In many application areas like numerical design, parameter estimation, and inverse problems, the multiple solution of a parametrized model is required. In these real-time and many query contexts short computational times become indispensable. Often however, already a single forward solution takes several hours. The reduced basis method allows to split up the solution process of a parametrized problem into an expensive offline and a cheap online step. In the offline step the reduced model is built self-adaptively and long computational times are accepted. In the online step the reduced model can then be solved in the order of seconds. Rigorous error estimation techniques assure the reliability of the reduced basis solutions.

Reduced basis technique

The basic idea of the reduced basis method is depicted in the following.

The set of possible solutions of a parametrized partial differential equation (PDE) is given as a manifold in the underlying high dimensional finite element space, which is used for discretization of the PDE. The reduced basis method constructs a low dimensional approximation to this manifold from so called snapshot solutions, which are rigorous solutions of the original problem for fixed parameters values. The original parametrized model is then projected onto this reduced basis which dramatically reduces the complexity of the problem and allows fast online solution.

Application example

An application of the reduced basis method is shape optimization of photomasks. Since optical systems act as low pass filters, corner rounding is introduced when a rectangle is imaged onto a waver. Optical proximity correction introduces assist features like serifs to the photomask in order to improve the quality of the image.

The geometrical optimization of such assist features is a typical many query application, which can be solved very fast with the reduced basis method.

• J. Pomplun, F. Schmidt
  Accelerated a posteriori error estimation for the reduced basis method with application to 3D electromagnetic scattering problems
  SIAM J. Sci. Comput. (accepted)
• J. Pomplun, L. Zschiedrich, S. Burger, F. Schmidt
  Reduced basis method for computational lithography
  In: Numerical Methods in Optical MetrologyProc. SPIE,  Vol. 7488 (2009)
• J. Pomplun, F. Schmidt
  Reduced basis method for fast and robust simulation of electromagnetic scattering problems
  In: Numerical Methods in Optical MetrologyProc. SPIE,  Vol. 7390 (2009)
• J. Pomplun, F. Schmidt
  Reduced Basis Method for Electromagnetic Field Computations
  In: Scientific Computing in Electrical Engineering, Proceedings of the SCEE-2008Springer (2008)