Wolter van der Veen - Academia.edu (original) (raw)
Papers by Wolter van der Veen
Journal of Computational and Applied Mathematics, 1998
PSIDE is a code for solving implicit differential equations on parallel computers. It is an imple... more PSIDE is a code for solving implicit differential equations on parallel computers. It is an implementation of the four-stage Radau IIA method. The nonlinear systems are solved by a modified Newton process, in which every Newton iterate itself is computed by an iteration process. This process is constructed such that the four stage values can be computed simultaneously. We describe
Journal of Computational and Applied Mathematics, 1998
PSIDE -- Parallel Software for Implicit Differential Equations -- is a code for solving implicit ... more PSIDE -- Parallel Software for Implicit Differential Equations -- is a code for solving implicit differential equations on shared memory parallel computers. In this paper we describe the user interface.
PSIDE is a code for solving implicit dierential equations on parallel computers. It is an impleme... more PSIDE is a code for solving implicit dierential equations on parallel computers. It is an implementation of the four-stage Radau IIA method. The nonlinear systems are solved by a modied Newton process, in which every Newton iterate itself is computed by an iteration process. This process is constructed such that the four stage values can be computed simultaneously. We describe here how PSIDE is set up as a modular system and what control strategies have been chosen. 1991 Mathematics Subject Classication: Primary: 65-04, Secondary: 65L05, 65Y05. 1991 Computing Reviews Classication System: G.1.7, G.4. Keywords and Phrases: numerical software, parallel computers, IVP, IDE, ODE, DAE. Note: The maintenance of PSIDE belongs to the project MAS2.2: 'Parallel Software for Implicit Dierential Equations'. Acknowledgements: This work is supported nancially by the 'Technologiestichting STW' (Dutch Foundation for Technical Sciences), grants no. CWI.2703, CWI.4533. The use of s...
Journal of Computational and Applied Mathematics - J COMPUT APPL MATH, 1996
In this paper a collection of Initial Value test Problems for systems of Ordinary Differential Eq... more In this paper a collection of Initial Value test Problems for systems of Ordinary Differential Equations, Implicit Differential Equations and Differential-Algebraic Equations is presented. This test set is maintained by the project group for Parallel IVP Solvers of CWI, department of Numerical Mathematics. This group invites everyone to contribute new test problems to this test set. How new problems can be submitted can be found in this paper as well.
Numerical Algorithms, 1994
For the parallel integration of stiff initial value problems (IVPs), three main approaches can be... more For the parallel integration of stiff initial value problems (IVPs), three main approaches can be distinguished: approaches based on "parallelism across the problem", on "parallelism across the method" and on "parallelism across the steps". The first type of parallelism does not require special integration methods and can be exploited within any available IVP solver. The methodparallel approach received some attention in the case of Runge-Kutta based methods. For these methods, the required number of processors is roughly half the order of the generating Runge-Kutta method and the speed-up with respect to a good sequential IVP solver is about a factor 2. The third type of parallelism (step-parallelism) can be achieved in any IVP solver based on predictor-corrector iteration. Most step-parallel methods proposed so far employ a large number of processors, but lack the property of robustness, due to a poor convergence behaviour in the iteration process. Hence, the effective speed-up is rather poor. The step-parallel iteration process proposed in the present paper is less massively parallel, but turns out to be sufficiently robust to solve the four-stage Radau IIA corrector used in our experiments within a few effective iterations per step and to achieve speed-up factors up to 10 with respect to the best sequential codes.
Rendiconti del Seminario Matematico e Fisico di Milano, 1995
We construct and analyse three methods for solving initial value problems for implicit differenti... more We construct and analyse three methods for solving initial value problems for implicit differential equations (IDEs) on parallel computer systems. The first IDE method can be applied to general IDEs of higher index, the other two methods can be applied to partitioned (or semi-explicit) IDEs. The partitioned IDE methods both exploit the special form of the problem and often converge faster than the general IDE method. The first partitioned IDE method is suitable for higher-index problems, the second partitioned IDE method only applies to index 1 problems, but possesses more parallelism across the method. The convergence of these methods is illustrated by solving implicit IDEs of index 0 until 3 that are taken from the literature.
