Error estimation in reduced basis method for systems with time-varying and nonlinear boundary conditions

Abstract

Many physical phenomena, such as mass transport and heat transfer, are modeled by systems of partial differential equations with time-varying and nonlinear boundary conditions. Control inputs and disturbances typically affect the system dynamics at the boundaries and a correct numerical implementation of boundary conditions is therefore crucial. However, numerical simulations of high-order discretized partial differential equations are often too computationally expensive for real-time and many-query analysis. For this reason, model complexity reduction is essential. In this paper, it is shown that the classical reduced basis method is unable to incorporate time-varying and nonlinear boundary conditions. To address this issue, it is shown that, by using a modified surrogate formulation of the reduced basis ansatz combined with a feedback interconnection and a input-related term, the effects of the boundary conditions are accurately described in the reduced-order model. The results are compared with the classical reduced basis method. Unlike the classical method, the modified ansatz incorporates boundary conditions without generating unphysical results at the boundaries. Moreover, a new approximation of the bound and a new estimate for the error induced by model reduction are introduced. The effectiveness of the error measures is studied through simulation case studies and a comparison with existing error bounds and estimates is provided. The proposed approximate error bound gives a finite bound of the actual error, unlike existing error bounds that grow exponentially over time. Finally, the proposed error estimate is more accurate than existing error estimates.