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Linear Solvers for Reservoir Simulation Problems: An Overview and Recent developments

ARCHIVES OF COMPUTATIONAL methods IN ENGINEERING 2022年第6期29卷 4341-4378页

作者： Nardean, Stefano Ferronato, Massimiliano Abushaikha, Ahmad Hamad Bin Khalifa Univ Qatar Fdn Coll Sci & Engn Div Sustainable Dev Doha Qatar Univ Padua Dept Civil Environm & Architectural Engn Padua Italy

Linear solvers for reservoir simulation applications are the objective of this review. Specifically, we focus on techniques for Fully Implicit (FI) solution methods, in which the set of governing Partial Differential equations (PDEs) is properly discretized in time (usually by the Backward Euler scheme), and space, and tackled by assembling and linearizing a single system of equations to solve all the model unknowns simultaneously. Due to the usually large size of these systems arising from real-world models, iterative methods, specifically Krylov subspace solvers, have become conventional choices;nonetheless, their success largely revolves around the quality of the preconditioner that is supplied to accelerate their convergence. These two intertwined elements, i.e., the solver and the preconditioner, are the focus of our analysis, especially the latter, which is still the subject of extensive research. The progressive increase in reservoir model size and complexity, along with the introduction of additional physics to the classical flow problem, display the limits of existing solvers. Intensive usage of computational and memory resources are frequent drawbacks in practice, resulting in unpleasantly slow convergence rates. Developing efficient, robust, and scalable preconditioners, often relying on physics-based assumptions, is the way to avoid potential bottlenecks in the solving phase. In this work, we proceed in reviewing principles and state-of-the-art of such linear solution tools to summarize and discuss the main advances and research directions for reservoir simulation problems. We compare the available preconditioning options, showing the connections existing among the different approaches, and try to develop a general algebraic framework.

关键词： iterative methods

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IMPROVING BLOCK CONJUGATE GRADIENT METHOD BY RANDOMIZED SKETCHING FOR large-SCALE FREQUENCY-DOMAIN WRI 83

IMPROVING BLOCK CONJUGATE GRADIENT METHOD BY RANDOMIZED SKET...

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83rd EAGE Conference and Exhibition 2022

作者： Rezaei Nevisi, A. Aghamiry, H. Aghazade, K. Gholami, A. Operto, S. University of Tehran Iran University of Côte d'Azur Geoazur CNRS - IRD - OCA France

ISBN: (纸本)9781713859314

The wavefield reconstruction inversion (WRI) or its refined variant (IR-WRI) has provided outstanding results for imaging complex subsurface with rather remote initial models. The frequencydomain formulation of these methods requires solving iteratively a normal system of equations formulated as a weighted sum of the Gram matrices associated with the Helmholtz operator and the sampling operator. Due to their low memory complexity and their high scalability, block iterative methods are natural choices for solving these systems for large-scale problems with multiple sources (right-hand sides, RHSs). The block conjugate gradient (BCG) is among the most theoretically accurate methods. However, it may fail in practice when the RHSs are linearly dependent or when too many RHSs are processed jointly. This study proposes a method to overcome this issue with source sketching techniques, whereby the number of RHSs is reduced while dependency between them is decreased. We validate the proposed sketched-BCG with the additive Schwarz (AS) preconditioner to solve IR-WRI using the 2004 BP velocity model benchmark. Copyright© (2022) by the European Association of Geoscientists & Engineers (EAGE). All rights reserved.

