Optimization and Model Reduction of Time Dependent Pde-constrained Optimization Problems: Applications to Surface Acoustic Wave Driven Microfluidic Biochips

The optimal design of structures and systems described by partial differential equations (PDEs) often gives rise to large-scale optimization problems, in particular if the underlying system of PDEs represents a multiscale, multiphysics problem. Therefore, reduced order modeling techniques such as balanced truncation model reduction (BTMR), proper orthogonal decomposition (POD), or reduced basis methods (RB) are used to significantly decrease the computational complexity while maintaining the desired accuracy of the approximation. We are interested in such shape optimization problems where the design issue is restricted to a relatively small portion of the computational domain and in optimal control problems where the nonlinearity is local in nature. In these cases, it appears to be natural to rely on a full order model only in that specific part of the domain and to use a reduced order model elsewhere. A convenient methodology to realize this idea is a suitable combination of domain decomposition techniques and BTMR. We will consider such an approach for optimal control and shape optimization problems governed by advectiondiffusion equations and derive explicit error bounds for the modeling error. As an application in life sciences, we will be concerned with the optimal design of capillary barriers as part of a network of microchannels and reservoirs on surface acoustic wave driven microfluidic biochips. Here, the state equations represent a multiscale multiphysics problem consisting of the linearized equations of piezoelectricity and the compressible Navier-Stokes equations. The multiscale character is due to the occurrence of fluid flow on different time scales. A standard homogenization approach by means of a state parameter results in a first-order time periodic linearized compressible Navier-Stokes equations and a secondorder compressible Stokes system. The second-order compressible Stokes system provides