The eigenfunctions of a differential operator (or a discretized differential operator, or a matrix of a similar type) need not each be separately represented. A compact multilevel collective structure can be organized, in which many eigenfunctions have the same fine-level representation; they are distinguished only in the way in which that representation is mollified, where the mollification is a point-by-pont multiplication by a smooth function which is represented on the next coarser grid. At that coarser level, many of the mollification functions can again use a common representation, etc., so at increasingly coarser levels more and more eigensets are separated out, each becoming progressively more specific.
This makes it possible to calculate N eigenfunctions in just O(N log N) computer operations, using only O(N log N) storage. By comparison, a usual multigrid eigensolver would require at least O(N**3) operations and O(N**2) storage. Moreover the multiscale eigenbasis (MEB) structure allows to expand any given function in terms of the N eigenfunctions in again just O(N log N) operations. This is a vast generalization of the Fast Fourier Transform (FFT), which achieves the same complexity order (with a smaller prefactor), but is limited to the eigenbasis of differential operators with constant coefficients, periodic boundary conditions and uniform discretization.
The multiplicative Galerkin mode of coarsening the eigenproblem avoids certain flaws in classical multigrid eigensolvers: it yields coarse equations that have themselves the form of an eigenproblem, whose eigenvectors correspond one-to-one to fine eigenvectors; orthogonalizations need only be performed at coarse levels; and arbitrarily coarse grids can be employed even in calculating high eigenfunctions. This mode is particularly natural and useful for algebraic multigrid (AMG), since it involves construction of the intergrid interpolation operators.
The MEB structure also allows very efficient eigen calculations in unbounded domains for operators that have asymptotically constant or periodic coefficients, which is the usual situation in many applications, including molecular and condensed-matter electronic structure calculations.
The work is a collaboration in progress with Drs. Oren Livne and Ira Livshits.