关键词： Conjugate gradient method

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MPMNet: A Data-driven MPM Framework for Dynamic Fluid-solid Interaction

arXiv

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arXiv 2023年

作者： Li, Jin Gao, Yang Dai, Ju Li, Shuai Hao, Aimin Qin, Hong State Key Laboratory of Virtual Reality Technology and Systems Beijing Advanced Innovation Center for Biomedical Engineering Beihang University Beijing100191 China Chinese Academy of Medical Sciences Beijing100050 China Peng Cheng Laboratory Shenzhen518066 China Zhongguancun Laboratory Beijing100094 China Stony BrookNY11794-2424 United States

High-accuracy, high-efficiency physics-based fluid-solid interaction is essential for reality modeling and computer animation in online games or real-time Virtual Reality (VR) systems. However, the large-scale simulation of incompressible fluid and its interaction with the surrounding solid environment is either time-consuming or suffering from the reduced time/space resolution due to the complicated iterative nature pertinent to numerical computations of involved Partial Differential equations (PDEs). In recent years, we have witnessed significant growth in exploring a different, alternative data-driven approach to addressing some of the existing technical challenges in conventional model-centric graphics and animation methods. This paper showcases some of our exploratory efforts in this direction. One technical concern of our research is to address the central key challenge of how to best construct the numerical solver effectively and how to best integrate spatiotemporal/dimensional neural networks with the available MPM’s pressure solvers. In particular, we devise the MPMNet, a hybrid data-driven framework supporting the popular and powerful Material Point Method (MPM), to combine the comprehensive properties of MPM in numerically handling physical behaviors ranging from fluid to deformable solids and the high efficiency of data-driven models. At the architectural level, our MPMNet comprises three primary components: A data processing module to describe the physical properties by way of the input fields;A deep neural network group to learn the spatiotemporal features;And an iterative refinement process to continue to reduce possible numerical errors. The goal of these special technical developments is to aim at involved numerical acceleration while preserving physical accuracy, realizing efficient and accurate fluid-solid interactions in a data-driven fashion. The extensive experimental results verify that our MPMNet can tremendously speed up the computation comp

关键词： Numerical methods

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A highly scalable approach to solving linear systems using two-stage multisplitting

arXiv

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arXiv 2020年

作者： Brown, Nick Mark Bull, J. Bethune, Iain Edinburgh Parallel Computing Centre James Clerk Maxwell Building Kings Buildings Edinburgh United Kingdom

iterative methods for solving large sparse systems of linear equations are widely used in many HPC applications. Extreme scaling of these methods can be difficult, however, since global communication to form dot products is typically required at every iteration. To try to overcome this limitation we propose a hybrid approach, where the matrix is partitioned into blocks. Within each block, we use a highly optimised (parallel) conventional solver, but we then couple the blocks together using block Jacobi or some other multisplitting technique that can be implemented in either a synchronous or an asynchronous fashion. This allows us to limit the block size to the point where the conventional iterative methods no longer scale, and to avoid global communication (and possibly synchronisation) across all processes. Our block framework has been built to use PETSc, a popular scientific suite for solving sparse linear systems, as the synchronous intra-block solver, and we demonstrate results on up to 32768 cores of a Cray XE6 system. At this scale, the conventional solvers are still more efficient, though trends suggest that the hybrid approach may be beneficial at higher core counts. Copyright © 2020, The Authors. All rights reserved.

关键词： iterative methods

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类Hartree-Fock方程的数值方法

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计算数学 2019年第2期41卷 113-125页

作者：林霖加州大学伯克利分校数学系劳伦斯伯克利国家实验室美国伯克利加利福尼亚94720

本文的主要目的是介绍近年来大基组下的类Hartree-Fock方程数值求解的一些进展,类Hartree-Fock方程出现在Hartree-Fock理论和含杂化泛函的Kohn-Sham密度泛函理论中,是电子结构理论中一类重要的方程.该方程在复杂的化学和材料体系的电子... 详细信息

