Abstract
In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.
Highlights
Computational modelling is an indispensable tool for analysis, design, optimization and control
The empirical interpolation method interpolates the nonlinear function at selected spatial locations using an empirically derived basis, and is applied directly to the governing partial differential equation (PDE) [7], while the discrete empirical interpolation method (DEIM) is applicable to general ordinary differential equation (ODE) or algebraic systems arising from a spatial discretization [25]
We introduce an extension of proper orthogonal decomposition (POD) for dynamic, parametrized, linear and nonlinear PDEs
Summary
Computational modelling is an indispensable tool for analysis, design, optimization and control. The empirical interpolation method interpolates the nonlinear function at selected spatial locations using an empirically derived basis, and is applied directly to the governing PDE [7], while the discrete empirical interpolation method (DEIM) is applicable to general ODE or algebraic systems arising from a spatial discretization [25] Both methods construct a subspace for the approximation of the nonlinear term and use a greedy algorithm to select interpolation points. The Gauss–Newton with approximated tensors method operates at the fully discrete level in space and time, and is based on satisfying consistency and discrete-optimality conditions by solving a residual-minimization problem [27] This leads to a Petrov–Galerkin (rather than Galerkin) problem with a test basis that depends on the residual derivatives w.r.t. the state variable. In appendix A, we provide details of the method of snapshots and singular value decomposition (SVD), the latter of which we use in practice
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