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Recursive Biorthogonal Decomposition of Multivariate Functions and Nonlinear Partial Differential Equations


We develop a numerical method for the decomposition of multivariate functions based on recursively

applying biorthogonal decompositions in function spaces. The result is an approximation of

the multivariate function by sums of products of univariate functions. Decompositions of this type

can conveniently be visualized by binary trees and in some sense are a functional analog of the decompositions

in tensor numerical methods that are obtained through sequences of matrix reshaping

and singular value decomposition. The underlying theory of recursive biorthogonal decomposition

in function spaces is developed and computational aspects are discussed. This decomposition is

generalized to handle time dependence in such a way which allows for the decomposition and propagation

of solutions to nonlinear time dependent partial differential equations. In this way we obtain

a numerical solution for time dependent problems which remains on a low parametric manifold of

constant rank for all time. We also discuss the addition and removal of time dependent modes during

propagation to allow for robust adaptive solvers. Applications to prototype linear hyperbolic

problems are presented and discussed.

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