本文的主要目的是介绍近年来大基组下的类Hartree-Fock方程数值求解的一些进展,类Hartree-Fock方程出现在Hartree-Fock理论和含杂化泛函的Kohn-Sham密度泛函理论中,是电子结构理论中一类重要的方程.该方程在复杂的化学和材料体系的电子结构计算中有广泛地应用.由于计算代价的原因,类Hartree-Fock方程一般只被用在较小规模的量子体系(含几十到几百个电子)的计算.从数学角度上讲,类Hartree-Fock方程是一个非线性积分-微分方程组,其计算代价主要来自于积分算子的部分,也就是Fock交换算子.通过发展和结合自适应压缩交换算子方法(ACE),投影的C-DIIS方法(PC-DIIS)方法,以及插值可分密度近似方法(ISDF),我们大大降低了杂化泛函密度泛函理论的计算代价.以含1000个硅原子的体系为例,我们将平面波基组下的杂化泛函的计算代价降至接近不含Fock交换算子的半局域泛函计算的水平.同时,我们发现类Hartree-Fock方程的数学结构也为一类特征值问题的迭代求解提供了新的思路.

关键词：类Hartree-Fock方程非线性特征值问题积分-微分算子量子化学电子结构理论密度泛函理论

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Mathematical modeling for systems of large dimension through a modification of the method of iterative aggregation 2

Mathematical modeling for systems of large dimension through...

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2nd Young Scientist's International Workshop on trends in Information Processing, YSIP2 2017

作者： Grobova, Tatyana Troyanov, Alexey Lysov, Vladislav Antonov, Vladimir Department of Applied Mathematics and Computer Security North Caucasus University Stavropol Russia Department of Information Security North Caucasus University Stavropol Russia

This article provides a new method for accelerating the convergence such that is a synthesis of two methods: Seidel method and an iterative method for solving a linear system of equations. Using a new method accelerating the convergence for various grades of operator equations allows to construct a sequence of approximations that more convergence to solve the operator equations. In this paper, the method allows to significantly speed up the process of convergence to the precise solution. There is an implementation of the resulting method and sufficient conditions of convergence. Copyright © 2017 by the paper's authors.

关键词： iterative methods

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On the Convergence of iterative methods and Pseudoinverse Approaches in Global Meshless Collocation

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International Journal of Applied and Computational Mathematics 2017年第4期3卷 3987-4000页

作者： Mishra, Pankaj K. Nath, Sankar K. Department of Geology and Geophysics IIT Kharagpur Kharagpur India

In the context of numerical approximation of partial differential equations, meshless methods are recent developments, which have been reported to be relatively straightforward, yet provide better convergence and accuracy as compared to the conventional mesh-based approaches, for some specific problems like stress–strain analysis and modeling in a deforming media. Among several proposed schemes, strong-form collocation schemes using radial basis functions are easy-to-program, require no mesh in the domain or at the boundary, avoid numerical integration, and have similar formulations for all dimensions. Some of the radial basiss functions like the Gaussian, multiquadric and inverse-multiquadric are infinitely smooth. The standard global meshless formulations based on such radial basis functions are often found to lead towards ill-conditioned systems of linear equations. For such formulations, we investigate the degree of ill-conditioning against degrees of freedom for different radial basis functions. Also, the convergence of four iterative methods and pseudoinverse approaches has been tested for the solution of such ill-conditioned systems. In order to compute the pseudoinverse, two different approaches, i.e., singular value decomposition and full rank Cholesky factorization have been used. A set of numerical experiments, performed here, demonstrate that pseudoinverse approach based on singular value decomposition performs better than the iterative methods. Although, pseudoinverse computation via full rank Cholesky factorization is faster, it is not recommended in global meshless collocation schemes due to poor convergence at relatively large degrees of freedom. © 2016, Springer India Pvt. Ltd.

