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DOE OSTI2023

Sparse regression for plasma physics

Many scientific problems can be formulated as sparse regression, i.e., regression onto a set of parameters when there is a desire or expectation that some of the parameters are exactly zero or do not substantially contribute. This includes many problems in signal and image processing, system identification, optimization, and parameter estimation methods such as Gaussian process regression. Sparsity facilitates exploring high-dimensional spaces while finding parsimonious and interpretable solutions. In the present work, we illustrate some of the important ways in which sparse regression appears in plasma physics and point out recent contributions and remaining challenges to solving these problems in this field. Further, a brief review is provided for the optimization problem and the state-of-the-art solvers, especially for constrained and high-dimensional sparse regression.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2011

S-Preconditioner for Multi-fold Data Reduction with Guaranteed User-Controlled Accuracy

The growing gap between the massive amounts of data generated by petascale scientific simulation codes and the capability of system hardware and software to effectively analyze this data necessitates data reduction. Yet, the increasing data complexity challenges most, if not all, of the existing data compression methods. In fact, lossless compression techniques offer no more than 10% reduction on scientific data that we have experience with, which is widely regarded as effectively incompressible. To bridge this gap, in this paper, we advocate a transformative strategy that enables fast, accurate, and multi-fold reduction of double-precision floating-point scientific data. The intuition behind our method is inspired by an effective use of preconditioners for linear algebra solvers optimized for a particular class of computational dwarfs (e.g., dense or sparse matrices). Focusing on a commonly used multi-resolution wavelet compression technique as the underlying solver for data reduction we propose the S-preconditioner, which transforms scientific data into a form with high global regularity to ensure a significant decrease in the number of wavelet coefficients stored for a segment of data. Combined with the subsequent EQ-calibrator, our resultant method (called S-Preconditioned EQ-Calibrated Wavelets (SPEQC-WAVELETS)), robustly achieved a 4- to 5- fold data reduction while guaranteeing user-defined accuracy of reconstructed data to be within 1% point-by-point relative error, lower than 0:01 Normalized RMSE, and higher than 0:99 Pearson Correlation. In this paper, we show the results we obtained by testing our method on six petascale simulation codes including fusion, combustion, climate, astrophysics, and subsurface groundwater in addition to 13 publicly available scientific datasets. We also demonstrate that application-driven data mining tasks performed on decompressed variables or their derived quantities produce results of comparable quality with the ones for the original data.

Jin, Ye↗

DOE OSTI2024

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ-approximate inverse Cholesky factors of the kernel matrices with complexity O(N log d (N/ϵ)) in space and O(N log 2d (N/ϵ)) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Denoising Poisson phaseless measurements via orthogonal dictionary learning

Phaseless diffraction measurements recorded by CCD detectors are often affected by Poisson noise. In this paper, we propose a dictionary learning model by employing patches based sparsity in order to denoise such Poisson phaseless measurements. The model consists of three terms: (i) A representation term by an orthogonal dictionary, (ii) an L 0 pseudo norm of the coefficient matrix, and (iii) a Kullback-Leibler divergence term to fit phaseless Poisson data. Fast alternating minimization method (AMM) and proximal alternating linearized minimization (PALM) are adopted to solve the proposed model, and especially the theoretical guarantee of the convergence of PALM is provided. The subproblems for these two algorithms both have fast solvers, and indeed, the solutions for the sparse coding and dictionary updating both have closed forms due to the orthogonality of learned dictionaries. Numerical experiments for phase retrieval using coded diffraction and ptychographic patterns are conducted to show the efficiency and robustness of proposed methods, which, by preserving texture features, produce visually and quantitatively improved restored images compared with other phase retrieval algorithms without regularization and local sparsity promoting algorithms.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Towards Efficient Alternating Current Optimal Power Flow Analysis on Graphical Processing Units

We present a solution of sparse ACOPF analysis on GPU. In particular, we discuss the performance bottlenecks and detail our efforts to accelerate the linear solver, a core component of ACOPF that dominates the computational time. ACOPF solutions of two large-scale systems, synthetic Northeast (25,000 buses) and Eastern (70,000 buses) \cite{birchfield2017tamu-cases} on GPU show promising speed-up compared to CPU based solution using a state-of-the-art solver. To our knowledge, this is the first result demonstrating acceleration of sparse ACOPF on GPUs.