Journal of Engineering Mathematics, 1995
Take-down policy If you believe that this document breaches copyright please contact us providing... more Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
Journal of Computational and Applied Mathematics, 1995
UvA-DARE (Digital Academic Repository) Parallel Iteration across the steps of high-order Runge-Ku... more UvA-DARE (Digital Academic Repository) Parallel Iteration across the steps of high-order Runge-Kutta Methods for nonstiff initial value problems van der Houwen, P.J.; Sommeijer, B.P.; van der Veen, W.A.
Computers & Mathematics with Applications, 1995
For the parallel integration of stiff initial value problems, three types of parallelism can be e... more For the parallel integration of stiff initial value problems, three types of parallelism can be employed: "parallelism across the problem," "parallelism across the method" and "parallelism across the steps." Recently, methods based on Runge-Kutta schemes that use parallelism across the method have been proposed in [1,2]. These methods solve implicit Runge-Kutta schemes by means of the so-called diagonally iteration scheme and are called PDIRK methods. The experiments described in [1], show that the speedup factor of certain high-order PDIRK methods, is about 2 with respect to a good sequential code. However, a disadvantage of the high-order PDIRK methods is, that a relatively large number of iterations is needed for each step. This disadvantage can be compensated by employing step-parallelism. Step-parallel methods are methods in which a number of steps are treated simultaneously. This form of parallelism can be applied to any predictor-corrector method. A common feature of this approach is their poor convergence behaviour, unless the various strategies are carefully designed. In this paper, we describe two strategies for the PDIRK across the steps method. Example problems tested in this paper show for the best strategy, a speed-up factor ranging from 4 to 7 with respect to the best sequential codes.
Applied Numerical Mathematics, 1997
We construct and analyze parallel iterative solvers for the solution of the linear systems arisin... more We construct and analyze parallel iterative solvers for the solution of the linear systems arising in the application of Newton's method to ;;-stage implicit Runge-Kutta (RK) type discretizations of implicit differential equations (ID Es). These 1 incar solvers arc partly iterative and partly direct. Each linear system iteration again requires the solution of linear subsystems. hut now only of IDE dimension, which is ;; times less than the dimension of the linear system in Newton's melhod. Thus. the effective costs on a parallel computer system arc only one LU-decomposition of !DE dimension for each Jacobian update, yielding a considerable reduction of the effective LU-costs. The method parameters can he chosen such that only a few iterations by the linear solver are needed. The algorithmic properties arc illustrated hy solving the transistor problem (index 1) and the car axis problem (index 3) taken from the CW! test set. 1997 Elsevier Science B. V.
Applied Numerical Mathematics, 1995
One of the most powerful methods for solving initial value problems for ordinary differential equ... more One of the most powerful methods for solving initial value problems for ordinary differential equations is an implicit Runge-Kutta method such as the Radau IIA methods. These methods are both highly accurate and highly stable. However, the iterative scheme needed for solving the implicit RK equations requires a lot of computational effort. The arrival of parallel computer systems has changed the situation in the sense that the effective computational effort can be reduced to a large extent. One option is the application of the iteration scheme concurrently at a number of step points on the taxis. In this paper, we shall analyse the convergence of a special class of such step-parallel iteration methods.
UvA-DARE (Digital Academic Repository) Waveform relaxation methods for implicit differential equa... more UvA-DARE (Digital Academic Repository) Waveform relaxation methods for implicit differential equations van der Houwen, P.J.; van der Veen, W.A.
Journal of Computational and Applied Mathematics, 1997
This paper deals with solving stiff systems of differential equations by implicit Multistep Runge... more This paper deals with solving stiff systems of differential equations by implicit Multistep Runge-Kutta (MRK) methods. For this type of methods, nonlinear systems of dimension sd arise, where s is the number of Runge-Kutta stages and d the dimension of the problem. Applying a Newton process leads to linear systems of the same dimension, which can be very expensive to solve in practice. With a parallel iterative linear system solver, especially designed for MRK methods, we approximate these linear systems by s systems of dimension d, which can be solved in parallel on a computer with s processors. In terms of Jacobian evaluations and LU-decompositions, the k-step s-stage MRK applied with this technique is ons processors equally expensive as the widely used k-step Backward Differentiation Formula on 1 processor, whereas the stability properties are better than that of BDF. A simple implementation of both methods shows that, for the same number of Newton iterations, the accuracy delivered by the new method is higher than that of BDF.
