Optimally stable parallel predictors for Adams—Moulton correctors

Katz, Irving; Franklin, Mark A.; Sen, Abhilash

doi:10.1016/0898-1221(77)90097-9

Cited by 16 publications

(5 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…11. This study indicated that the more stable predictor form is obtained by shifting the predictor to the right, i.e., from n − 1 to n. This approximately restores the original AB's formulation.…”

Section: Parallelizing the Predictor-corrector Methodsmentioning

confidence: 61%

See 1 more Smart Citation

Parallel Predictor-Corrector Methods for the Solution of Ordinary Differential Equations Ii

Mazzone¹

1999

Int. J. Mod. Phys. C

View full text Add to dashboard Cite

This work presents parallel multistep methods for the solution of ordinary differential equations. The characteristic of parallel computing is that there is a "front" and the computation at points ahead of the front depends only on information behind it. This requires a resetting of serial algorithms and may also lead to numerical errors and instabilities. The analysis of positive and negative aspects of parallel computing is the subject of this paper. Some of the methods presented below are uncommon in the literature on mathematical computing. Others have been elaborated for this study on the basis of the traditional Adams-Bashforth multistep methods.A performance comparison of the methods is made by numerical testing in molecular dynamics calculations. The increase of the number of processors m appears to seriously deteriorate the stability of the calculations and the use of m larger than 2 seems impractical.

show abstract

Section: Parallelizing the Predictor-corrector Methodsmentioning

confidence: 61%

“…The computers have to exchange information after the predictor equation to construct the term F p n+1 and F p n+2 on the right of Eqs. (11). PBM and PPM represent an interesting contrast.…”

Section: Block Methodsmentioning

confidence: 99%

Parallel Predictor-Corrector Methods for the Solution of Ordinary Differential Equations Ii

Mazzone¹

1999

Int. J. Mod. Phys. C

View full text Add to dashboard Cite

show abstract

“…From a stability point of view, this algorithm should not behave markedly differently between a serial or parallel implementation, unlike algorithms of the predictor-corrector type where it has been shown [Katz et al, 1977] that the stability interval for a fourth order parallel predictor-corrector algorithm is about 0.846 as compared with 1.285 for the fourth-order serial Adams-Bashforth, Adams-Moulton pair. The reason there is that in the parallel case the predictor and corrector equations are spread over two different time steps.…”

Section: Error and Stability Questionsmentioning

confidence: 93%

“…problem is typical of the test differential equations used byKatz, Franklin and Sen [1977] to test the error and stability properties of parallel algorithms.…”

mentioning

confidence: 99%

Methods for system simulation on a restricted data flow architecture

Damodaran¹,

Hazra²

1981

Proceedings of the ACM '81 Conference on - ACM 81

View full text Add to dashboard Cite

“…is means that, at any point, we can find the solution numerically using any number of the previous points. e Adams-Moulton method [62,63,65,66] is a modification of the Adams-Bashforth method, which was Fractional tumor cells kill by chemotherapy [40,42].…”

Section: Methods Of Solutionmentioning

confidence: 99%

Mathematical Modelling for the Role of CD4⁺T Cells in Tumor-Immune Interactions

Makhlouf

El-Shennawy

Elkaranshawy

2020

Computational and Mathematical Methods in Medicine

View full text Add to dashboard Cite

Mathematical modelling has been used to study tumor-immune cell interaction. Some models were proposed to examine the effect of circulating lymphocytes, natural killer cells, and CD8+T cells, but they neglected the role of CD4+T cells. Other models were constructed to study the role of CD4+T cells but did not consider the role of other immune cells. In this study, we propose a mathematical model, in the form of a system of nonlinear ordinary differential equations, that predicts the interaction between tumor cells and natural killer cells, CD4+T cells, CD8+T cells, and circulating lymphocytes with or without immunotherapy and/or chemotherapy. This system is stiff, and the Runge–Kutta method failed to solve it. Consequently, the “Adams predictor-corrector” method is used. The results reveal that the patient’s immune system can overcome small tumors; however, if the tumor is large, adoptive therapy with CD4+T cells can be an alternative to both CD8+T cell therapy and cytokines in some cases. Moreover, CD4+T cell therapy could replace chemotherapy depending upon tumor size. Even if a combination of chemotherapy and immunotherapy is necessary, using CD4+T cell therapy can better reduce the dose of the associated chemotherapy compared to using combined CD8+T cells and cytokine therapy. Stability analysis is performed for the studied patients. It has been found that all equilibrium points are unstable, and a condition for preventing tumor recurrence after treatment has been deduced. Finally, a bifurcation analysis is performed to study the effect of varying system parameters on the stability, and bifurcation points are specified. New equilibrium points are created or demolished at some bifurcation points, and stability is changed at some others. Hence, for systems turning to be stable, tumors can be eradicated without the possibility of recurrence. The proposed mathematical model provides a valuable tool for designing patients’ treatment intervention strategies.

show abstract

Optimally stable parallel predictors for Adams—Moulton correctors

Cited by 16 publications

References 4 publications

Parallel Predictor-Corrector Methods for the Solution of Ordinary Differential Equations Ii

Parallel Predictor-Corrector Methods for the Solution of Ordinary Differential Equations Ii

Methods for system simulation on a restricted data flow architecture

Mathematical Modelling for the Role of CD4⁺T Cells in Tumor-Immune Interactions

Contact Info

Product

Resources

About

Optimally stable parallel predictors for Adams—Moulton correctors

Cited by 16 publications

References 4 publications

Parallel Predictor-Corrector Methods for the Solution of Ordinary Differential Equations Ii

Parallel Predictor-Corrector Methods for the Solution of Ordinary Differential Equations Ii

Methods for system simulation on a restricted data flow architecture

Mathematical Modelling for the Role of CD4+T Cells in Tumor-Immune Interactions

Contact Info

Product

Resources

About

Mathematical Modelling for the Role of CD4⁺T Cells in Tumor-Immune Interactions