We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.
Purpose
The purpose of this paper is to present the implementation of a balanced domain decomposition approach for the numerical simulation of large electro-quasistatic (EQS) systems in biology. The numerical scheme is analyzed and first applications are discussed.
Design/methodology/approach
The scheme is based on a finite element discretization of the individual domains obtained by decomposition and a physically consistent inter-domain coupling realized via Robin boundary conditions. The proposed algorithms can efficiently be implemented on a highly parallelized computing grid.
Findings
The feasibility and applicability of the method is proven. Further, a couple of technical details are found that increase the efficiency of the method.
Originality/value
The presented method offers an enhanced geometrical flexibility and extensibility to simulate larger cell systems with higher model resolution compared to other methods presented in the literature. The presented analysis provides an understanding of the balanced coupling scheme for large EQS systems.
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