Journal of Computational and Applied Mathematics, 1998
PSIDE is a code for solving implicit differential equations on parallel computers. It is an imple... more PSIDE is a code for solving implicit differential equations on parallel computers. It is an implementation of the four-stage Radau IIA method. The nonlinear systems are solved by a modified Newton process, in which every Newton iterate itself is computed by an iteration process. This process is constructed such that the four stage values can be computed simultaneously. We describe
Journal of Computational and Applied Mathematics, 1998
PSIDE -- Parallel Software for Implicit Differential Equations -- is a code for solving implicit ... more PSIDE -- Parallel Software for Implicit Differential Equations -- is a code for solving implicit differential equations on shared memory parallel computers. In this paper we describe the user interface.
PSIDE is a code for solving implicit dierential equations on parallel computers. It is an impleme... more PSIDE is a code for solving implicit dierential equations on parallel computers. It is an implementation of the four-stage Radau IIA method. The nonlinear systems are solved by a modied Newton process, in which every Newton iterate itself is computed by an iteration process. This process is constructed such that the four stage values can be computed simultaneously. We describe here how PSIDE is set up as a modular system and what control strategies have been chosen. 1991 Mathematics Subject Classication: Primary: 65-04, Secondary: 65L05, 65Y05. 1991 Computing Reviews Classication System: G.1.7, G.4. Keywords and Phrases: numerical software, parallel computers, IVP, IDE, ODE, DAE. Note: The maintenance of PSIDE belongs to the project MAS2.2: 'Parallel Software for Implicit Dierential Equations'. Acknowledgements: This work is supported nancially by the 'Technologiestichting STW' (Dutch Foundation for Technical Sciences), grants no. CWI.2703, CWI.4533. The use of s...
Journal of Computational and Applied Mathematics - J COMPUT APPL MATH, 1996
In this paper a collection of Initial Value test Problems for systems of Ordinary Differential Eq... more In this paper a collection of Initial Value test Problems for systems of Ordinary Differential Equations, Implicit Differential Equations and Differential-Algebraic Equations is presented. This test set is maintained by the project group for Parallel IVP Solvers of CWI, department of Numerical Mathematics. This group invites everyone to contribute new test problems to this test set. How new problems can be submitted can be found in this paper as well.
Numerical Algorithms, 1994
For the parallel integration of stiff initial value problems (IVPs), three main approaches can be... more For the parallel integration of stiff initial value problems (IVPs), three main approaches can be distinguished: approaches based on "parallelism across the problem", on "parallelism across the method" and on "parallelism across the steps". The first type of parallelism does not require special integration methods and can be exploited within any available IVP solver. The methodparallel approach received some attention in the case of Runge-Kutta based methods. For these methods, the required number of processors is roughly half the order of the generating Runge-Kutta method and the speed-up with respect to a good sequential IVP solver is about a factor 2. The third type of parallelism (step-parallelism) can be achieved in any IVP solver based on predictor-corrector iteration. Most step-parallel methods proposed so far employ a large number of processors, but lack the property of robustness, due to a poor convergence behaviour in the iteration process. Hence, the effective speed-up is rather poor. The step-parallel iteration process proposed in the present paper is less massively parallel, but turns out to be sufficiently robust to solve the four-stage Radau IIA corrector used in our experiments within a few effective iterations per step and to achieve speed-up factors up to 10 with respect to the best sequential codes.
Rendiconti del Seminario Matematico e Fisico di Milano, 1995
We construct and analyse three methods for solving initial value problems for implicit differenti... more We construct and analyse three methods for solving initial value problems for implicit differential equations (IDEs) on parallel computer systems. The first IDE method can be applied to general IDEs of higher index, the other two methods can be applied to partitioned (or semi-explicit) IDEs. The partitioned IDE methods both exploit the special form of the problem and often converge faster than the general IDE method. The first partitioned IDE method is suitable for higher-index problems, the second partitioned IDE method only applies to index 1 problems, but possesses more parallelism across the method. The convergence of these methods is illustrated by solving implicit IDEs of index 0 until 3 that are taken from the literature.
