The Experts below are selected from a list of 8421 Experts worldwide ranked by ideXlab platform
Weng Cho Chew - One of the best experts on this subject based on the ideXlab platform.
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Error Minimization of Multipole Expansion
SIAM Journal on Scientific Computing, 2005Co-Authors: Shinichiro Ohnuki, Weng Cho ChewAbstract:In this paper, we focus on the truncation error of the Multipole Expansion for the fast Multipole method and the multilevel fast Multipole algorithm. When the buffer size is large enough, the error can be controlled and minimized by using the conventional selection rules. On the other hand, if the buffer size is small, the conventional selection rules no longer hold, and the new approach which we have recently proposed is needed. However, this method is still not sufficient to minimize the error for small buffer cases. We clarify this fact and show that the information about the placement of true worst-case interaction is needed. A novel algorithm to minimize the truncation error is proposed.
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Truncation Error Analysis of Multipole Expansion
SIAM Journal on Scientific Computing, 2004Co-Authors: Shinichiro Ohnuki, Weng Cho ChewAbstract:The multilevel fast Multipole algorithm is based on the Multipole Expansion, which has numerical error sources such as truncation of the addition theorem, numerical integration, and interpolation/anterpolation. Of these, we focus on the truncation error and discuss its control precisely. The conventional selection rule fails when the buffer size is small compared to the desired numerical accuracy. We propose a new approach and show that the truncation error can be controlled and predicted regardless of the number of buffer sizes.
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Numerical accuracy of Multipole Expansion for 2D MLFMA
IEEE Transactions on Antennas and Propagation, 2003Co-Authors: Shinichiro Ohnuki, Weng Cho ChewAbstract:A numerical study of the Multipole Expansion for the multilevel fast Multipole algorithm (MLFMA) is presented. In the numerical implementation of MLFMA, the error comes from three sources: the truncation of the addition theorem; the approximation of the integration; the aggregation and disaggregation process. These errors are due to the factorization of the Green's function which is the mathematical core of the algorithm. Among the three error sources, we focus on the truncation error and a new approach of selecting truncation numbers for the addition theorem is proposed. Using this approach, the error prediction and control can be improved for the small buffer sizes and high accuracy requirements.
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a new approach for controlling truncation error of the Multipole Expansion
IEEE Antennas and Propagation Society International Symposium, 2002Co-Authors: Shinichiro Ohnuki, Weng Cho ChewAbstract:The purpose of this paper is to clarify how to control the truncation error of the Multipole Expansion for FMM (fast Multipole method) and MLFMA (multilevel fast Multipole algorithm). The choice of truncation number still remains a complex issue. We perform the numerical experiment carefully and improve the control and prediction accuracy. The past selection rule is to use the excess bandwidth formula which becomes worse for small buffer sizes and high accuracy requirements. To improve this, a new approach is proposed.
N R Heckenberg - One of the best experts on this subject based on the ideXlab platform.
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Multipole Expansion of strongly focussed laser beams
arXiv: Optics, 2003Co-Authors: Timo A Nieminen, Halina Rubinszteindunlop, N R HeckenbergAbstract:Multipole Expansion of an incident radiation field - that is, representation of the fields as sums of vector spherical wavefunctions - is essential for theoretical light scattering methods such as the T-matrix method and generalised Lorenz-Mie theory (GLMT). In general, it is theoretically straightforward to find a vector spherical wavefunction representation of an arbitrary radiation field. For example, a simple formula results in the useful case of an incident plane wave. Laser beams present some difficulties. These problems are not a result of any deficiency in the basic process of spherical wavefunction Expansion, but are due to the fact that laser beams, in their standard representations, are not radiation fields, but only approximations of radiation fields. This results from the standard laser beam representations being solutions to the paraxial scalar wave equation. We present an efficient method for determining the Multipole representation of an arbitrary focussed beam.
