A new Green's function Monte Carlo algorithm for the estimation of the derivative of the solution of Helmholtz equation subject to Neumann and mixed boundary conditions
The objective of this paper is the extension and application of a newly-developed Green's function Monte Carlo (GFMC) algorithm to the estimation of the derivative of the solution of the one-dimensional (1D) Helmholtz equation subject to Neumann and mixed boundary conditions problems. The tradi...
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| Published in: | Journal of computational physics Vol. 315; pp. 264 - 272 |
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| Format: | Journal Article |
| Language: | English |
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15.06.2016
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| ISSN: | 0021-9991, 1090-2716 |
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| Abstract | The objective of this paper is the extension and application of a newly-developed Green's function Monte Carlo (GFMC) algorithm to the estimation of the derivative of the solution of the one-dimensional (1D) Helmholtz equation subject to Neumann and mixed boundary conditions problems. The traditional GFMC approach for the solution of partial differential equations subject to these boundary conditions involves “reflecting boundaries” resulting in relatively large computational times. My work, inspired by the work of K.K. Sabelfeld is philosophically different in that there is no requirement for reflection at these boundaries. The underlying feature of this algorithm is the elimination of the use of reflecting boundaries through the use of novel Green's functions that mimic the boundary conditions of the problem of interest. My past work has involved the application of this algorithm to the estimation of the solution of the 1D Laplace equation, the Helmholtz equation and the modified Helmholtz equation. In this work, this algorithm has been adapted to the estimation of the derivative of the solution which is a very important development. In the traditional approach involving reflection, to estimate the derivative at a certain number of points, one has to a priori estimate the solution at a larger number of points. In the case of a one-dimensional problem for instance, to obtain the derivative of the solution at a point, one has to obtain the solution at two points, one on each side of the point of interest. These points have to be close enough so that the validity of the first-order approximation for the derivative operator is justified and at the same time, the actual difference between the solutions at these two points has to be at least an order of magnitude higher than the statistical error in the estimation of the solution, thus requiring a significantly larger number of random-walks than that required for the estimation of the solution. In this new approach, identical amount of computational resources is needed irrespective of if we are trying to estimate the solution or the derivative. This becomes very significant in electromagnetic problems where the scalar/vector potential is the unknown in the PDE of interest, but the quantity of interest is the electric/magnetic field or in heat conduction problems where temperature of an object is the unknown variable in a PDE, but the quantity of interest is the spatial/temporal variation of the temperature. In this work, this algorithm is applied to the estimation of the derivative of the solution of the 1D Helmholtz equation which is the frequency domain version of both Maxwell's equations and the heat conduction equation. As a result the algorithm is an important first step in the development of computationally efficient GFMC algorithms for Neumann and mixed boundary condition problems. The numerical results have been validated by an exact, analytical solution and very good agreement has been observed. The long-term goal of this research is the application of this methodology to the numerical solution of the F region ionization problem in space plasma modeling. |
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| AbstractList | The objective of this paper is the extension and application of a newly-developed Green's function Monte Carlo (GFMC) algorithm to the estimation of the derivative of the solution of the one-dimensional (1D) Helmholtz equation subject to Neumann and mixed boundary conditions problems. The traditional GFMC approach for the solution of partial differential equations subject to these boundary conditions involves “reflecting boundaries” resulting in relatively large computational times. My work, inspired by the work of K.K. Sabelfeld is philosophically different in that there is no requirement for reflection at these boundaries. The underlying feature of this algorithm is the elimination of the use of reflecting boundaries through the use of novel Green's functions that mimic the boundary conditions of the problem of interest. My past work has involved the application of this algorithm to the estimation of the solution of the 1D Laplace equation, the Helmholtz equation and the modified Helmholtz equation. In this work, this algorithm has been adapted to the estimation of the derivative of the solution which is a very important development. In the traditional approach involving reflection, to estimate the derivative at a certain number of points, one has to a priori estimate the solution at a larger number of points. In the case of a one-dimensional problem for instance, to obtain the derivative of