On the use of parallel processors for implicit Runge-Kutta methods

Cooper, G. J.; Vignesvaran, R.

doi:10.1007/bf02243848

Cited by 6 publications

(4 citation statements)

References 27 publications

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“…The dependence of tttj on e indicates that the mantissa length when computing the e-solution is determined in accordance with the bound (6). The dependence of tttj on e indicates that the mantissa length when computing the e-solution is determined in accordance with the bound (6).…”

Section: ~I " X(t~ -X/ "A == X~ -X T (T I = C T + "A)mentioning

confidence: 95%

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Computational complexity of integrating ordinary differential equations

Besarab¹

1996

Cybern Syst Anal

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1. The Cauchy problem for ordinary differential equations (ODE) is widely used as a mathematical model in various fields of research. In some cases, numerical solution of problems may involve a very large volume of computations, and it is therefore relevant to consider the issue of obtaining a numerical solution in acceptable time. In other cases, the problem must be solved in real time, which imposes rigid constraints on the machine time available for its solution.Construction of a numerical solution of the Cauchy problem with specified quality characteristics (accuracy of numerical solution, computer time and memory requirements to obtain an approximate solution) is therefore a highly topical issue [1,2].Various aspects of time complexity of numerical integration of ODE have been investigated in [3-6] and elsewhere. The notion of e-complexity of the problem [7] plays an important role in comparing the speed of algorithms that compute the e-solution of the problem. Here e-complexity is the exact lower bound of time complexity on the set of all methods for computing the solution with accuracy e (given a computational model and the input data specification).The construction of exact lower bounds and corresponding algorithms is a very difficult undertaking, and we therefore often consider approximate forms of the problem to obtain time complexity bounds and examine the possibility of lowering the upper complexity bounds.In this paper, we consider time bounds for algorithms that compute an e-solution by traditional numerical methods, and also by methods that implement a special form of input data and use parallelized computation.2. Consider the Cauchy problem X(O=/(t,X), X(O)=X o, t~ I0, t~where X, fare m-dimensional vectors, f is a sufficiently smooth function of the variables t, X, 0 < t o < oo. Assume that for this problem we have generated by some numerical stepwise method of order p [2] the table x,,,tr l-1 ..... ~,.wheret i E ,x,A:0 < t 1 < t 2... < t N=t~ Table (2) is called a numerical e-solution of problem (1) ifwhere p is some measure of closeness of the exact and the numerical solutions, 9 is a given positive number.The value e i = X(ti) --X i is called the accumulated computational error, and its components ~i " X(t~ -X/, "a == X~ -X t (t i ,= c t + "a)are the method error (et, i) and the rounding error (eri). Here {Xi*} is a table of the form (2) corresponding to exact execution of arithmetic operations (without rounding).

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Section: ~I " X(t~ -X/ "A == X~ -X T (T I = C T + "A)mentioning

confidence: 95%

“…Various aspects of time complexity of numerical integration of ODE have been investigated in [3][4][5][6] and elsewhere. The notion of e-complexity of the problem [7] plays an important role in comparing the speed of algorithms that compute the e-solution of the problem.…”

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confidence: 99%

Computational complexity of integrating ordinary differential equations

Besarab¹

1996

Cybern Syst Anal

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“…Knowing that forces in nature vary constantly, these may be different from the simulated lapse of the time differential, where they are considered as constants. There are other simulation methods, different ways of simulation error control, and numerical optimizations [3][13] [5], but we are going to use the basic method for the analysis of different optimization and parallelization approaches.…”

Section: Introductionmentioning

confidence: 99%

Sequential Optimization and Shared and Distributed Memory Parallelization in Clusters: N-Body/Particle Simulation

Tinetti¹,

Martín²

2012

Parallel and Distributed Computing and Systems / 790: Software Engineering and Applications

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The particle-particle method for N-Body problems is one of the most commonly used methods in computer driven physics simulation. These algorithms are, in general, very simple to design and code, and highly parallelizable. In this article, we present the most important approaches for the application of the three performance improvement areas on these algorithms when executed on high performance computing (HPC) clusters: 1) sequential optimization (a single core in a node of the cluster), 2) shared memory parallelism (in a single node with multiple CPUs available, just like a multiprocessor), and 3) distributed memory parallelism (in the whole cluster). For each one of the improvement areas we present the employed techniques and the obtained performance gain. Also, we will show how some (sequential/classical) code optimizations are almost essential for obtaining at least acceptable parallel performance and scalability.

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“…i,k , for i = 1, 2. Since these two formulas are independent of each other, they can be calculated in parallel [13]. Additionally, the limited memory vectors, s i,k and y i,k for i = 1, 2 can be updated in parallel.…”

Section: Hybrid Algorithm Of Order Onementioning

confidence: 99%

Hybrid Runge-Kutta and quasi-Newton methods for unconstrained nonlinear optimization

Mohr¹

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Finding a local minimizer in unconstrained nonlinear optimization and a fixed point of a gradient system of ordinary differential equations (ODEs) are two closely related problems. Quasi-Newton algorithms are widely used in unconstrained nonlinear optimization while Runge-Kutta methods are widely used for the numerical integration of ODEs. In this thesis, hybrid algorithms combining low-order implicit Runge-Kutta methods for gradient systems and quasi-Newton type updates of the Jacobian matrix such as the BFGS update are considered. These hybrid algorithms numerically approximate the gradient flow, but the exact Jacobian matrix is not used to solve the nonlinear system at each step. Instead, a quasi-Newton matrix is used to approximate the Jacobian matrix and matrix-vector multiplications are performed in a limited memory setting to reduce storage, computations, and the need to calculate Jacobian information. For hybrid algorithms based on Runge-Kutta methods of order at least two, a curve search is implemented instead of the standard line search used in quasi-Newton algorithms. Stepsize control techniques are also performed to control the stepsize associated with the underlying Runge-Kutta method. These hybrid algorithms are tested on a variety of test problems and their performance is compared with that of the limited memory BFGS algorithm.

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On the use of parallel processors for implicit Runge-Kutta methods

Cited by 6 publications

References 27 publications

Computational complexity of integrating ordinary differential equations

Computational complexity of integrating ordinary differential equations

Sequential Optimization and Shared and Distributed Memory Parallelization in Clusters: N-Body/Particle Simulation

Hybrid Runge-Kutta and quasi-Newton methods for unconstrained nonlinear optimization

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