基于非稳态不可压缩流的可扩张并行算法研究
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摘要
PDE约束的优化问题是指由偏微分方程组(PDEs)约束的优化系统。仿真问题是指在给定合适的数据(例如几何形状,系数,边界条件,初始条件,源函数)的情况下,求解偏微分方程组中的例如位移,速度,温度,电场,磁场,种浓度等状态变量。而和仿真问题相对应的是优化问题,优化问题的目的是在满足一个目标函数和约束条件下,寻求一些例如状态变量等的决策变量,其中这些约束常常是基于仿真问题所对应的偏微分方程组。PDE约束优化问题的规模,复杂性和兀限维性质对目前通用的优化算法产生了重大挑战。正是由于该类问题的这些特征,从而往往需要使用正规化,迭代求解器,预处理,全局化技术,不精确求解和针对问题的特点来构造基本并行算子等方法来求解该类问题。
随着并行计算机的技术和规模的发展,可以解决非常复杂的数值问题,包括由非线性偏微分方程组(PDEs)约束的优化问题。这种计算复杂性的上升趋势就要求我们设计可扩张的并行算法和采用现代软件工程的先进数值库技术。本文介绍了运用并行计算方法求解PDE约束优化问题和相应的仿真问题,提出了一类求解该类问题的新的全耦合全空间并行算法。本博士论文由以下几部分组成：
在第2章中,本博士论文探讨一类由Navier-Stokes方程组得到的非定常不可压缩流的仿真问题的全耦合的,并行的牛顿-克雷洛夫-施瓦兹(NKS)算法。该算法包括两个主要部分：外部迭代的非线性牛顿法和线性迭代部分的一个Two-level施瓦兹预条件子。现有方法中,计算此类仿真问题一般都是基于所谓的减空间法,该方法更容易实现,但是可能在算法的收敛性上有一些问题和缺陷,从而导致算法常常是不收敛的。因此,本论文采用一类新型算法：全空间方法。将状态变量耦合到一个单一的大的非线性方程组系统中。该耦合系统相较于其子系统变得更病态,尽管如此,在NKS方法的强效下,可以在大规模并行计算机上有效地解决这些困难的系统。数值结果显示了该并行算法的牛顿迭代次数和线性迭代次数同时独立于网格数,内核个数和雷诺数,并且在超过两千个内核的情况下,验证了该并行算法的可扩展性。
在第3章中,本博士论文构造了一种全隐格式的并行区域分解算法来求解带时间项的非线性偏微分方程组约束的优化问题。特别地,研究了非定长不可压缩Navier-Stokes方程的边界控制问题。在时间隐式离散后,一个全耦合非线性稀疏的子优化问题需要在每个时间步进行求解。本论文使用一类全空间的拉格朗日-牛顿-克雷洛夫-施瓦兹(LNKS)算法用来求解该子优化问题。在优化算法中,全隐全空间方法被认为是最简单考虑到而又最难实现的一类方法。而本文的数值结果表明含有限制加性施瓦兹预条件子的LNKS算法是求解这些高难问题的一类有效方法。为了展示该算法的可扩展性和鲁棒性,本论文在一些不同的雷诺数和时间步长,计算规模涉及几百万个未知量并且在超过二千个内核的情况下研究其计算性能。
在第4章中,本博士论文提出了一类并行半光滑-牛顿-克雷洛夫-施瓦兹(SNKS)算法求解一类不等式约束的优化问题：互补问题。在半光滑-牛顿-克雷洛夫-施瓦兹算法中包含了非精确半光滑牛顿法,克雷洛夫子空间法和施瓦兹预处理技术。通过使用半光滑函数,此优化问题的解可以通过求解一个大型稀疏非线性系统的代数方程组而得到。数值结果表明了该方法的有效性。
最后在第5章,基于牛顿-克雷洛夫-施瓦兹并行算法,本博士论文模拟了液力变矩器内流场分布。
本论文编写的并行程序是在由Argonne国家实验室开发的软件"Portable, Ex-tensible Toolkit for Scientific computation (PETSc)"的基础上研发的。因此,在附录中,简单地描述PETSc的一些使用方法,正是基于此面向对象的软件,编写了基于优化控制问题的并行程序,并进行了数值实验。
PDE-constrained optimization problem refers to the optimization of systemsgoverned by partial diferential equations (PDEs). The simulation problem is tosolve the PDEs for the state variables (e.g. displacement, velocity, temperature,electric feld, magnetic feld, species concentration), given appropriate data (e.g.geometry, coefcients, boundary conditions, initial conditions, source functions).The optimization problem seeks to determine some of these data the decisionvariables given performance goals in the form of an objective function and pos-sibly inequality or equality constraints on the behavior of the system. Since thebehavior of the system is modeled by the PDEs, they appear as (usually equal-ity) constraints in the optimization problem. The size, complexity, and infnite-dimensional nature of PDE-constrained optimization problems present signifcantchallenges for general-purpose optimization algorithms. These features often re-quire regularization, iterative solvers, preconditioning, globalization, inexactness,and parallel implementation that are tailored to the structure of the underlyingoperators.
