FAU Erlangen-Nuremberg

Multilevel Based All-At-Once Methods in PDE Constrained Optimization with Applications to Shape Optimization of Active Microfluidic Biochips

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Project leaders

Lehrstuhl für Angewandte Analysis mit Schwerpunkt Numerik
Uni Augsburg

Tel: 0821 598-2194
Tel: 0821 598-2192 (Sekr.)
Fax: 0821 598-2193

Email: rohop@math.uh.edu


Institut fuer Mathematik
Uni Augsburg

Tel: 0821 598-2190
Fax: 0821 598-2193

Email: siebert@math.uni-augsburg.de


Institut für Physik, Lehrstuhl für Experimentalphysik I
Uni Augsburg

Tel: 0821 598-3300
Tel: 0821 598-3327 (Sekr.)
Fax: 0821 598-3225

Email: achim.wixforth@physik.uni-augsburg.de


DFG funded assistants

Thomas Frommelt (Uni Augsburg)

Dr. Svetozara Petrova (Uni Augsburg)



This project within the area of PDE constrained optimization focuses on the development, analysis and implementation of optimization algorithms that combine efficient solution techniques from the numerics of PDEs, namely multilevel iterative solvers, and state-of-the-art optimization approaches, the so-called `all-at-once' optimization methods. It is well-known that multilevel techniques provide efficient PDE solvers of optimal algorithmic complexity. On the other hand, optimization methods within the all-at-once approach, such as sequential quadratic programming (SQP) methods and primal-dual Newton interior-point methods, have the appealing feature that in contrast to more traditional approaches, the numerical solution of the state equations is an integral part of the optimization routine. This is realized by incorporating the PDEs as constraints into the optimization routine. These strategies allow to save a considerable amount of computational work compared to methods that treat the PDE solution as an implicit function of the control/design variables. Moreover, the proper combination of multilevel techniques and optimization algorithms makes it possible to extract essential structural information from the originally infinite dimensional optimization problem.

This can not be done with respect to a single grid. We aim to develop and analyze multilevel preconditioners for optimization subproblems arising in SQP and primal-dual Newton interior-point methods including strategies to control the level of inexactness allowable in optimization subproblems, when using iterative subproblem solvers. Moreover, we will investigate strategies to use multilevel methods for detection of negative curvature and in path following methods. The developed PDE constrained optimization algorithms will be applied to the optimal design of microfluidic biochips with emphasis on the optimization of the geometry of the devices (shape optimization) in order to achieve an optimal operational behavior. Here, the state equations consist of a system of partial differential equations describing the transport of microfluids driven by piezoelectrically agitated surface acoustic waves. During the last couple of years, such biochips have attracted a considerable amount of interest, since pharmacology, molecular biology, and clinical diagnostics require the precise handling of precious, tiny samples and costly reagents in amounts of nanoliters. Biochips can transport such volumes and perform biochemical analysis of the samples.

Microfluidic biochips and microarrays are used in pharmaceutical, medical and forensic applications as well as in academic research and development for high throughput screening, genotyping and sequencing by hybridization in genomics, protein profiling in proteomics, and cytometry in cell analysis. Traditional technologies rely on fluorescent dyes, radioactive markers, or nanoscale gold-beads based on positive hybridization processes. However, these methods only allow a relatively small number of DNA probes per assay, and they only yield endpoint results and do not provide information about the kinetics of the processes. With the need for better sensitivity, flexibility, cost-effectiveness and a significant speed-up of hybridization, the current technological trend is obtained by the integration of the microfluidics on the chips itself. A new type of nanotechnological devices are surface acoustic wave driven microfluidic biochips. The experimental technique is based on piezoelectrically actuated surface acoustic waves on the surface of a chip which transport the droplet containing probe along a lithographically produced network channels to marker molecules placed at prespecified surface locations. Hence, these microfluidic biochips allow the in-situ investigation of the dynamics of hybridization processes with extremely high time resolution. The scientific merits of the proposal are the design, analysis and implementation of efficient algorithmic tools for a class of challenging problems in nonlinear optimization and the demonstration of their performance by the application to a real-life problem that has a significant impact on material sciences and life sciences. The broader technological impact of the project is a better design of microfluidic biochips with an improved performance for biochemical applications.

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