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Multipole Expansion of strongly focussed laser beams
Journal of Quantitative Spectroscopy & Radiative Transfer, 2003Co-Authors: Timo A Nieminen, Halina Rubinszteindunlop, N R HeckenbergAbstract:Multipole Expansion of an incident radiation field-that is, representation of the fields as sums of vector spherical wavefunctions-is essential for theoretical light scattering methods such as the T-matrix method and generalised Lorenz-Mie theory (GLMT). In general, it is theoretically straightforward to find a vector spherical wavefunction representation of an arbitrary radiation field. For example, a simple formula results in the useful case of an incident plane wave. Laser beams present some difficulties. These problems are not a result of any deficiency in the basic process of spherical wavefunction Expansion, but are due to the fact that laser beams, in their standard representations, are not radiation fields, but only approximations of radiation fields. This results from the standard laser beam representations being solutions to the paraxial scalar wave equation. We present an efficient method for determining the Multipole representation of an arbitrary focussed beam. (C) 2003 Elsevier Science Ltd. All rights reserved.
Bogumil Jeziorski - One of the best experts on this subject based on the ideXlab platform.
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convergence properties of the Multipole Expansion of the exchange contribution to the interaction energy
Molecular Physics, 2016Co-Authors: Piotr Gniewek, Bogumil JeziorskiAbstract:ABSTRACTThe conventional surface integral formula Jsurf[Φ] and an alternative volume integral formula Jvar[Φ] are used to compute the asymptotic exchange splitting of the interaction energy of the hydrogen atom and a proton employing the primitive function Φ in the form of its truncated Multipole Expansion. Closed-form formulas are obtained for the asymptotics of Jsurf[ΦN] and Jvar[ΦN], where ΦN is the Multipole Expansion of Φ truncated after the 1/RN term, R being the internuclear separation. It is shown that the obtained sequences of approximations converge to the exact result with the rate corresponding to the convergence radius equal to 2 and 4 when the surface and the volume integral formulas are used, respectively. When the Multipole Expansion of a truncated, Kth order polarisation function is used to approximate the primitive function, the convergence radius becomes equal to unity in the case of Jvar[Φ]. At low order, the observed convergence of Jvar[ΦN] is, however, geometric and switches to harmo...
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Convergence properties of the Multipole Expansion of the exchange contribution to the interaction energy
Molecular Physics, 2016Co-Authors: Piotr Gniewek, Bogumil JeziorskiAbstract:The conventional surface integral formula $J_{\rm surf}[\Phi]$ and an alternative volume integral formula $J_{\rm var}[\Phi]$ are used to compute the asymptotic exchange splitting of the interaction energy of the hydrogen atom and a proton employing the primitive function $\Phi$ in the form of its truncated Multipole Expansion. Closed-form formulas are obtained for the asymptotics of $J_{\rm surf}[\Phi_N]$ and $J_{\rm var}[\Phi_N]$, where $\Phi_N$ is the Multipole Expansion of $\Phi$ truncated after the $1/R^N$ term, $R$ being the internuclear separation. It is shown that the obtained sequences of approximations converge to the exact results with the rate corresponding to the convergence radius equal to 2 and 4 when the surface and the volume integral formulas are used, respectively. When the Multipole Expansion of a truncated, $K$th order polarization function is used to approximate the primitive function the convergence radius becomes equal to unity in the case of $J_{\textrm{var}}[\Phi]$. At low order the observed convergence of $J_{\rm var}[\Phi_N]$ is, however, geometric and switches to harmonic only at certain value of $N=N_c$ dependent on $K$. An equation for $N_c$ is derived which very well reproduces the observed $K$-dependent convergence pattern. The results shed new light on the convergence properties of the conventional SAPT Expansion used in applications to many-electron diatomics.