As the technology and size of parallel computing systems advance, so does thedetails at which we can solve very complex numerical problems, including optimiza-tion problems constrained by nonlinear partial diferential equations (PDEs). Thistrend of increasing computational complexity demands both the design of scalableparallel algorithms and the adoption of modern software engineering techniquesfor the developments of numerical libraries. This thesis presents the developmentthe robust parallel numerical methods for solving PDE-constrained optimizationproblems and the corresponding simulation problem. We proposed a new class offully coupled full space parallel algorithms. This thesis consists of the followingparts.
In Chapter2, we investigate some fully coupled parallel Newton-Krylov-Schwarz (NKS) algorithms for the simulation problem of unsteady incompress-ible flows governed by the Navier-Stokes equations. The algorithms include twomajor parts: a nonlinear Newton method for the outer iteration and a two-levelSchwarz preconditioner for the linear part of the problem. Most of the existingapproaches for this kind of simulation problems are based on the so-called reducedspace method which is easier to implement but may have convergence issues insome situations. In the full space approach we couple the state variables in a sin- gle large system of nonlinear equations. The coupled system is considerably moreill-conditioned than its sub-systems, however, with the powerful NKS approach, weare able to solve these difcult systems efciently on large scale parallel comput-ers. We show numerically that such an approach is scalable in the sense that thenumber of Newton iterations and the number of linear iterations are both nearlyindependent of the grid size, the number of processors, and the Reynolds numbers.We present numerical experiments obtained on supercomputers with more thantwo thousand processors.
In Chapter3, We develop a parallel fully implicit domain decomposition algo-rithm for solving optimization problems constrained by time dependent nonlinearpartial diferential equations. In particular, we study the boundary control of un-steady incompressible Navier-Stokes equations. After an implicit discretization intime, a fully coupled sparse nonlinear optimization problem needs to be solved ateach time step. The class of full space Lagrange-Newton-Krylov-Schwarz (LNKS)algorithms is used to solve the sequence of optimization problems. Among op-timization algorithms, the fully implicit full space approach is considered to bethe easiest to formulate and the hardest to solve. We show that LNKS, with arestricted additive Schwarz preconditioner, is an efcient class of methods for solv-ing these hard problems. To demonstrate the scalability and robustness of thealgorithm, we consider several problems with a wide range of Reynolds numbersand time step sizes, and we present numerical results for large scale calculationsinvolving several millions unknowns obtained on machines with more than twothousand processors.
In Chapter4, we present some parallel semismooth Newton-Krylov-Schwarz(NKS) algorithm for solving a kind of optimization problems constrained by in-equality nonlinear partial diferential equations: complementarity problems. Thefamily of Semismooth-Newton-Krylov-Schwarz methods is based on a semismoothinexact Newton method, Krylov subspace method, and overlapping Schwarz pre-conditioner. Using semismooth function, the solution of the optimization problemcan be obtained by solving a large sparse nonlinear system of algebraic equations.Numerical results show that the efciency can be achieved by the proposed method.