the solution at a point, one has to obtain the solution at two points, one on each side of the point of interest. These points have to be close enough so that the validity of the first-order approximation for the derivative operator is justified and at the same time, the actual difference between the solutions at these two points has to be at least an order of magnitude higher than the statistical error in the estimation of the solution, thus requiring a significantly larger number of random-walks than that required for the estimation of the solution. In this new approach, identical amount of computational resources is needed irrespective of if we are trying to estimate the solution or the derivative. This becomes very significant in electromagnetic problems where the scalar/vector potential is the unknown in the PDE of interest, but the quantity of interest is the electric/magnetic field or in heat conduction problems where temperature of an object is the unknown variable in a PDE, but the quantity of interest is the spatial/temporal variation of the temperature. In this work, this algorithm is applied to the estimation of the derivative of the solution of the 1D Helmholtz equation which is the frequency domain version of both Maxwell's equations and the heat conduction equation. As a result the algorithm is an important first step in the development of computationally efficient GFMC algorithms for Neumann and mixed boundary condition problems. The numerical results have been validated by an exact, analytical solution and very good agreement has been observed. The long-term goal of this research is the application of this methodology to the numerical solution of the F region ionization problem in space plasma modeling. |
| Author | Chatterjee, Kausik |
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| Keywords | Space plasma modeling Mixed boundary conditions Parallel algorithm Monte Carlo Random walk Neumann boundary conditions Helmholtz equation Green's function |
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| References | Chatterjee, Ananthapadmanabhan (br0050) 2012; 18 Chatterjee, Sandora, Mitchell, Stefan, Nummey, Poggie (br0060) 2010; 25 Sabelfeld (br0040) 1991 Haberman (br0020) 1998 Mikhailov (br0070) 1995 Schunk, Nagy (br0080) 2009 Dimov (br0010) 2008 Haji-Sheikh (br0030) 1965 Chatterjee (10.1016/j.jcp.2016.02.075_br0050) 2012; 18 Sabelfeld (10.1016/j.jcp.2016.02.075_br0040) 1991 Schunk (10.1016/j.jcp.2016.02.075_br0080) 2009 Haberman (10.1016/j.jcp.2016.02.075_br0020) 1998 Chatterjee (10.1016/j.jcp.2016.02.075_br0060) 2010; 25 Mikhailov (10.1016/j.jcp.2016.02.075_br0070) 1995 Dimov (10.1016/j.jcp.2016.02.075_br0010) 2008 Haji-Sheikh (10.1016/j.jcp.2016.02.075_br0030) 1965 |
| References_xml | – year: 1998 ident: br0020 article-title: Elementary Applied Partial Differential Equations – start-page: 106 year: 1965 end-page: 108 ident: br0030 article-title: Application of Monte Carlo methods to thermal conduction problems – volume: 25 start-page: 373 year: 2010 end-page: 380 ident: br0060 article-title: A new software and hardware parallelized floating random-walk algorithm for the modified Helmholtz equation subject to Neumann and mixed boundary conditions publication-title: Appl. Comput. Electromagn. Soc. J. – year: 2008 ident: br0010 article-title: Monte Carlo Methods for Applied Scientists – start-page: 159 year: 1991 end-page: 176 ident: br0040 article-title: Monte Carlo Methods in Boundary Value Problems – volume: 18 start-page: 265 year: 2012 end-page: 273 ident: br0050 article-title: A Green's function Monte Carlo algorithm for the Helmholtz equation subject to Neumann and mixed boundary conditions: validation with a 1D benchmark problem publication-title: Monte Carlo Methods Appl. – start-page: 608 year: 2009 end-page: 613 ident: br0080 article-title: Ionospheres: Physics, Plasma Physics and Chemistry – year: 1995 ident: br0070 article-title: New Monte Carlo Methods with Estimating Derivatives – start-page: 159 year: 1991 ident: 10.1016/j.jcp.2016.02.075_br0040 – volume: 18 start-page: 265 year: 2012 ident: 10.1016/j.jcp.2016.02.075_br0050 article-title: A Green's function Monte Carlo algorithm for the Helmholtz equation subject to Neumann and mixed boundary conditions: validation with a 1D benchmark problem publication-title: Monte Carlo Methods Appl. doi: 10.1515/mcma-2012-0009 – start-page: 608 year: 2009 ident: 10.1016/j.jcp.2016.02.075_br0080 – year: 2008 ident: 10.1016/j.jcp.2016.02.075_br0010 – start-page: 106 year: 1965 ident: 10.1016/j.jcp.2016.02.075_br0030 – volume: 25 start-page: 373 year: 2010 ident: 10.1016/j.jcp.2016.02.075_br0060 article-title: A new software and hardware parallelized floating random-walk algorithm for the modified Helmholtz equation subject to Neumann and mixed boundary conditions publication-title: Appl. Comput. Electromagn. Soc. J. – year: 1998 ident: 10.1016/j.jcp.2016.02.075_br0020 – year: 1995 ident: 10.1016/j.jcp.2016.02.075_br0070 |
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| SubjectTerms | Algorithms Boundaries Boundary conditions Derivatives Estimates Green's function Helmholtz equation Helmholtz equations Mathematical analysis Mathematical models Mixed boundary conditions Monte Carlo Neumann boundary conditions Parallel algorithm Random walk Space plasma modeling |
| Title | A new Green's function Monte Carlo algorithm for the estimation of the derivative of the solution of Helmholtz equation subject to Neumann and mixed boundary conditions |
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