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Description:
A main objective of numerical analysis and modeling is the simulation of complex technological problems, arising in engineering and natural sciences. Numerical simulations help to understand, design and optimize, or control and characterize systems or components. Usually the behaviour of a system is described by physical quantities like temperature, stress, or electromagnetic fields. These fields are solutions to partial differential equations (PDEs), which are stated on the domain of interest with appropriate boundary conditions. Since in general the analytical solution to a PDE is unavailable, a discretization procedure such as finite element, finite, discontinuous Galerkin, or finite volume method has to be applied. The discretized system is then solved numerically. For real world problems the numerical solution is usually expensive, regarding computational resources and time. Computational times can be of the order of seconds, up to hours and days, and even many problems can not be solved at all with reasonable effort due to their complexity. In engineering applications like optimization or parameter estimation the discretized models have to be solved multiply for different configurations of the system under consideration, for example, regarding geometrical or material parameters. Hence, a large number of solutions for different parameters are required in reasonable time (many-query context), or a single solution has to be computed very fast (real-time context). Even for moderate problems these requirements can often not be met with above discretization methods. In applications usually the output of interest is not the solution of the PDE itself, but some derived quantities. Hence, a method for fast and reliable evaluation of input-output relationships is desirable. The input are, for example, geometrical or material parameters of the system under consideration. The output is given implicitly as a functional of the field variable, which is the solution to the input parameter dependent PDE. The reduced basis ...