Finally, in Chapter5, we simulate the flow feld distribution within the torqueconverter based on Newton-Krylov-Schwarz algorithms.
Our algorithms are implemented based on the Portable Extensible Toolkitfor Scientifc computation (PETSc). In Appendix A, we describe the high level use of PETSc, which is the object-oriented software we have developed for theimplementation of the parallel algorithms based one the proposed optimizationproblem and for the execuation of our parallel numerical experiments.
随着并行计算机的技术和规模的发展,可以解决非常复杂的数值问题,包括由非线性偏微分方程组(PDEs)约束的优化问题。这种计算复杂性的上升趋势就要求我们设计可扩张的并行算法和采用现代软件工程的先进数值库技术。本文介绍了运用并行计算方法求解PDE约束优化问题和相应的仿真问题,提出了一类求解该类问题的新的全耦合全空间并行算法。本博士论文由以下几部分组成：
在第2章中,本博士论文探讨一类由Navier-Stokes方程组得到的非定常不可压缩流的仿真问题的全耦合的,并行的牛顿-克雷洛夫-施瓦兹(NKS)算法。该算法包括两个主要部分：外部迭代的非线性牛顿法和线性迭代部分的一个Two-level施瓦兹预条件子。现有方法中,计算此类仿真问题一般都是基于所谓的减空间法,该方法更容易实现,但是可能在算法的收敛性上有一些问题和缺陷,从而导致算法常常是不收敛的。因此,本论文采用一类新型算法：全空间方法。将状态变量耦合到一个单一的大的非线性方程组系统中。该耦合系统相较于其子系统变得更病态,尽管如此,在NKS方法的强效下,可以在大规模并行计算机上有效地解决这些困难的系统。数值结果显示了该并行算法的牛顿迭代次数和线性迭代次数同时独立于网格数,内核个数和雷诺数,并且在超过两千个内核的情况下,验证了该并行算法的可扩展性。
在第3章中,本博士论文构造了一种全隐格式的并行区域分解算法来求解带时间项的非线性偏微分方程组约束的优化问题。特别地,研究了非定长不可压缩Navier-Stokes方程的边界控制问题。在时间隐式离散后,一个全耦合非线性稀疏的子优化问题需要在每个时间步进行求解。本论文使用一类全空间的拉格朗日-牛顿-克雷洛夫-施瓦兹(LNKS)算法用来求解该子优化问题。在优化算法中,全隐全空间方法被认为是最简单考虑到而又最难实现的一类方法。而本文的数值结果表明含有限制加性施瓦兹预条件子的LNKS算法是求解这些高难问题的一类有效方法。为了展示该算法的可扩展性和鲁棒性,本论文在一些不同的雷诺数和时间步长,计算规模涉及几百万个未知量并且在超过二千个内核的情况下研究其计算性能。
在第4章中,本博士论文提出了一类并行半光滑-牛顿-克雷洛夫-施瓦兹(SNKS)算法求解一类不等式约束的优化问题：互补问题。在半光滑-牛顿-克雷洛夫-施瓦兹算法中包含了非精确半光滑牛顿法,克雷洛夫子空间法和施瓦兹预处理技术。通过使用半光滑函数,此优化问题的解可以通过求解一个大型稀疏非线性系统的代数方程组而得到。数值结果表明了该方法的有效性。
最后在第5章,基于牛顿-克雷洛夫-施瓦兹并行算法,本博士论文模拟了液力变矩器内流场分布。
本论文编写的并行程序是在由Argonne国家实验室开发的软件"Portable, Ex-tensible Toolkit for Scientific computation (PETSc)"的基础上研发的。因此,在附录中,简单地描述PETSc的一些使用方法,正是基于此面向对象的软件,编写了基于优化控制问题的并行程序,并进行了数值实验。
PDE-constrained optimization problem refers to the optimization of systemsgoverned by partial diferential equations (PDEs). The simulation problem is tosolve the PDEs for the state variables (e.g. displacement, velocity, temperature,electric feld, magnetic feld, species concentration), given appropriate data (e.g.geometry, coefcients, boundary conditions, initial conditions, source functions).The optimization problem seeks to determine some of these data the decisionvariables given performance goals in the form of an objective function and pos-sibly inequality or equality constraints on the behavior of the system. Since thebehavior of the system is modeled by the PDEs, they appear as (usually equal-ity) constraints in the optimization problem. The size, complexity, and infnite-dimensional nature of PDE-constrained optimization problems present signifcantchallenges for general-purpose optimization algorithms. These features often re-quire regularization, iterative solvers, preconditioning, globalization, inexactness,and parallel implementation that are tailored to the structure of the underlyingoperators.
