A multiwavelet approach to the direct solution of the Poisson equation: implementation and optimization

This work is presenting a fully numerical approach for the direct solution of the Poisson equation for the electrostatic potential given by an arbitrary charge density. Efficient solution of this equation is important in many fields of science, where the current work is dealing with nuclear and electronic potential calculations, used in the field of computational chemistry. The equation is solved using the mathematical framework of multiwavelets, which is a theory that enables us to represent functions and operators with rigorous error control. The multiwavelet basis is well suited to treat the multiple length scales present in the calculation of electronic structure. A prototype implementation of the solution of the Poisson equation using the multiwavelet formalism has previously been obtained by the computational chemistry group at the University of Tromso¸, and the current work is dealing with optimization of the existing code and with the development of the code into new areas of applicability, specifically to the calculation of electronic structure using the framework of density functional theory

Kinetic energy-free Hartree–Fock equations: an integral formulation

We have implemented a self-consistent feld solver for Hartree–Fock calculations, by making use of Multiwavelets and Multiresolution Analysis. We show how such a solver is inherently a preconditioned steepest descent method and therefore a good starting point for rapid convergence. A distinctive feature of our implementation is the absence of any reference to the kinetic energy operator. This is desirable when Multiwavelets are employed, because diferential operators such as the Laplacian in the kinetic energy are challenging to represent correctly. The theoretical framework is described in detail and the implemented algorithm is both presented in the paper and made available as a Python notebook. Two simple examples are presented, highlighting the main features of our implementation: arbitrary predefned precision, rapid and robust convergence, absence of the kinetic energy operator

An operator-customized wavelet-finite element approach for the adaptive solution of second-order partial differential equations on unstructured meshes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Civil and Environmental Engineering, 2005.Includes bibliographical references (p. 139-142).Unlike first-generation wavelets, second-generation wavelets can be constructed on any multi-dimensional unstructured mesh. Instead of limiting ourselves to the choice of primitive wavelets, effectively HB detail functions, we can tailor the wavelets to gain additional qualities. In particular, we propose to customize our wavelets to the problem's operator. For any given linear elliptic second-order PDE, and within a Lagrangian FE space of any given order, we can construct a basis of compactly supported wavelets that are orthogonal to the coarser basis functions with respect to the weak form of the PDE. We expose the connection between the wavelet's vanishing moment properties and the requirements for operator-orthogonality in multiple dimensions. We give examples in which we successfully eliminate all scale-coupling in the problem's multi-resolution stiffness matrix. Consequently, details can be added locally to a coarser solution without having to re-compute the coarser solution.The Finite Element Method (FEM) is a widely popular method for the numerical solution of Partial Differential Equations (PDE), on multi-dimensional unstructured meshes. Lagrangian finite elements, which preserve C⁰ continuity with interpolating piecewise-polynomial shape functions, are a common choice for second-order PDEs. Conventional single-scale methods often have difficulty in efficiently capturing fine-scale behavior (e.g. singularities or transients), without resorting to a prohibitively large number of variables. This can be done more effectively with a multi-scale method, such as the Hierarchical Basis (HB) method. However, the HB FEM generally yields a multi-resolution stiffness matrix that is coupled across scales. We propose a powerful generalization of the Hierarchical Basis: a second-generation wavelet basis, spanning a Lagrangian finite element space of any given polynomial order.by Stefan F. D'Heedene.Ph.D