| Summary: | Wavelet-based algorithms are developed for solution of quantum and electromagnetic differential equations. Wavelets offer orthonormal localized bases with built-in multiscale properties for the representation of functions, differential operators, and multiplicative operators. The work described here is part of a series of tools for use in the ultimate goal of general, efficient, accurate and automated wavelet-based algorithms for solution of differential equations.
The most recent work, and the focus here, is the elimination of operator matrices in wavelet bases. For molecular quantum eigenvalue and dynamics calculations in multiple dimensions, it is the coupled potential energy matrices that generally dominate storage requirements. A Coefficient Product Approximation (CPA) for the potential operator and wave function wavelet expansions dispenses with the matrix, reducing storage and coding complexity. New developments are required, however. It is determined that the CPA is most accurate for specific choices of wavelet families, and these are given here. They have relatively low approximation order (number of vanishing wavelet function moments), which would ordinarily be thought to compromise both wavelet reconstruction and differentiation accuracy. Higher-order convolutional coefficient filters are determined that overcome both apparent problems. The result is a practical wavelet method where the effect of applying the Hamiltonian matrix to a coefficient vector can be calculated accurately without constructing the matrix.
The long-familiar Lanczos propagation algorithm, wherein one constructs and diagonalizes a symmetric tridiagonal matrix, uses both eigenvalues and eigenvectors. We show here that time-reversal-invariance for Hermitian Hamiltonians allows a new algorithm that avoids the usual need to keep a number Lanczos vectors around. The resulting Conjugate Symmetric Lanczos (CSL) method, which will apply for wavelets or other choices of basis or grid discretization, is simultaneously low-operation-count and low-storage. A modified CSL algorithm is used for solution of Maxwell's time-domain equations in Hamiltonian form for non-lossy media. The matrix-free algorithm is expected to complement previous work and to decrease both storage and computational overhead. It is expected- that near-field electromagnetic solutions around nanoparticles will benefit from these wavelet-based tools. Such systems are of importance in plasmon-enhanced spectroscopies.
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