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exchange splitting of the interaction energy and the Multipole Expansion of the wave function
Journal of Chemical Physics, 2015Co-Authors: Piotr Gniewek, Bogumil JeziorskiAbstract:The exchange splitting J of the interaction energy of the hydrogen atom with a proton is calculated using the conventional surface-integral formula Jsurf[Φ], the volume-integral formula of the symmetry-adapted perturbation theory JSAPT[Φ], and a variational volume-integral formula Jvar[Φ]. The calculations are based on the Multipole Expansion of the wave function Φ, which is divergent for any internuclear distance R. Nevertheless, the resulting approximations to the leading coefficient j0 in the large-R asymptotic series J(R) = 2e(-R-1)R(j0 + j1R(-1) + j2R(-2) + ⋯) converge with the rate corresponding to the convergence radii equal to 4, 2, and 1 when the Jvar[Φ], Jsurf[Φ], and JSAPT[Φ] formulas are used, respectively. Additionally, we observe that also the higher jk coefficients are predicted correctly when the Multipole Expansion is used in the Jvar[Φ] and Jsurf[Φ] formulas. The symmetry adapted perturbation theory formula JSAPT[Φ] predicts correctly only the first two coefficients, j0 and j1, gives a wrong value of j2, and diverges for higher jn. Since the variational volume-integral formula can be easily generalized to many-electron systems and evaluated with standard basis-set techniques of quantum chemistry, it provides an alternative for the determination of the exchange splitting and the exchange contribution of the interaction potential in general.
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exchange splitting of the interaction energy and the Multipole Expansion of the wave function
arXiv: Chemical Physics, 2015Co-Authors: Piotr Gniewek, Bogumil JeziorskiAbstract:The exchange splitting $J$ of the interaction energy of the hydrogen atom with a proton is calculated using the conventional surface-integral formula $J_{\textrm{surf}}[\varphi]$, the volume-integral formula of the symmetry-adapted perturbation theory $J_{\textrm{SAPT}}[\varphi]$, and a variational volume-integral formula $J_{\textrm{var}}[\varphi]$. The calculations are based on the Multipole Expansion of the wave function $\varphi$, which is divergent for any internuclear distance $R$. Nevertheless, the resulting approximations to the leading coefficient $j_0$ in the large-$R$ asymptotic series $J(R) = 2 e^{-R-1} R ( j_0 + j_1 R^{-1} + j_2 R^{-2} +\cdots ) $ converge, with the rate corresponding to the convergence radii equal to 4, 2, and 1 when the $J_{\textrm{var}}[\varphi]$, $J_{\textrm{surf}}[\varphi]$, and $J_{\textrm{SAPT}}[\varphi]$ formulas are used, respectively. Additionally, we observe that also the higher $j_k$ coefficients are predicted correctly when the Multipole Expansion is used in the $J_{\textrm{var}}[\varphi]$ and $J_{\textrm{surf}}[\varphi]$ formulas. The SAPT formula $J_{\textrm{SAPT}}[\varphi]$ predicts correctly only the first two coefficients, $j_0$ and $j_1$, gives a wrong value of $j_2$, and diverges for higher $j_n$. Since the variational volume-integral formula can be easily generalized to many-electron systems and evaluated with standard basis-set techniques of quantum chemistry, it provides an alternative for the determination of the exchange splitting and the exchange contribution of the interaction potential in general.
Shinichiro Ohnuki - One of the best experts on this subject based on the ideXlab platform.
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Error Minimization of Multipole Expansion
SIAM Journal on Scientific Computing, 2005Co-Authors: Shinichiro Ohnuki, Weng Cho ChewAbstract:In this paper, we focus on the truncation error of the Multipole Expansion for the fast Multipole method and the multilevel fast Multipole algorithm. When the buffer size is large enough, the error can be controlled and minimized by using the conventional selection rules. On the other hand, if the buffer size is small, the conventional selection rules no longer hold, and the new approach which we have recently proposed is needed. However, this method is still not sufficient to minimize the error for small buffer cases. We clarify this fact and show that the information about the placement of true worst-case interaction is needed. A novel algorithm to minimize the truncation error is proposed.