Journal of Engineering Mathematics, 1995
Take-down policy If you believe that this document breaches copyright please contact us providing... more Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
Journal of Computational and Applied Mathematics, 1995
UvA-DARE (Digital Academic Repository) Parallel Iteration across the steps of high-order Runge-Ku... more UvA-DARE (Digital Academic Repository) Parallel Iteration across the steps of high-order Runge-Kutta Methods for nonstiff initial value problems van der Houwen, P.J.; Sommeijer, B.P.; van der Veen, W.A.
Computers & Mathematics with Applications, 1995
For the parallel integration of stiff initial value problems, three types of parallelism can be e... more For the parallel integration of stiff initial value problems, three types of parallelism can be employed: "parallelism across the problem," "parallelism across the method" and "parallelism across the steps." Recently, methods based on Runge-Kutta schemes that use parallelism across the method have been proposed in [1,2]. These methods solve implicit Runge-Kutta schemes by means of the so-called diagonally iteration scheme and are called PDIRK methods. The experiments described in [1], show that the speedup factor of certain high-order PDIRK methods, is about 2 with respect to a good sequential code. However, a disadvantage of the high-order PDIRK methods is, that a relatively large number of iterations is needed for each step. This disadvantage can be compensated by employing step-parallelism. Step-parallel methods are methods in which a number of steps are treated simultaneously. This form of parallelism can be applied to any predictor-corrector method. A common feature of this approach is their poor convergence behaviour, unless the various strategies are carefully designed. In this paper, we describe two strategies for the PDIRK across the steps method. Example problems tested in this paper show for the best strategy, a speed-up factor ranging from 4 to 7 with respect to the best sequential codes.
Applied Numerical Mathematics, 1997
We construct and analyze parallel iterative solvers for the solution of the linear systems arisin... more We construct and analyze parallel iterative solvers for the solution of the linear systems arising in the application of Newton's method to ;;-stage implicit Runge-Kutta (RK) type discretizations of implicit differential equations (ID Es). These 1 incar solvers arc partly iterative and partly direct. Each linear system iteration again requires the solution of linear subsystems. hut now only of IDE dimension, which is ;; times less than the dimension of the linear system in Newton's melhod. Thus. the effective costs on a parallel computer system arc only one LU-decomposition of !DE dimension for each Jacobian update, yielding a considerable reduction of the effective LU-costs. The method parameters can he chosen such that only a few iterations by the linear solver are needed. The algorithmic properties arc illustrated hy solving the transistor problem (index 1) and the car axis problem (index 3) taken from the CW! test set. 1997 Elsevier Science B. V.
Applied Numerical Mathematics, 1995
One of the most powerful methods for solving initial value problems for ordinary differential equ... more One of the most powerful methods for solving initial value problems for ordinary differential equations is an implicit Runge-Kutta method such as the Radau IIA methods. These methods are both highly accurate and highly stable. However, the iterative scheme needed for solving the implicit RK equations requires a lot of computational effort. The arrival of parallel computer systems has changed the situation in the sense that the effective computational effort can be reduced to a large extent. One option is the application of the iteration scheme concurrently at a number of step points on the taxis. In this paper, we shall analyse the convergence of a special class of such step-parallel iteration methods.
UvA-DARE (Digital Academic Repository) Waveform relaxation methods for implicit differential equa... more UvA-DARE (Digital Academic Repository) Waveform relaxation methods for implicit differential equations van der Houwen, P.J.; van der Veen, W.A.
Journal of Computational and Applied Mathematics, 1997
This paper deals with solving stiff systems of differential equations by implicit Multistep Runge... more This paper deals with solving stiff systems of differential equations by implicit Multistep Runge-Kutta (MRK) methods. For this type of methods, nonlinear systems of dimension sd arise, where s is the number of Runge-Kutta stages and d the dimension of the problem. Applying a Newton process leads to linear systems of the same dimension, which can be very expensive to solve in practice. With a parallel iterative linear system solver, especially designed for MRK methods, we approximate these linear systems by s systems of dimension d, which can be solved in parallel on a computer with s processors. In terms of Jacobian evaluations and LU-decompositions, the k-step s-stage MRK applied with this technique is ons processors equally expensive as the widely used k-step Backward Differentiation Formula on 1 processor, whereas the stability properties are better than that of BDF. A simple implementation of both methods shows that, for the same number of Newton iterations, the accuracy delivered by the new method is higher than that of BDF.