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Model reduction for parametric and nonlinear PDEs by matrix interpolation

An approach is proposed to construct parametrized reduced-order model (ROM) of the nonlinear and parametric partial differential equations (PDEs). The approach is based on the Kriging interpolation among local reduced-order matrices and the Discrete Empirical Interpolation Method (DEIM) of non-linear terms. These reduced-order matrices are first constructed and stored in appropriate matrix manifolds from several local original models in the parameter space. This step is done in the offline stage. The interpolation of the matrix manifolds is then performed in the online stage. The effectiveness of the approach is demonstrated by two numerical examples of a nonlinear Burgers equation and an unsteady contaminant transport problem.