As the technology and size of parallel computing systems advance, so does thedetails at which we can solve very complex numerical problems, including optimiza-tion problems constrained by nonlinear partial diferential equations (PDEs). Thistrend of increasing computational complexity demands both the design of scalableparallel algorithms and the adoption of modern software engineering techniquesfor the developments of numerical libraries. This thesis presents the developmentthe robust parallel numerical methods for solving PDE-constrained optimizationproblems and the corresponding simulation problem. We proposed a new class offully coupled full space parallel algorithms. This thesis consists of the followingparts.
In Chapter2, we investigate some fully coupled parallel Newton-Krylov-Schwarz (NKS) algorithms for the simulation problem of unsteady incompress-ible flows governed by the Navier-Stokes equations. The algorithms include twomajor parts: a nonlinear Newton method for the outer iteration and a two-levelSchwarz preconditioner for the linear part of the problem. Most of the existingapproaches for this kind of simulation problems are based on the so-called reducedspace method which is easier to implement but may have convergence issues insome situations. In the full space approach we couple the state variables in a sin- gle large system of nonlinear equations. The coupled system is considerably moreill-conditioned than its sub-systems, however, with the powerful NKS approach, weare able to solve these difcult systems efciently on large scale parallel comput-ers. We show numerically that such an approach is scalable in the sense that thenumber of Newton iterations and the number of linear iterations are both nearlyindependent of the grid size, the number of processors, and the Reynolds numbers.We present numerical experiments obtained on supercomputers with more thantwo thousand processors.
In Chapter3, We develop a parallel fully implicit domain decomposition algo-rithm for solving optimization problems constrained by time dependent nonlinearpartial diferential equations. In particular, we study the boundary control of un-steady incompressible Navier-Stokes equations. After an implicit discretization intime, a fully coupled sparse nonlinear optimization problem needs to be solved ateach time step. The class of full space Lagrange-Newton-Krylov-Schwarz (LNKS)algorithms is used to solve the sequence of optimization problems. Among op-timization algorithms, the fully implicit full space approach is considered to bethe easiest to formulate and the hardest to solve. We show that LNKS, with arestricted additive Schwarz preconditioner, is an efcient class of methods for solv-ing these hard problems. To demonstrate the scalability and robustness of thealgorithm, we consider several problems with a wide range of Reynolds numbersand time step sizes, and we present numerical results for large scale calculationsinvolving several millions unknowns obtained on machines with more than twothousand processors.
In Chapter4, we present some parallel semismooth Newton-Krylov-Schwarz(NKS) algorithm for solving a kind of optimization problems constrained by in-equality nonlinear partial diferential equations: complementarity problems. Thefamily of Semismooth-Newton-Krylov-Schwarz methods is based on a semismoothinexact Newton method, Krylov subspace method, and overlapping Schwarz pre-conditioner. Using semismooth function, the solution of the optimization problemcan be obtained by solving a large sparse nonlinear system of algebraic equations.Numerical results show that the efciency can be achieved by the proposed method.
Finally, in Chapter5, we simulate the flow feld distribution within the torqueconverter based on Newton-Krylov-Schwarz algorithms.
Our algorithms are implemented based on the Portable Extensible Toolkitfor Scientifc computation (PETSc). In Appendix A, we describe the high level use of PETSc, which is the object-oriented software we have developed for theimplementation of the parallel algorithms based one the proposed optimizationproblem and for the execuation of our parallel numerical experiments.
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