Residual Minimizing Model Reduction for Parameterized Nonlinear Dynamical Systems. (arXiv:1012.0351v1 [math.NA])

Residual Minimizing Model Reduction for Parameterized Nonlinear Dynamical Systems. (arXiv:1012.0351v1 [math.NA]): "

We present a method for approximating the solution of a parameterized,
nonlinear dynamical (or static) system using an affine combination of solutions
computed at other points in the input parameter space. The coefficients of the
affine combination are computed with a nonlinear least squares procedure that
minimizes the residual of the dynamical system. The approximation properties of
this residual minimizing scheme are comparable to existing reduced basis and
POD-Galerkin model reduction methods, but its implementation requires only
independent evaluations of the nonlinear forcing function. We prove some
interesting characteristics of the scheme including uniqueness and an
interpolatory property, and we present heuristics for mitigating the effects of
the ill-conditioning and reducing the overall cost of the method. We apply the
method to representative numerical examples from kinetics - a three state
system with one parameter controlling the stiffness - and groundwater modeling
- a nonlinear parabolic PDE with a random field model for the hydraulic
conductivity.

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