Title: Towards information optimal simulation of partial differential equations,
Abstract: Most simulation schemes for partial differential equations (PDEs) focus on
minimizing a simple error norm of a discretized version of a field. This paper
takes a fundamentally different approach; the discretized field is interpreted
as data providing information about a real physical field that is unknown. This
information is sought to be conserved by the scheme as the field evolves in
time. Such an information theoretic approach to simulation was pursued before
by information field dynamics (IFD). In this paper we work out the theory of
IFD for nonlinear PDEs in a noiseless Gaussian approximation. The result is an
action that can be minimized to obtain an informationally optimal simulation
scheme. It can be brought into a closed form using field operators to calculate
the appearing Gaussian integrals. The resulting simulation schemes are tested
numerically in two instances for the Burgers equation. Their accuracy surpasses
finite-difference schemes on the same resolution. The IFD scheme, however, has
to be correctly informed on the subgrid correlation structure. In certain
limiting cases we recover well-known simulation schemes like spectral Fourier
Galerkin methods. We discuss implications of the approximations made.