Fully adaptive algorithms for multivariate integral equations using the non-standard form and multiwavelets with applications to the Poisson and bound-state Helmholtz kernels in three dimensions

We have developed and implemented a general formalism for fast numerical solution of time-independent linear partial differential equations as well as integral equations through the application of numerically separable integral operators in d ≥ 1 dimensions using the non-standard (NS) form. The proposed formalism is universal, compact and oriented towards the practical implementation into a working code using multiwavelets. The formalism is applied to the case of Poisson and bound-state Helmholtz operators in d = 3. Our algorithms are fully adaptive in the sense that the grid supporting each function is obtained on the fly while the function is being computed. In particular, when the function g = O f is obtained by applying an integral operator O, the corresponding grid is not obtained by transferring the grid from the input function f. This aspect has significant implications that will be discussed in the numerical section. The operator kernels are represented in a separated form with finite but arbitrary precision using Gaussian functions. Such a representation combined with the NS form allows us to build a sparse, banded representation of Green’s operator kernel. We have implemented a code for the application of such operators in a separated NS form to a multivariate function in a finite but, in principle, arbitrary number of dimensions. The error of the method is controlled, while the low complexity of the numerical algorithm is kept. The implemented code explicitly computes all the 22d components of the d-dimensional operator. Our algorithms are described in detail in the paper through pseudo-code examples. The final goal of our work is to be able to apply this method to build a fast and accurate Kohn–Sham solver for density functional theory.

[1]  M. Ratner.  Molecular electronic-structure theory , 2000 .

[2]  Wolfgang Dahmen,et al.  Adaptive application of operators in standard representation , 2006, Adv. Comput. Math..

[3]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[4]  李幼升,et al.  Ph , 1989 .

[5]  B. M. Fulk.  MATH , 1992 .

[6]  Gregory Beylkin,et al.  Fast adaptive algorithms in the non-standard form for multidimensional problems ✩ , 2007, 0706.0747.

[7]  Boris N. Khoromskij,et al.  Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part I. Separable Approximation of Multi-variate Functions , 2005, Computing.

[8]  Gregory Beylkin,et al.  Multiresolution quantum chemistry: basic theory and initial applications. , 2004, The Journal of chemical physics.

[9]  Gregory Beylkin,et al.  LU Factorization of Non-standard Forms and Direct Multiresolution Solvers , 1998 .

[10]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods II—Beyond the Elliptic Case , 2002, Found. Comput. Math..

[11]  C. Ross.  Found , 1869, The Dental register.

[12]  Bradley K. Alpert,et al.  Adaptive solution of partial di erential equations in multiwavelet bases , 2002 .

[13]  Wolfgang Dahmen,et al.  Adaptive methods for boundary integral equations: Complexity and convergence estimates , 2007, Math. Comput..

[14]  Andrew G. Glen,et al.  APPL , 2001 .

[15]  B. Alpert.  A class of bases in L 2 for the sparse representations of integral operators , 1993 .

[16]  Ronald R. Coifman,et al.  Wavelet-Like Bases for the Fast Solution of Second-Kind Integral Equations , 1993, SIAM J. Sci. Comput..

[17]  Gregory Beylkin,et al.  On the Adaptive Numerical Solution of Nonlinear Partial Differential Equations in Wavelet Bases , 1997 .

[18]  Wolfgang Dahmen,et al.  Adaptive Wavelet Schemes for Elliptic Problems - Implementation and Numerical Experiments , 2001, SIAM J. Sci. Comput..

[19]  A. Gilbert,et al.  A Multiresolution Method for Numerical Reduction and Homogenization of Nonlinear ODEs , 1998 .

[20]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[21]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[22]  Reinhold Schneider,et al.  Daubechies wavelets as a basis set for density functional pseudopotential calculations. , 2008, The Journal of chemical physics.

[23]  Wolfgang Dahmen,et al.  Compression Techniques for Boundary Integral Equations - Asymptotically Optimal Complexity Estimates , 2006, SIAM J. Numer. Anal..

[24]  Alberto Torchinsky,et al.  Real-Variable Methods in Harmonic Analysis , 1986 .

[25]  Martin J. Mohlenkamp,et al.  Algorithms for Numerical Analysis in High Dimensions , 2005, SIAM J. Sci. Comput..

[26]  Martin J. Mohlenkamp,et al.  Numerical operator calculus in higher dimensions , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[27]  P. Hohenberg,et al.  Inhomogeneous Electron Gas , 1964 .

[28]  Edward F. Valeev,et al.  Low-order tensor approximations for electronic wave functions: Hartree-Fock method with guaranteed precision. , 2011, The Journal of chemical physics.

[29]  Gregory Beylkin,et al.  Multiresolution quantum chemistry in multiwavelet bases: Hartree-Fock exchange. , 2004, The Journal of chemical physics.

[30]  M. H. Kalos,et al.  Monte Carlo Calculations of the Ground State of Three- and Four-Body Nuclei , 1962 .