Learning Preconditioner for Conjugate Gradient PDE Solvers

Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given precision level. One challenge in PCG solvers is the selection of preconditioners, as different problem-dependent systems can benefit from different preconditioners. We present a new method to introduce \emph{inductive bias} in preconditioning conjugate gradient algorithm. Given a system matrix and a set of solution vectors arise from an underlying distribution, we train a graph neural network to obtain an approximate decomposition to the system matrix to be used as a preconditioner in the context of PCG solvers. We conduct extensive experiments to demonstrate the efficacy and generalizability of our proposed approach in solving various 2D and 3D linear second-order PDEs

Neural incomplete factorization: learning preconditioners for the conjugate gradient method

Finding suitable preconditioners to accelerate iterative solution methods, such as the conjugate gradient method, is an active area of research. In this paper, we develop a computationally efficient data-driven approach to replace the typically hand-engineered algorithms with neural networks. Optimizing the condition number of the linear system directly is computationally infeasible. Instead, our method generates an incomplete factorization of the matrix and is, therefore, referred to as neural incomplete factorization (NeuralIF). For efficient training, we utilize a stochastic approximation of the Frobenius loss which only requires matrix-vector multiplications. At the core of our method is a novel messagepassing block, inspired by sparse matrix theory, that aligns with the objective of finding a sparse factorization of the matrix. By replacing conventional preconditioners used within the conjugate gradient method by data-driven models based on graph neural networks, we accelerate the iterative solving procedure. We evaluate our proposed method on both a synthetic and a real-world problem arising from scientific computing and show its ability to reduce the solving time while remaining computationally efficient.Comment: Under review. 18 pages, 8 figure

Implementation of the conjugate gradient algorithm for heterogeneous systems

Lattice QCD calculations require significant computational effort, with the dominant fraction of resources typically spent in the numerical inversion of the Dirac operator. One of the simplest methods to solve such large and sparse linear systems is the conjugate gradient (CG) approach. In this work we present an implementation of CG that can be executed on different devices, including CPUs, GPUs, and FPGAs. This is achieved by using the SYCL/DPC++ framework, which allows the execution of the same source code on heterogeneous systems

Implementation of the conjugate gradient algorithm for heterogeneous systems

Lattice QCD calculations require significant computational effort, with the dominant fraction of resources typically spent in the numerical inversion of the Dirac operator. One of the simplest methods to solve such large and sparse linear systems is the conjugate gradient (CG) approach. In this work we present an implementation of CG that can be executed on different devices, including CPUs, GPUs, and FPGAs. This is achieved by using the SYCL/DPC++ framework, which allows the execution of the same source code on heterogeneous systems

Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest

Numerical evaluation of aerodynamic roughness of the built environment and complex terrain

Aerodynamic drag in the atmospheric boundary layer (ABL) is affected by the structure and density of obstacles (surface roughness) and nature of the terrain (topography). In building codes and standards, average roughness is usually determined somewhat subjectively by examination of aerial photographs. For detailed wind mapping, boundary layer wind tunnel (BLWT) testing is usually recommended. This may not be cost effective for many projects, in which case numerical studies become good alternatives. This thesis examines Computational Fluid Dynamics (CFD) for evaluation of aerodynamic roughness of the built environment and complex terrain. The present study started from development of an in-house CFD software tailored for ABL simulations. A three-dimensional finite-volume code was developed using flexible polyhedral elements as building blocks. The program is parallelized using MPI to run on clusters of processors so that micro-scale simulations can be conducted quickly. The program can also utilize the power of latest technology in high performance computing, namely GPUs. Various turbulence models including mixing-length, RANS, and LES models are implemented, and their suitability for ABL simulations assessed. Then the effect of surface roughness alone on wind profiles is assessed using CFD. Cases with various levels of complexity are considered including simplified models with roughness blocks of different arrangement, multiple roughness patches, semi-idealized urban model, and real built environment. Comparison with BLWT data for the first three cases showed good agreement thereby justifying explicit three-dimensional numerical approach. Due to lack of validation data, the real built environment case served only to demonstrate use of CFD for such purposes. Finally, the effect of topographic features on wind profiles was investigated using CFD. This work extends prior work done by the research team on multiple idealized two-dimensional topographic features to more elaborate three-dimensional simulations. It is found that two-dimensional simulations overestimate speed up over crests of hills and also show larger recirculation zones. The current study also emphasized turbulence characterization behind hills. Finally a real complex terrain case of the well-known Askervein hill was simulated and the results validated against published field observations. In general the results obtained from the current simulations compared well with those reported in literature

Computer Science Research Institute 2004 annual report of activities.

This report summarizes the activities of the Computer Science Research Institute (CSRI) at Sandia National Laboratories during the period January 1, 2004 to December 31, 2004. During this period the CSRI hosted 166 visitors representing 81 universities, companies and laboratories. Of these 65 were summer students or faculty. The CSRI partially sponsored 2 workshops and also organized and was the primary host for 4 workshops. These 4 CSRI sponsored workshops had 140 participants--74 from universities, companies and laboratories, and 66 from Sandia. Finally, the CSRI sponsored 14 long-term collaborative research projects and 5 Sabbaticals

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal