Performance-Based Numerical Solver Selection in the Lighthouse Framework

Efficient approximate solution of sparse linear systems

The DEFLATED-GMRES(m, k) method is one of the major iterative solvers for the large sparse linear systems of equations, x = b. This algorithm assembles a preconditioner adaptively for the GMRES(m) method based on eigencomponents gathered from the Arnoldi process during iterations. It is usually known that if a restarted GMRES(m) method is used to solve linear systems of equations, the information of the smallest eigencomponents is lost at each restart and the super-linear convergence may also be lost. In this paper, we propose an adaptive procedure that combines the DEFLATED-GMRES(m, k) algorithm and the determination of a restart frequency automatically. It is shown that a new algorithm combining elements of both will reduce the negative effects of the restarted procedure. The numerical experiments are presented on three test problems by using the MIMD parallel machine AP3000. From these numerical results, we show that the proposed algorithm leads to faster convergence than the conventional DEFLATED-GMRES(m, k) method. Copyright © 2000 John Wiley & Sons, Ltd.

.Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov subspace methods, which are among the most popular iterative solvers for large sparse linear systems. We develop a deflation strategy for symmetric saddle point matrices by taking advantage of their underlying block structure. The vectors used for deflation come from an elliptic singular value decomposition relying on the generalized Golub–Kahan bidiagonalization process. The block targeted by deflation is the off-diagonal one since it features a problematic singular value distribution for certain applications. One example is the Stokes flow in elongated channels, where the off-diagonal block has several small, isolated singular values, depending on the length of the channel. Applying deflation to specific parts of the saddle point system is important when using solvers such as CRAIG, which operates on individual blocks rather than the whole system. The theory is developed by extending the existing framework for deflating square matrices before applying a Krylov subspace method such as MINRES. Numerical experiments confirm the merits of our strategy and lead to interesting questions about using approximate vectors for deflation.KeywordsGolub–Kahan bidiagonalizationeigenvalue deflationsingular value decompositionsaddle point problemsStokes equationMSC codes15A1835P1565F1065F1565N22

Large sparse linear algebraic systems can be found in a variety of scientific and engineering fields and many scientists strive to solve them in an efficient and robust manner. In this paper, we propose an interpretable neural solver, the Fourier neural solver (FNS), to address them. FNS is based on deep learning and a fast Fourier transform. Because the error between the iterative solution and the ground truth involves a wide range of frequency modes, the FNS combines a stationary iterative method and frequency space correction to eliminate different components of the error. Local Fourier analysis shows that the FNS can pick up on the error components in frequency space that are challenging to eliminate with stationary methods. Numerical experiments on the anisotropic diffusion equation, convection–diffusion equation, and Helmholtz equation show that the FNS is more efficient and more robust than the state-of-the-art neural solver.

The simulation of several physical phenomena, arising from a wide class of scientific fields, requires solving large sparse linear systems effectively. In order to exploit appropriately the available supercomputing infrastructures, designing efficient and scalable solvers for large sparse linear systems is necessary. A hybrid parallel algebraic iterative method based on the Schur complement method and multiprojection techniques that utilizes semiaggregated subdomains is proposed. The graph corresponding to the coefficient matrix of the sparse linear system is partitioned by a graph partitioning algorithm and the Schur complement problem is solved by a preconditioned Krylov subspace method. The preconditioning scheme is based on domain decomposition and the resulting subdomains consist of fine and aggregated components from the inner boundaries. Moreover, two variants for approximating the local Schur complements, derived from the semiaggregation procedure, are proposed in order to improve memory requirements and preprocessing time. The proposed method is suitable for distributed memory systems with multicore nodes and has improved convergence behavior for a large number of workstations. Numerical and comparative results concerning the applicability, the convergence behavior, and the parallel performance of the proposed scheme are given.

The development of efficient, general-purpose software for the iterative solution of sparse linear systems on parallel MIMD computers depends on recent results from a wide variety of research areas. Parallel graph heuristics, convergence analysis, and basic linear algebra implementation issues must all be considered.

Artificial intelligence has been applied in wildfire science and management since the 1990s, with early applications including neural networks and expert systems. Since then, the field has rapidly progressed congruently with the wide adoption of machine learning (ML) methods in the environmental sciences. Here, we present a scoping review of ML applications in wildfire science and management. Our overall objective is to improve awareness of ML methods among wildfire researchers and managers, as well as illustrate the diverse and challenging range of problems in wildfire science available to ML data scientists. To that end, we first present an overview of popular ML approaches used in wildfire science to date and then review the use of ML in wildfire science as broadly categorized into six problem domains, including (i) fuels characterization, fire detection, and mapping; (ii) fire weather and climate change; (iii) fire occurrence, susceptibility, and risk; (iv) fire behavior prediction; (v) fire effects; and (vi) fire management. Furthermore, we discuss the advantages and limitations of various ML approaches relating to data size, computational requirements, generalizability, and interpretability, as well as identify opportunities for future advances in the science and management of wildfires within a data science context. In total, to the end of 2019, we identified 300 relevant publications in which the most frequently used ML methods across problem domains included random forests, MaxEnt, artificial neural networks, decision trees, support vector machines, and genetic algorithms. As such, there exists opportunities to apply more current ML methods — including deep learning and agent-based learning — in the wildfire sciences, especially in instances involving very large multivariate datasets. We must recognize, however, that despite the ability of ML models to learn on their own, expertise in wildfire science is necessary to ensure realistic modelling of fire processes across multiple scales, while the complexity of some ML methods such as deep learning requires a dedicated and sophisticated knowledge of their application. Finally, we stress that the wildfire research and management communities play an active role in providing relevant, high-quality, and freely available wildfire data for use by practitioners of ML methods.

