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Papers by Boris Faleichik

arXiv (Cornell University), Aug 21, 2019

The purpose of this work is to introduce a new idea of how to avoid the factorization of large ma... more The purpose of this work is to introduce a new idea of how to avoid the factorization of large matrices during the solution of stiff systems of ODEs. Starting from the general form of an explicit linear multistep method we suggest to adaptively choose its coefficients on each integration step in order to minimize the norm of the residual of an implicit BDF formula. Thereby we reduce the number of unknowns on each step from n to O(1), where n is the dimension of the ODE system. We call this type of methods Minimal Residual Multistep (MRMS) methods. In the case of linear non-autonomous problem, besides the evaluations of the right-hand side of ODE, the resulting numerical scheme additionally requires one solution of a linear least-squares problem with a thin matrix per step. We show that the order of the method and its zero-stability properties coincide with those of the used underlying BDF formula. For the simplest analog of the implicit Euler method the properties of linear stability are investigated. Though the classical absolute stability analysis is not fully relevant to the MRMS methods, it is shown that this one-step method is applicable in stiff case. In the numerical experiment section we consider the fixed-step integration of a two-dimensional non-autonomous heat equation using the MRMS methods and their classical BDF counterparts. The starting values are taken from a preset slowly-varying exact solution. The comparison showed that both methods give similar numerical solutions, but in the case of large systems the MRMS methods are faster, and their advantage considerably increases with the growth of dimension. Python code with the experimantal code can be downloaded from the GitHub repository https://github.com/bfaleichik/mrms.

Computational methods in applied mathematics, 2008

arXiv (Cornell University), Dec 12, 2020

AIP Conference Proceedings, 2009

Journal of Computational and Applied Mathematics

arXiv (Cornell University), Mar 28, 2023

AIP Conference Proceedings, 2009

Computational Methods in Applied Mathematics, 2008

The goal of the research is to construct practicable numerical algorithms for stiff systems of or... more The goal of the research is to construct practicable numerical algorithms for stiff systems of ordinary differential equations which let you increase the accuracy of the approximate solution without decreasing the length of the time interval. To achieve this goal, we have constructed a family of new iterative analytic processes generalising the Picard process. For a basic representative of this family, we demon-strate its better convergence properties on a scalar linear problem in comparison with the classical Picard process. For the general form of such iterative processes, we discuss their connection with existing methods for operator equations and propose a method for choosing their parameters. The efficiency of this parameter determination method is justified with a numerical experiment. In conclusion we propose a general approach to the construction of numerical algorithms which is based on the discretisation of the constructed iterative analytic processes.

Journal of Computational and Applied Mathematics

The purpose of this work is to introduce a new idea of how to avoid the factorization of large ma... more The purpose of this work is to introduce a new idea of how to avoid the factorization of large matrices during the solution of stiff systems of ODEs. Starting from the general form of an explicit linear multistep method we suggest to adaptively choose its coefficients on each integration step in order to minimize the norm of the residual of an implicit BDF formula. Thereby we reduce the number of unknowns on each step from n to O(1), where n is the dimension of the ODE system. We call this type of methods Minimal Residual Multistep (MRMS) methods. In the case of linear non-autonomous problem, besides the evaluations of the right-hand side of ODE, the resulting numerical scheme additionally requires one solution of a linear least-squares problem with a thin matrix per step. We show that the order of the method and its zero-stability properties coincide with those of the used underlying BDF formula. For the simplest analog of the implicit Euler method the properties of linear stability are investigated. Though the classical absolute stability analysis is not fully relevant to the MRMS methods, it is shown that this one-step method is applicable in stiff case. In the numerical experiment section we consider the fixed-step integration of a two-dimensional non-autonomous heat equation using the MRMS methods and their classical BDF counterparts. The starting values are taken from a preset slowly-varying exact solution. The comparison showed that both methods give similar numerical solutions, but in the case of large systems the MRMS methods are faster, and their advantage considerably increases with the growth of dimension. Python code with the experimantal code can be downloaded from the GitHub repository https://github.com/bfaleichik/mrms.

Journal of the Belarusian State University. Mathematics and Informatics

In this work we present explicit Adams-type multi-step methods with extended stability intervals,... more In this work we present explicit Adams-type multi-step methods with extended stability intervals, which are analogous to the stabilised Chebyshev Runge – Kutta methods. It is proved that for any k ≥ 1 there exists an explicit k-step Adams-type method of order one with stability interval of length 2k. The first order methods have remarkably simple expressions for their coefficients and error constant. A damped modification of these methods is derived. In the general case, to construct a k-step method of order p it is necessary to solve a constrained optimisation problem in which the objective function and p constraints are second degree polynomials in k variables. We calculate higher-order methods up to order six numerically and perform some numerical experiments to confirm the accuracy and stability of the methods.

