Automatic Synthesis of Low-Complexity Translation Operators for the Fast Multipole Method

We demonstrate a new, hybrid symbolic-numerical method for the automatic synthesis of all families of translation operators required for the execution of the Fast Multipole Method (FMM). Our method is applicable in any dimensionality and to any translation-invariant kernel. The Fast Multipole Method, of course, is the leading approach for attaining linear complexity in the evaluation of long-range (e.g. Coulomb) many-body interactions. Low complexity in translation operators for the Fast Multipole Method (FMM) is usually achieved by algorithms specialized for a potential obeying a specific partial differential equation (PDE). Absent a PDE or specialized algorithms, Taylor series based FMMs or kernel-independent FMM have been used, at asymptotically higher expense. When symbolically provided with a constant-coefficient elliptic PDE obeyed by the potential, our algorithm can automatically synthesize translation operators requiring $\mathrm{O}(p^d)$ operations, where $p$ is the expansion order and $d$ is dimension, compared with $\mathrm{O}(p^{2d})$ operations in a naive approach carried out on (Cartesian) Taylor expansions. This is achieved by using a compression scheme that asymptotically reduces the number of terms in the Taylor expansion and then operating directly on this ``compressed'' representation. Judicious exploitation of shared subexpressions permits formation, translation, and evaluation of local and multipole expansions to be performed in $\mathrm{O}(p^{d})$ operations, while an FFT-based scheme permits multipole-to-local translations in $\mathrm{O}(p^{d-1}\log(p))$ operations. We demonstrate computational scaling of code generation and evaluation as well as numerical accuracy through numerical experiments on a number of potentials from classical physics

Matrix compression by common subexpression elimination

Abstract In this report a method for common subexpression elimination in matrices is explored. The method is applied to several types of matrices occurring in numerical simulations. In all cases, the cost of a matrix-vector multiplication is reduced by a significant amount. The amount of storage required for the eliminated matrices is also less than that required for the original matrices. When the proposed method is applied to the Fourier transform matrix, the output is equivalent to the fast Fourier transform. For some matrices used in the fast multipole method for dislocation dynamics, the cost of a matrix-vector multiplication is reduced from O(p 6 ) to O(p 4.5 ), where p is the expansion order. Using an expansion order of 5, one can expect a factor of four speedup of the fast multipole part of a dislocation dynamics code

Matrix compression by common subexpression elimination

In this report a method for common subexpression elimination in matrices isexplored. The method is applied to several types of matrices occurring in numericalsimulations. In all cases, the cost of a matrix-vector multiplication is reduced by asignificant amount. The amount of storage required for the eliminated matrices isalso less than that required for the original matrices. When the proposed method isapplied to the Fourier transform matrix, the output is equivalent to the fast Fouriertransform. For some matrices used in the fast multipole method for dislocationdynamics, the cost of a matrix-vector multiplication is reduced from O(p^6) to O(p^4.5),where p is the expansion order. Using an expansion order of 5, one can expect a factorof four speedup of the fast multipole part of a dislocation dynamics code.QC 2010080

Matrix compression by common subexpression elimination

In this report a method for common subexpression elimination in matrices isexplored. The method is applied to several types of matrices occurring in numericalsimulations. In all cases, the cost of a matrix-vector multiplication is reduced by asignificant amount. The amount of storage required for the eliminated matrices isalso less than that required for the original matrices. When the proposed method isapplied to the Fourier transform matrix, the output is equivalent to the fast Fouriertransform. For some matrices used in the fast multipole method for dislocationdynamics, the cost of a matrix-vector multiplication is reduced from O(p^6) to O(p^4.5),where p is the expansion order. Using an expansion order of 5, one can expect a factorof four speedup of the fast multipole part of a dislocation dynamics code.QC 2010080

Matrix compression by common subexpression elimination

In this report a method for common subexpression elimination in matrices isexplored. The method is applied to several types of matrices occurring in numericalsimulations. In all cases, the cost of a matrix-vector multiplication is reduced by asignificant amount. The amount of storage required for the eliminated matrices isalso less than that required for the original matrices. When the proposed method isapplied to the Fourier transform matrix, the output is equivalent to the fast Fouriertransform. For some matrices used in the fast multipole method for dislocationdynamics, the cost of a matrix-vector multiplication is reduced from O(p^6) to O(p^4.5),where p is the expansion order. Using an expansion order of 5, one can expect a factorof four speedup of the fast multipole part of a dislocation dynamics code.QC 2010080