Power grid analysis, GPU↗

DOE OSTI2015

Acceleration of GPU-based Krylov solvers via data transfer reduction

Krylov subspace iterative solvers are often the method of choice when solving large sparse linear systems. At the same time, hardware accelerators such as graphics processing units continue to offer significant floating point performance gains for matrix and vector computations through easy-to-use libraries of computational kernels. However, as these libraries are usually composed of a well optimized but limited set of linear algebra operations, applications that use them often fail to reduce certain data communications, and hence fail to leverage the full potential of the accelerator. In this study, we target the acceleration of Krylov subspace iterative methods for graphics processing units, and in particular the Biconjugate Gradient Stabilized solver that significant improvement can be achieved by reformulating the method to reduce data-communications through application-specific kernels instead of using the generic BLAS kernels, e.g. as provided by NVIDIA’s cuBLAS library, and by designing a graphics processing unit specific sparse matrix-vector product kernel that is able to more efficiently use the graphics processing unit’s computing power. Furthermore, we derive a model estimating the performance improvement, and use experimental data to validate the expected runtime savings. Finally, considering that the derived implementation achieves significantly higher performance, we assert that similar optimizations addressing algorithm structure, as well as sparse matrix-vector, are crucial for the subsequent development of high-performance graphics processing units accelerated Krylov subspace iterative methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1990

High-performance equation solvers and their impact on finite element analysis

The role of equation solvers in modern structural analysis software is described. Direct and iterative equation solvers which exploit vectorization on modern high-performance computer systems are described and compared. The direct solvers are two Cholesky factorization methods. The first method utilizes a novel variable-band data storage format to achieve very high computation rates and the second method uses a sparse data storage format designed to reduce the number of operations. The iterative solvers are preconditioned conjugate gradient methods. Two different preconditioners are included; the first uses a diagonal matrix storage scheme to achieve high computation rates and the second requires a sparse data storage scheme and converges to the solution in fewer iterations that the first. The impact of using all of the equation solvers in a common structural analysis software system is demonstrated by solving several representative structural analysis problems.

Poole, Eugene L.↗

NASA NTRS1992

High-performance equation solvers and their impact on finite element analysis

The role of equation solvers in modern structural analysis software is described. Direct and iterative equation solvers which exploit vectorization on modern high-performance computer systems are described and compared. The direct solvers are two Cholesky factorization methods. The first method utilizes a novel variable-band data storage format to achieve very high computation rates and the second method uses a sparse data storage format designed to reduce the number od operations. The iterative solvers are preconditioned conjugate gradient methods. Two different preconditioners are included; the first uses a diagonal matrix storage scheme to achieve high computation rates and the second requires a sparse data storage scheme and converges to the solution in fewer iterations that the first. The impact of using all of the equation solvers in a common structural analysis software system is demonstrated by solving several representative structural analysis problems.

Poole, Eugene L.↗

NASA NTRS1993

Implementing abstract multigrid or multilevel methods

Multigrid methods can be formulated as an algorithm for an abstract problem that is independent of the partial differential equation, domain, and discretization method. In such an abstract setting, problems not arising from partial differential equations can be treated. A general theory exists for linear problems. The general theory was motivated by a series of abstract solvers (Madpack). The latest version was motivated by the theory. Madpack now allows for a wide variety of iterative and direct solvers, preconditioners, and interpolation and projection schemes, including user callback ones. It allows for sparse, dense, and stencil matrices. Mildly nonlinear problems can be handled. Also, there is a fast, multigrid Poisson solver (two and three dimensions). The type of solvers and design decisions (including language, data structures, external library support, and callbacks) are discussed. Based on the author's experiences with two versions of Madpack, a better approach is proposed. This is based on a mixed language formulation (C and FORTRAN + preprocessor). Reasons for not using FORTRAN, C, or C++ (individually) are given. Implementing the proposed strategy is not difficult.