Lecture Notes in Computer Science, 2015

Journal of Computational and Applied Mathematics, 2014

Journal of Computational and Applied Mathematics, 2014

arXiv (Cornell University), Aug 21, 2019

The purpose of this work is to introduce a new idea of how to avoid the factorization of large ma... more The purpose of this work is to introduce a new idea of how to avoid the factorization of large matrices during the solution of stiff systems of ODEs. Starting from the general form of an explicit linear multistep method we suggest to adaptively choose its coefficients on each integration step in order to minimize the norm of the residual of an implicit BDF formula. Thereby we reduce the number of unknowns on each step from n to O(1), where n is the dimension of the ODE system. We call this type of methods Minimal Residual Multistep (MRMS) methods. In the case of linear non-autonomous problem, besides the evaluations of the right-hand side of ODE, the resulting numerical scheme additionally requires one solution of a linear least-squares problem with a thin matrix per step. We show that the order of the method and its zero-stability properties coincide with those of the used underlying BDF formula. For the simplest analog of the implicit Euler method the properties of linear stability are investigated. Though the classical absolute stability analysis is not fully relevant to the MRMS methods, it is shown that this one-step method is applicable in stiff case. In the numerical experiment section we consider the fixed-step integration of a two-dimensional non-autonomous heat equation using the MRMS methods and their classical BDF counterparts. The starting values are taken from a preset slowly-varying exact solution. The comparison showed that both methods give similar numerical solutions, but in the case of large systems the MRMS methods are faster, and their advantage considerably increases with the growth of dimension. Python code with the experimantal code can be downloaded from the GitHub repository https://github.com/bfaleichik/mrms.

Computational methods in applied mathematics, 2008

arXiv (Cornell University), Dec 12, 2020

AIP Conference Proceedings, 2009

Journal of Computational and Applied Mathematics

arXiv (Cornell University), Mar 28, 2023

AIP Conference Proceedings, 2009

Computational Methods in Applied Mathematics, 2008

The goal of the research is to construct practicable numerical algorithms for stiff systems of or... more The goal of the research is to construct practicable numerical algorithms for stiff systems of ordinary differential equations which let you increase the accuracy of the approximate solution without decreasing the length of the time interval. To achieve this goal, we have constructed a family of new iterative analytic processes generalising the Picard process. For a basic representative of this family, we demon-strate its better convergence properties on a scalar linear problem in comparison with the classical Picard process. For the general form of such iterative processes, we discuss their connection with existing methods for operator equations and propose a method for choosing their parameters. The efficiency of this parameter determination method is justified with a numerical experiment. In conclusion we propose a general approach to the construction of numerical algorithms which is based on the discretisation of the constructed iterative analytic processes.

Journal of Computational and Applied Mathematics

The purpose of this work is to introduce a new idea of how to avoid the factorization of large ma... more The purpose of this work is to introduce a new idea of how to avoid the factorization of large matrices during the solution of stiff systems of ODEs. Starting from the general form of an explicit linear multistep method we suggest to adaptively choose its coefficients on each integration step in order to minimize the norm of the residual of an implicit BDF formula. Thereby we reduce the number of unknowns on each step from n to O(1), where n is the dimension of the ODE system. We call this type of methods Minimal Residual Multistep (MRMS) methods. In the case of linear non-autonomous problem, besides the evaluations of the right-hand side of ODE, the resulting numerical scheme additionally requires one solution of a linear least-squares problem with a thin matrix per step. We show that the order of the method and its zero-stability properties coincide with those of the used underlying BDF formula. For the simplest analog of the implicit Euler method the properties of linear stability are investigated. Though the classical absolute stability analysis is not fully relevant to the MRMS methods, it is shown that this one-step method is applicable in stiff case. In the numerical experiment section we consider the fixed-step integration of a two-dimensional non-autonomous heat equation using the MRMS methods and their classical BDF counterparts. The starting values are taken from a preset slowly-varying exact solution. The comparison showed that both methods give similar numerical solutions, but in the case of large systems the MRMS methods are faster, and their advantage considerably increases with the growth of dimension. Python code with the experimantal code can be downloaded from the GitHub repository https://github.com/bfaleichik/mrms.

Journal of the Belarusian State University. Mathematics and Informatics

In this work we present explicit Adams-type multi-step methods with extended stability intervals,... more In this work we present explicit Adams-type multi-step methods with extended stability intervals, which are analogous to the stabilised Chebyshev Runge – Kutta methods. It is proved that for any k ≥ 1 there exists an explicit k-step Adams-type method of order one with stability interval of length 2k. The first order methods have remarkably simple expressions for their coefficients and error constant. A damped modification of these methods is derived. In the general case, to construct a k-step method of order p it is necessary to solve a constrained optimisation problem in which the objective function and p constraints are second degree polynomials in k variables. We calculate higher-order methods up to order six numerically and perform some numerical experiments to confirm the accuracy and stability of the methods.

Lecture Notes in Computer Science, 2015

Journal of Computational and Applied Mathematics, 2014

Journal of Computational and Applied Mathematics, 2014