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    Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems

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    We present a method for approximating the solution of a parameterized, nonlinear dynamical 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 governing equations. 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. It is particularly appropriate when one wishes to approximate the states at a few points in time without time marching from the initial conditions. We prove some interesting characteristics of the scheme including 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 conductive heat transfer - a nonlinear parabolic PDE with a random field model for the thermal conductivity.Comment: 28 pages, 8 figures, 2 table

    Application of Fredholm integral equations inverse theory to the radial basis function approximation problem

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    This paper reveals and examines the relationship between the solution and stability of Fredholm integral equations and radial basis function approximation or interpolation. The underlying system (kernel) matrices are shown to have a smoothing property which is dependent on the choice of kernel. Instead of using the condition number to describe the ill-conditioning, hence only looking at the largest and smallest singular values of the matrix, techniques from inverse theory, particularly the Picard condition, show that it is understanding the exponential decay of the singular values which is critical for interpreting and mitigating instability. Results on the spectra of certain classes of kernel matrices are reviewed, verifying the exponential decay of the singular values. Numerical results illustrating the application of integral equation inverse theory are also provided and demonstrate that interpolation weights may be regarded as samplings of a weighted solution of an integral equation. This is then relevant for mapping from one set of radial basis function centers to another set. Techniques for the solution of integral equations can be further exploited in future studies to find stable solutions and to reduce the impact of errors in the data
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