Gregory Beylkin : Solving Equations using nonlinear approximations

The idea of using nonlinear approximations as a tool for solving equations is as natural as that of using bases and, in fact, was proposed in 1960 in the context of quantum chemistry. The usual approach to solving partial differential and integral equations is to select a basis (possibly a multiresolution basis) or a grid, project equations onto such basis and solve the resulting discrete equations. The nonlinear alternative is to look for the solution within a large lass of functions (larger than any basis) by constructing optimal or near optimal approximations at every step of an algorithm for solving the equations. While this approach can theoretically be very efficient, the difficulties of constructing optimal approximations prevented any significant use of it in practice. However, during the last 15 years, nonlinear approximations have been successfully used to approximate operator kernels via exponentials or Gaussians to any user-specified accuracy, thus enabling a number of multidimensional multiresolution algorithms. In a new development several years ago, we constructed a fast and accurate reduction algorithm for optimal approximation of functions via exponentials or Gaussians (or, in a dual form, by rational functions) than can be used for solving partial differential and integral equations equations. We present two examples of the resulting solvers: one for the viscous Burgers' equation and another for solving the Hartree-Fock equations of quantum chemistry. Burgers' equation is often used as a testbed for numerical methods: if the viscosity \vu; is small, its solutions develop sharp (moving) transition regions of width O (\vu) presenting significant challenges for numerical methods. Using nonlinear approximations for solving the Hartree-Fock equations is the first step to a wider use of the approach in quantum chemistry. We maintain a functional form for the spatial orbitals as a linear combinations of products of decaying exponentials and spherical harmonics entered at the nuclear cusps. While such representations are similar to the classial Slater-type orbitals, in the course of computation we optimize both the exponents and the coefficients in order to achieve an efficient representation of solutions and to obtain guaranteed error bounds.

• Category: Applied Math and Analysis
• Duration: 01:34:49
• Date:  February 22, 2016 at 4:25 PM
• Views: 100
• Tags:   seminar, Applied Math And Analysis Seminar