Optimization and model reduction of time dependent PDE-constrained optimization problems: Applications to surface acoustic wave driven microfluidic biochips.

摘要

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 advection-diffusion equations and derive explicit error bounds for the modeling error.;Another application considered is the shape optimization of an aorto-coronaric bypass. Finally, in order to address environmental issues, we present an optimal control problem where our aim is to reduce the water pollution in a region of choice.;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 second-order compressible Stokes system. The second-order compressible Stokes system provides an appropriate model for the optimal design of the capillary barriers.