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A typical approach to solving partial differential or integral equations is to approximate their solution via a fixed (perhaps, an adaptive) basis and then find the coefficients of the representation. Nonlinear approximations express solutions via linear combinations of functions chosen from a set much larger than a basis. The problem then is to find an optimal (or near optimal) representation of the solution in this form while satisfying the underlying equations. Nonlinear approximations to find solutions of quantum chemistry problems (within oneparticle theories, e.g., HartreeFock) has its origins in seminal papers of Boys, Longstaff and Singer (1960). The authors used linear combinations of Gaussians and optimized their exponents and coefficients via Newton's method. Unfortunately, such approach turned out to be practical only for very small molecules. Consequently, construction of spatial orbitals has been performed offline and the resulting sets of functions used as a fixed basis. More recent multiresolution approaches (e.g., via multiwavelets), use dynamically refined selection of basis functions to represent spatial orbitals. However, since the basis functions do not resemble the spatial orbitals, such multiresolution methods require a relatively large number of parameters in resulting representations. Another example of nonlinear approximations are separated representations of multivariate functions. In a paper with M. Mohlenkamp and F. Prez (2008), we used such representations to solve the multiparticle Schrödinger equation. Since the wavefunction is antisymmetric, it is natural to approximate it as a sum of Slater determinants. Many current methods do so, but they impose additional structural constraints on the determinants, such as orthogonality between orbitals or an excitation pattern. Our method has no such constraints and instead, for a specified accuracy, we are looking for the best separated approximation to the wavefunction. Using new algorithms for optimizing nonlinear approximations via exponentials (or Gaussians), T. Haut and I recently developed an adaptive method for oneparticle theories. Compared to previous multiresolution approaches, we do not use bases and, as a result, significantly reduce the number of independent parameters needed to represent solutions. We are planning to use these functional forms within a separated representation to accelerate the algorithm for solving the Schrödinger equation (whose current implementation using multiwavelet bases is too slow to be competitive with oneparticle methods).
The talk will describe relevant nonlinear approximations, algorithms for constructing them, current results and, more generally, challenges in solving multivariate problems.
