^{1,a)}

### Abstract

We present a two-stage error estimation scheme for the fast multipole method (FMM). This scheme can be applied to any particle system. It incorporates homogeneous as well as inhomogeneous distributions. The FMM error as a consequence of the finite representation of the multipole expansions and the operator error is correlated with an absolute or relative user-requested energy threshold. Such a reliable error control is the basis for making reliable simulations in computational physics. Our FMM program on the basis of the two-stage error estimation scheme is available on request.

I. INTRODUCTION

A. Laser plasma interaction

II. THEORY

A. First stage of the error estimation scheme

B. Second stage of the error estimation scheme

III. RESULTS

IV. SUMMARY

### Key Topics

- Interpolation
- 5.0
- Plasma expansion
- 5.0
- Polynomials
- 5.0
- Coulomb explosion
- 4.0
- Laser plasma interactions
- 4.0

## Figures

The positions of the two box centers. Subscript 1 refers to box 1 and subscript 2 to box 2, respectively.

The positions of the two box centers. Subscript 1 refers to box 1 and subscript 2 to box 2, respectively.

as a function of the level of poles.

as a function of the level of poles.

The minimal and maximal ratio of the exact far field energy and the multipole approximation of for . The minimal ratio is illustrated in (a). The maximal ratio is illustrated in (b). The charges and are located in box 1 with origin and box 2 with origin , respectively.

The minimal and maximal ratio of the exact far field energy and the multipole approximation of for . The minimal ratio is illustrated in (a). The maximal ratio is illustrated in (b). The charges and are located in box 1 with origin and box 2 with origin , respectively.

The terms and as functions of .

The terms and as functions of .

The positions of the two box centers along the axis. Subscript 1 refers to box 1 and subscript 2 to box 2, respectively.

The positions of the two box centers along the axis. Subscript 1 refers to box 1 and subscript 2 to box 2, respectively.

2D representation of a particle system as a result of a laser-induced Coulomb explosion. The two axes range (a) from 0 to 1, all 114 537 particles in 100% of the volume, (b) from 0.4375 to 0.5625, 78 946 particles (68.9% of all particles) in 1/512 of the volume, (c) from 0.492 187 5 to 0.507 812 5, 41 684 particles (36.4% of all particles) in 1/262 144 of the volume, and (d) from 0.499 023 437 5 to 0.500 976 562 5, 21 892 particles (19.1% of all particles) in 1/134 217 728 of the volume of the simulation box are shown.

2D representation of a particle system as a result of a laser-induced Coulomb explosion. The two axes range (a) from 0 to 1, all 114 537 particles in 100% of the volume, (b) from 0.4375 to 0.5625, 78 946 particles (68.9% of all particles) in 1/512 of the volume, (c) from 0.492 187 5 to 0.507 812 5, 41 684 particles (36.4% of all particles) in 1/262 144 of the volume, and (d) from 0.499 023 437 5 to 0.500 976 562 5, 21 892 particles (19.1% of all particles) in 1/134 217 728 of the volume of the simulation box are shown.

Levels of poles for three different error estimators depending on user-requested absolute energy errors for a system consisting of 100 000 particles each with charge 1 distributed along the axis . The solid line shows the level of poles as a result of our FMM error estimation scheme. The dashed and dot-dashed line show the level of poles due to the truncation of the expansions and the use of operator , respectively (Ref. 4). The FMM separation criterion is equal to 1. The solid and dashed line show a crossover point at a user-requested absolute energy error of .

Levels of poles for three different error estimators depending on user-requested absolute energy errors for a system consisting of 100 000 particles each with charge 1 distributed along the axis . The solid line shows the level of poles as a result of our FMM error estimation scheme. The dashed and dot-dashed line show the level of poles due to the truncation of the expansions and the use of operator , respectively (Ref. 4). The FMM separation criterion is equal to 1. The solid and dashed line show a crossover point at a user-requested absolute energy error of .

## Tables

depending on the level of poles for .

depending on the level of poles for .

The terms and for .

The terms and for .

Comparison of user-requested absolute energy errors with the absolute energy errors of FMM calculations for a system consisting of 100 000 particles each with charge 1 distributed along the axis . .