Multigrid is widely used as an efficient solver for sparse linear systems arising from the discretization of elliptic boundary value problems. Linear relaxation methods such as Gauss–Seidel and Red–Black Gauss–Seidel form the principal computational component of multigrid, and thus affect its efficiency. In the context of multigrid, these iterative solvers are executed for a small number of iterations (2–8). We exploit this property of the algorithm to develop a cache-efficient multigrid method, by focusing on improving the memory behavior of the linear relaxation methods. The efficiency in our cache-efficient linear relaxation algorithm comes from two sources: reducing the number of data cache and TLB misses, and reducing the number of memory references by keeping values registerresident. Our optimizations are applicable to multigrid applied to linear systems arising from constant coefficient elliptic PDEs on structured grids. Experiments on five modern computing platforms show a performance improvement of 1.15–2.7 times over a standard implementation of Full Multigrid V-Cycle.

A dose deposition matrix (DDM) prediction method using several voxel features and a machine learning (ML) approach is proposed for plan optimization in radiation therapy. Head and lung cases with the inhomogeneous medium are used as training and testing data. The prediction model is a cascade forward backprop neural network where the input is the features of the voxel, including 1) voxel to body surface distance along the beamlet axis, 2) voxel to beamlet axis distance, 3) voxel density, 4) heterogeneity corrected voxel to body surface distance, 5) heterogeneity corrected voxel to beamlet axis, and (6) the dose of voxel obtained from the pencil beam (PB) algorithm. The output is the predicted voxel dose corresponding to a beamlet. The predicted DDM was used for plan optimization (ML method) and compared with the dose of MC-based plan optimization (MC method) and the dose of pencil beam-based plan optimization (PB method). The mean absolute error (MAE) value was calculated for full volume relative to the dose of the MC method to evaluate the overall dose performance of the final plan. For patient with head tumor, the ML method achieves MAE value 0.49 × 10-4 and PB has MAE 1.86 × 10-4. For patient with lung tumor, the ML method has MAE 1.42 × 10-4 and PB has MAE 3.72 × 10-4. The maximum percentage difference in PTV dose coverage (D98) between ML and MC methods is no more than 1.2% for patient with head tumor, while the difference is larger than 10% using the PB method. For patient with lung tumor, the maximum percentage difference in PTV dose coverage (D98) between ML and MC methods is no more than 2.1%, while the difference is larger than 16% using the PB method. In this work, a reliable DDM prediction method is established for plan optimization by applying several voxel features and the ML approach. The results show that the ML method based on voxel features can obtain plans comparable to the MC method and is better than the PB method in achieving accurate dose to the patient, which is helpful for rapid plan optimization and accurate dose calculation. Establishment of a new machine learning method based on the relationship between the voxel and beamlet features for dose deposition matrix prediction in radiation therapy.

The adaptive s-step CG algorithm is a solver for sparse symmetric positive definite linear systems designed to reduce the synchronization cost per iteration while still achieving a user-specified accuracy requirement. In this work, we improve the adaptive s-step conjugate gradient algorithm by the use of iteratively updated estimates of the largest and smallest Ritz values, which give approximations of the largest and smallest eigenvalues of A, using a technique due to G.Meurant and P.Tichý (2018). The Ritz value estimates are used to dynamically update parameters for constructing Newton or Chebyshev polynomials so that the conditioning of the s-step bases can be continuously improved throughout the iterations. These estimates are also used to automatically set a variable related to the ratio of the sizes of the error and residual, which was previously treated as an input parameter. We show through numerical experiments that in many cases the new algorithm improves upon the previous adaptive s-step approach both in terms of numerical behavior and reduction in number of synchronizations.

New features of our DSC system for distributing a symbolic computation task over a network of processors are described. A new scheduler sends parallel subtasks to those compute nodes that are best suited in handling the added load of CPU usage and memory. Furthermore, a subtask can communicate back to the process that spawned it by a co-routine style calling mechanism. Two large experiments are described in this improved setting. We have implemented an algorithm that can prove a number of more than 1,000 decimal digits prime in about 2 months elapsed time on some 20 computers. A parallel version of a sparse linear system solver is used to compute the solution of sparse linear systems over finite fields. We are able to find the solution of a 100,000 by 100,000 linear system with about 10.3 million non-zero entries over the Galois field with 2 elements using 3 computers in about 54 hours CPU time.

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A systematic literature review of machine learning methods applied to predictive maintenance

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