Douglas, Craig C.↗

DOE OSTI2013

Revisiting Parallel Cyclic Reduction and Parallel Prefix-Based Algorithms for Block Tridiagonal System of Equations

Simulations that require solutions of block tridiagonal systems of equations rely on fast parallel solvers for runtime efficiency. Leading parallel solvers that are highly effective for general systems of equations, dense or sparse, are limited in scalability when applied to block tridiagonal systems. This paper presents scalability results as well as detailed analyses of two parallel solvers that exploit the special structure of block tridiagonal matrices to deliver superior performance, often by orders of magnitude. A rigorous analysis of their relative parallel runtimes is shown to reveal the existence of a critical block size that separates the parameter space spanned by the number of block rows, the block size and the processor count, into distinct regions that favor one or the other of the two solvers. Dependence of this critical block size on the above parameters as well as on machine-specific constants is established. These formal insights are supported by empirical results on up to 2,048 cores of a Cray XT4 system. To the best of our knowledge, this is the highest reported scalability for parallel block tridiagonal solvers to date.

Seal, Sudip K↗

NASA NTRS1990

A manual for PARTI runtime primitives

Primitives are presented that are designed to help users efficiently program irregular problems (e.g., unstructured mesh sweeps, sparse matrix codes, adaptive mesh partial differential equations solvers) on distributed memory machines. These primitives are also designed for use in compilers for distributed memory multiprocessors. Communications patterns are captured at runtime, and the appropriate send and receive messages are automatically generated.

Berryman, Harry↗

NASA NTRS1991

A manual for PARTI runtime primitives, revision 1

Das, Raja↗

DOE OSTI2014

Optimizing Sparse Matrix-Multiple Vectors Multiplication for Nuclear Configuration Interaction Calculations

Obtaining highly accurate predictions on the properties of light atomic nuclei using the configuration interaction (CI) approach requires computing a few extremal Eigen pairs of the many-body nuclear Hamiltonian matrix. In the Many-body Fermion Dynamics for nuclei (MFDn) code, a block Eigen solver is used for this purpose. Due to the large size of the sparse matrices involved, a significant fraction of the time spent on the Eigen value computations is associated with the multiplication of a sparse matrix (and the transpose of that matrix) with multiple vectors (SpMM and SpMM-T). Existing implementations of SpMM and SpMM-T significantly underperform expectations. Thus, in this paper, we present and analyze optimized implementations of SpMM and SpMM-T. We base our implementation on the compressed sparse blocks (CSB) matrix format and target systems with multi-core architectures. We develop a performance model that allows us to understand and estimate the performance characteristics of our SpMM kernel implementations, and demonstrate the efficiency of our implementation on a series of real-world matrices extracted from MFDn. In particular, we obtain 3-4 speedup on the requisite operations over good implementations based on the commonly used compressed sparse row (CSR) matrix format. The improvements in the SpMM kernel suggest we may attain roughly a 40% speed up in the overall execution time of the block Eigen solver used in MFDn.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

A Parallel Solver for Graph Laplacians

Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as problem sizes increase and single core speeds stagnate, parallelism is essential to solve such problems quickly. We present an unsmoothed aggregation multigrid method for solving graph Laplacians in a distributed memory setting. We introduce new parallel aggregation and low degree elimination algorithms targeted specifically at irregular degree graphs. These algorithms are expressed in terms of sparse matrix-vector products using generalized sum and product operations. This formulation is amenable to linear algebra using arbitrary distributions and allows us to operate on a 2D sparse matrix distribution, which is necessary for parallel scalability. Our solver outperforms the natural parallel extension of the current state of the art in an algorithmic comparison. We demonstrate scalability to 576 processes and graphs with up to 1.7 billion edges.

