unstructured Meshes

The Advisory Group for Aerospace Research and Development (AGARD) has requested my participation in the lecture series entitled Parallel Computing in Computational Fluid Dynamics to be held at the von Karman Institute in Brussels, Belgium on May 15-19, 1995. In addition, a request has been made from the US Coordinator for AGARD at the Pentagon for NASA Ames to hold a repetition of the lecture series on October 16-20, 1995. I have been asked to be a local coordinator for the Ames event. All AGARD lecture series events have attendance limited to NATO allied countries. A brief of the lecture series is provided in the attached enclosure. Specifically, I have been asked to give two lectures of approximately 75 minutes each on the subject of parallel solution techniques for the fluid flow equations on unstructured meshes. The title of my lectures is "Parallel CFD Algorithms for Aerodynamical Flow Solvers on Unstructured Meshes" (Parts I-II). The contents of these lectures will be largely review in nature and will draw upon previously published work in this area. Topics of my lectures will include: (1) Mesh partitioning algorithms. Recursive techniques based on coordinate bisection, Cuthill-McKee level structures, and spectral bisection. (2) Newton's method for large scale CFD problems. Size and complexity estimates for Newton's method, modifications for insuring global convergence. (3) Techniques for constructing the Jacobian matrix. Analytic and numerical techniques for Jacobian matrix-vector products, constructing the transposed matrix, extensions to optimization and homotopy theories. (4) Iterative solution algorithms. Practical experience with GIVIRES and BICG-STAB matrix solvers. (5) Parallel matrix preconditioning. Incomplete Lower-Upper (ILU) factorization, domain-decomposed ILU, approximate Schur complement strategies.

Barth, Timothy J.↗
NASA NTRS1997

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Several stabilized discretization procedures for conservation law equations on triangulated domains will be considered. Specifically, numerical schemes based on upwind finite volume, fluctuation splitting, Galerkin least-squares, and space discontinuous Galerkin discretization will be considered in detail. A standard energy analysis for several of these methods will be given via entropy symmetrization. Next, we will present some relatively new theoretical results concerning congruence relationships for left or right symmetrized equations. These results suggest new variants of existing FV, DG, GLS and FS methods which are computationally more efficient while retaining the pleasant theoretical properties achieved by entropy symmetrization. In addition, the task of Jacobian linearization of these schemes for use in Newton's method is greatly simplified owing to exploitation of exact symmetries which exist in the system. These variants have been implemented in the "ELF" library for which example calculations will be shown. The FV, FS and DG schemes also permit discrete maximum principle analysis and enforcement which greatly adds to the robustness of the methods. Some prevalent limiting strategies will be reviewed. Next, we consider embedding these nonlinear space discretizations into exact and inexact Newton solvers which are preconditioned using a nonoverlapping (Schur complement) domain decomposition technique. Elements of nonoverlapping domain decomposition for elliptic problems will be reviewed followed by the present extension to hyperbolic and elliptic-hyperbolic problems. Other issues of practical relevance such the meshing of geometries, code implementation, turbulence modeling, global convergence, etc. will be addressed as needed.

Barth, Timothy↗
NASA NTRS1998

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Several stabilized demoralization procedures for conservation law equations on triangulated domains will be considered. Specifically, numerical schemes based on upwind finite volume, fluctuation splitting, Galerkin least-squares, and space discontinuous Galerkin demoralization will be considered in detail. A standard energy analysis for several of these methods will be given via entropy symmetrization. Next, we will present some relatively new theoretical results concerning congruence relationships for left or right symmetrized equations. These results suggest new variants of existing FV, DG, GLS, and FS methods which are computationally more efficient while retaining the pleasant theoretical properties achieved by entropy symmetrization. In addition, the task of Jacobean linearization of these schemes for use in Newton's method is greatly simplified owing to exploitation of exact symmetries which exist in the system. The FV, FS and DG schemes also permit discrete maximum principle analysis and enforcement which greatly adds to the robustness of the methods. Discrete maximum principle theory will be presented for general finite volume approximations on unstructured meshes. Next, we consider embedding these nonlinear space discretizations into exact and inexact Newton solvers which are preconditioned using a nonoverlapping (Schur complement) domain decomposition technique. Elements of nonoverlapping domain decomposition for elliptic problems will be reviewed followed by the present extension to hyperbolic and elliptic-hyperbolic problems. Other issues of practical relevance such the meshing of geometries, code implementation, turbulence modeling, global convergence, etc, will. be addressed as needed.

Barth, Timothy J.↗
DOE OSTI2023

A scalable domain decomposition method for FEM discretizations of nonlocal equations of integrable and fractional type

Nonlocal models allow for the description of phenomena which cannot be captured by classical partial differential equations. The availability of efficient solvers is one of the main concerns for the use of nonlocal models in real world engineering applications. Here, we present a domain decomposition solver that is inspired by substructuring methods for classical local equations. In numerical experiments involving finite element discretizations of scalar and vectorial nonlocal equations of integrable and fractional type, we observe improvements in solution time of up to 14.6x compared to commonly used solver strategies.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A two-level GPU-accelerated incomplete LU preconditioner for general sparse linear systems

This paper presents a parallel preconditioning approach based on incomplete LU (ILU) factorizations in the framework of Domain Decomposition (DD) for general sparse linear systems. We focus on distributed memory parallel architectures, specifically, those that are equipped with graphic processing units (GPUs). In addition to block-Jacobi, we present general purpose two-level ILU Schur complement-based approaches, where different strategies are presented to solve the coarse-level reduced system. These strategies are combined with modified ILU methods in the construction of the coarse-level operator, in order to effectively remove smooth errors by targeting an algebraically smooth vector. We leverage available GPU-based sparse matrix kernels to accelerate the setup and the solve phases of the proposed ILU preconditioner. We evaluate the efficiency of the proposed methods as a smoother for algebraic multigrid (AMG) and as a preconditioner for Krylov subspace methods on challenging anisotropic diffusion problems and a collection of general sparse matrices.

97 MATHEMATICS AND COMPUTING↗