Comparison of user-requested absolute energy errors with the absolute energy errors of FMM calculations for a system consisting of 100 000 particles each with charge 1 distributed along the axis . .

Comparison of user-requested absolute energy errors with the absolute energy errors of FMM calculations for a system consisting of 114 537 inhomogeneously distributed positive charges. .

Comparison of user-requested absolute energy errors with the absolute energy errors of FMM calculations for a system consisting of 114 537 inhomogeneously distributed positive charges. .

Levels of poles for three different error estimators depending on user-requested absolute energy errors for a system consisting of 100 000 particles each with charge 1 distributed along the axis . The second column shows the levels of poles as a result of our FMM error estimation scheme. The next two columns show the levels of poles due to the truncation of the expansions and the use of operator , respectively, (Ref. 4). The FMM separation criterion is equal to 1.

Levels of poles for three different error estimators depending on user-requested absolute energy errors for a system consisting of 100 000 particles each with charge 1 distributed along the axis . The second column shows the levels of poles as a result of our FMM error estimation scheme. The next two columns show the levels of poles due to the truncation of the expansions and the use of operator , respectively, (Ref. 4). The FMM separation criterion is equal to 1.

Levels of poles for two different error estimators depending on user-requested relative energy errors and the resulting ratios of the number of floating point operations in the multipole-to-local translations of the rotation based FMM for homogeneously distributed particles each with charge 1. The Cartesian coordinates are given by .

Levels of poles for two different error estimators depending on user-requested relative energy errors and the resulting ratios of the number of floating point operations in the multipole-to-local translations of the rotation based FMM for homogeneously distributed particles each with charge 1. The Cartesian coordinates are given by .

The first coefficients of the continuous Chebyshev expansion.

The first coefficients of the continuous Chebyshev expansion.

Levels of poles for two different error estimators depending on user-requested relative energy errors and the resulting ratios of the number of floating point operations in the multipole-to-local translations of the rotation based FMM for homogeneously distributed particles each with charge 1. The standard error estimation is improved by Chebyshev economization. The Cartesian coordinates are given by .

Levels of poles for two different error estimators depending on user-requested relative energy errors and the resulting ratios of the number of floating point operations in the multipole-to-local translations of the rotation based FMM for homogeneously distributed particles each with charge 1. The standard error estimation is improved by Chebyshev economization. The Cartesian coordinates are given by .

Comparison of the levels of poles determined by the first and second stage of the FMM error estimation scheme for homogeneously distributed particles each with charge 1. The requested errors are user-requested relative energy errors. The Cartesian coordinates are given by .

Comparison of the levels of poles determined by the first and second stage of the FMM error estimation scheme for homogeneously distributed particles each with charge 1. The requested errors are user-requested relative energy errors. The Cartesian coordinates are given by .

Scaling of the two stages of the FMM error estimation scheme with respect to the number of particles for homogeneously distributed particles each with charge 1. The numbers in columns 2–4 show the increase in the number of floating point operations with respect to the eight times smaller particle system. The scaling is .

Scaling of the two stages of the FMM error estimation scheme with respect to the number of particles for homogeneously distributed particles each with charge 1. The numbers in columns 2–4 show the increase in the number of floating point operations with respect to the eight times smaller particle system. The scaling is .

Comparison of relative errors of two FMM calculations, one with use of the operator (local-to-local translation) and the second one without for single, double, and quadruple precision. The threshold depends on the length of the mantissa in the binary floating point representation.

Comparison of relative errors of two FMM calculations, one with use of the operator (local-to-local translation) and the second one without for single, double, and quadruple precision. The threshold depends on the length of the mantissa in the binary floating point representation.

The utilization of precision increase of the binary floating point representation. The relative errors of energies, potential, and gradient decrease in the same manner as the length of the mantissa increases from 52 (double precision ) to 112 (quadruple precision ).

The utilization of precision increase of the binary floating point representation. The relative errors of energies, potential, and gradient decrease in the same manner as the length of the mantissa increases from 52 (double precision ) to 112 (quadruple precision ).

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