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Truncation Error Analysis of Multipole Expansion
SIAM Journal on Scientific Computing, 2004Co-Authors: Shinichiro Ohnuki, Weng Cho ChewAbstract:The multilevel fast Multipole algorithm is based on the Multipole Expansion, which has numerical error sources such as truncation of the addition theorem, numerical integration, and interpolation/anterpolation. Of these, we focus on the truncation error and discuss its control precisely. The conventional selection rule fails when the buffer size is small compared to the desired numerical accuracy. We propose a new approach and show that the truncation error can be controlled and predicted regardless of the number of buffer sizes.
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Numerical accuracy of Multipole Expansion for 2D MLFMA
IEEE Transactions on Antennas and Propagation, 2003Co-Authors: Shinichiro Ohnuki, Weng Cho ChewAbstract:A numerical study of the Multipole Expansion for the multilevel fast Multipole algorithm (MLFMA) is presented. In the numerical implementation of MLFMA, the error comes from three sources: the truncation of the addition theorem; the approximation of the integration; the aggregation and disaggregation process. These errors are due to the factorization of the Green's function which is the mathematical core of the algorithm. Among the three error sources, we focus on the truncation error and a new approach of selecting truncation numbers for the addition theorem is proposed. Using this approach, the error prediction and control can be improved for the small buffer sizes and high accuracy requirements.
-
a new approach for controlling truncation error of the Multipole Expansion
IEEE Antennas and Propagation Society International Symposium, 2002Co-Authors: Shinichiro Ohnuki, Weng Cho ChewAbstract:The purpose of this paper is to clarify how to control the truncation error of the Multipole Expansion for FMM (fast Multipole method) and MLFMA (multilevel fast Multipole algorithm). The choice of truncation number still remains a complex issue. We perform the numerical experiment carefully and improve the control and prediction accuracy. The past selection rule is to use the excess bandwidth formula which becomes worse for small buffer sizes and high accuracy requirements. To improve this, a new approach is proposed.
Timo A Nieminen - One of the best experts on this subject based on the ideXlab platform.
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Multipole Expansion of strongly focussed laser beams
arXiv: Optics, 2003Co-Authors: Timo A Nieminen, Halina Rubinszteindunlop, N R HeckenbergAbstract:Multipole Expansion of an incident radiation field - that is, representation of the fields as sums of vector spherical wavefunctions - is essential for theoretical light scattering methods such as the T-matrix method and generalised Lorenz-Mie theory (GLMT). In general, it is theoretically straightforward to find a vector spherical wavefunction representation of an arbitrary radiation field. For example, a simple formula results in the useful case of an incident plane wave. Laser beams present some difficulties. These problems are not a result of any deficiency in the basic process of spherical wavefunction Expansion, but are due to the fact that laser beams, in their standard representations, are not radiation fields, but only approximations of radiation fields. This results from the standard laser beam representations being solutions to the paraxial scalar wave equation. We present an efficient method for determining the Multipole representation of an arbitrary focussed beam.
-
Multipole Expansion of strongly focussed laser beams
Journal of Quantitative Spectroscopy & Radiative Transfer, 2003Co-Authors: Timo A Nieminen, Halina Rubinszteindunlop, N R HeckenbergAbstract:Multipole Expansion of an incident radiation field-that is, representation of the fields as sums of vector spherical wavefunctions-is essential for theoretical light scattering methods such as the T-matrix method and generalised Lorenz-Mie theory (GLMT). In general, it is theoretically straightforward to find a vector spherical wavefunction representation of an arbitrary radiation field. For example, a simple formula results in the useful case of an incident plane wave. Laser beams present some difficulties. These problems are not a result of any deficiency in the basic process of spherical wavefunction Expansion, but are due to the fact that laser beams, in their standard representations, are not radiation fields, but only approximations of radiation fields. This results from the standard laser beam representations being solutions to the paraxial scalar wave equation. We present an efficient method for determining the Multipole representation of an arbitrary focussed beam. (C) 2003 Elsevier Science Ltd. All rights reserved.