关键词： Generalized inverse Ill-conditioning iterative methods Meshless methods Radial basis function

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Problems of Parallel Solution of large systems of Linear Algebraic equations

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Journal of Mathematical Sciences (United States) 2016年第6期216卷 795-804页

作者： Il’in, V.P. Institute of Computational Mathematics and Mathematical Geophysics SO RAN and Novosibirsk State University Novosibirsk Russian Federation

The paper considers some modern problems arising in developing parallel algorithms for solving large systems of linear algebraic equations with sparse matrices occurring in mathematical modeling of real-life processes and phenomena on a multiprocessor computer system (MCS). Two main requirements to methods and technologies under consideration are fast convergence of iterations and scalable parallelism, which are intrinsically contradictory and need a special investigation. The paper analyzes main trends is developing preconditioned iterative methods in Krylov’s subspaces based on algebraic domain decomposition and principles of their program implementation on a heterogeneous MCS with hierarchical memory structure. © 2016, Springer Science+Business Media New York.

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Enhancements to Chebyshev-Picard iteration efficiency for generally perturbed orbits and constrained dynamical systems

Enhancements to Chebyshev-Picard iteration efficiency for ge...

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作者： Macomber, Brent David Texas A&M University

学位级别：Ph.D.

Modified Chebyshev Picard Iteration (MCPI) is an iterative numerical method for solution of Ordinary Differential equations (ODEs). This dissertation presents a body of work that serves to enhance the overall performance and the algorithmic automation of MCPI, applied to the problem of perturbed orbit propagation. Additionally, an MCPI framework is derived that greatly improves MCPI performance for ODE systems that intrinsically have associated conserved quantities. Leveraging these developments, software libraries are presented that are designed to make MCPI more accessible and automated, both for the problem of orbit propagation, and for general ODE systems. The work outlined in this document is the result of an effort to promote MCPI from an algorithm of academic interest to a broadly applicable toolset for general use by researchers worldwide in all disciplines. MCPI is able to numerically propagate perturbed orbits to arbitrarily high solution accuracy, bounded by the limits of numerical precision. The improvements to MCPI for orbit propagation are focused on decreasing the computational cost of high-accuracy propagation in a two-fold manner; by reducing the number of required iterations necessary to achieve convergence, and decreasing the computational cost per iteration. Typically, the spherical harmonic gravity function evaluations are the most computationally expensive part of perturbed orbit propagation, so the strategies for reducing the cost per iteration focus on techniques for reducing the cost of gravity series evaluations. Additionally, automated tuning parameter selection logic is introduced to enable MCPI to propagate large batches of perturbed orbits, without the necessity of a user in the loop. By making use of an associated conserved quantity within applicable ODE systems, MCPI is shown to be able to achieve much higher performance. A first order and a second order constrained MCPI formulation are developed that are able to vastly reduce the req

关键词： Numerical methods Ordinary Differential equations Parallel Computing Orbit Propagation Space Situational Awareness Thesis

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On iterative techniques for computing flow in large two-dimensional discrete fracture networks

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2012年第18期236卷 4712-4724页

作者： Parashar, Rishi Reeves, Donald M. Desert Res Inst Div Hydrol Sci Reno NV 89512 USA

Computation of flow in discrete fracture networks often involves solving for hydraulic head values at all intersection points of a large number of stochastically generated fractures inside a bounded domain. For large systems, this approach leads to the generation of problems involving highly sparse matrices which must be solved iteratively. Distributions of fracture lengths spanning over several orders of magnitude, and the randomness of fracture orientations and locations, lead to coefficient matrices that are devoid of any regular structure in the sparsity pattern. In addition to the rapid increase in computational effort with increase in the size of the fracture network, the spread in the distribution of fracture parameters, such as length and transmissivity, dramatically influences the convergence behavior of the system of linear equations. An overview of the discrete fracture network (DFN) methodology for computation of flow is presented along with a comparative study of various Krylov subspace iterative methods for the resulting class of sparse matrices. The rate of convergence of the iterative techniques is found to exhibit a systematic pattern with respect to changes in statistical parameters of the stochastically generated fracture networks. Salient features of the observed trends in the convergence pattern are discussed and guidelines for design of DFN algorithms are provided. Published by Elsevier B.V.

关键词： Discrete fracture networks Sparse matrices DFN methodology Krylov subspace iterative methods

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