Konolige, Tristan↗

DOE OSTI2020

A matrix-free approach for finite-strain hyperelastic problems using geometric multigrid

This paper investigates matrix-free algorithms for problems in quasi-static finite-strain hyperelasticity. Iterative solvers with matrix-free operator evaluation have emerged as an attractive alternative to sparse matrices in the fluid dynamics and wave propagation communities because they significantly reduce the memory traffic, the limiting factor in classical finite element solvers. Specifically, we study different matrix-free realizations of the finite element tangent operator and determine whether generalized methods of incorporating complex constitutive behavior might be feasible. Furthermore, in order to improve the convergence behavior of iterative solvers, we also propose a method by which to construct level tangent operators and employ them to define a geometric multigrid preconditioner. Additionally, the performance of the matrix-free operator and the geometric multigrid preconditioner is compared to the matrix-based implementation with an algebraic multigrid (AMG) preconditioner on a single node for a representative numerical example of a heterogeneous hyperelastic material in two and three dimensions. We find that matrix-free methods for finite-strain solid mechanics are very promising, outperforming linear matrix-based schemes by two to five times, and that it is possible to develop numerically efficient implementations that are independent of the hyperelastic constitutive law.

42 ENGINEERING↗

NASA NTRS2000

Three-Dimensional Nacelle Aeroacoustics Code With Application to Impedance Education

A three-dimensional nacelle acoustics code that accounts for uniform mean flow and variable surface impedance liners is developed. The code is linked to a commercial version of the NASA-developed General Purpose Solver (for solution of linear systems of equations) in order to obtain the capability to study high frequency waves that may require millions of grid points for resolution. Detailed, single-processor statistics for the performance of the solver in rigid and soft-wall ducts are presented. Over the range of frequencies of current interest in nacelle liner research, noise attenuation levels predicted from the code were in excellent agreement with those predicted from mode theory. The equation solver is memory efficient, requiring only a small fraction of the memory available on modern computers. As an application, the code is combined with an optimization algorithm and used to reduce the impedance spectrum of a ceramic liner. The primary problem with using the code to perform optimization studies at frequencies above I1kHz is the excessive CPU time (a major portion of which is matrix assembly). The research recommends that research be directed toward development of a rapid sparse assembler and exploitation of the multiprocessor capability of the solver to further reduce CPU time.

Watson, Willie R.↗

DOE OSTI2026

Block encoding of the three-dimensional heterogeneous Poisson equation with application to fracture flow

Quantum linear system (QLS) algorithms offer the potential to solve large-scale linear systems exponentially faster than classical methods. However, applying QLS algorithms to real-world problems remains challenging due to issues such as state preparation, data loading, and efficient information extraction. In this work, we study the feasibility of applying QLS algorithms to solve discretized three-dimensional (3D) heterogeneous Poisson equations, with specific examples relating to groundwater flow through geologic fracture networks. We explicitly construct a block encoding for the 3D heterogeneous Poisson matrix by leveraging the sparse local structure of the discretized operator. While classical solvers benefit from preconditioning, we show that block encoding the system matrix and preconditioner separately does not improve the effective condition number that dominates the QLS run-time. This differs from classical approaches where the preconditioner and the system matrix can often be implemented independently. Nevertheless, due to the structure of the problem in three dimensions, the quantum algorithm achieves a run-time of 𝑂⁡(𝑁 2/3 polylog 𝑁 ⋅log (1/𝜖)), outperforming the best classical methods (with run times of 𝑂⁡(𝑁⁢log 𝑁 ⋅log (1/𝜖))) and offering exponential memory savings. These results highlight both the promise and limitations of QLS algorithms for practical scientific computing, and point to effective condition-number reduction as a key barrier in achieving quantum advantages.

58 GEOSCIENCES↗

DOE OSTI2019

hypre

hypre is a library of high performance preconditioners and solvers featuring multigrid methods for the solution of large, sparse linear systems of equations on massively parallel computers.

Falgout, RobertD↗