Lecture SS 25 Numerical Simulation
High-dimensional approximation
With the numerical solution of partial differential equations understood in many situations, a key challenge for engineers nowadays is the emergence of many-query applications. These are situations in which the repeated solution of partial differential equations with different input data is required. To accelerate these applications, surrogate models are essential, referring to sufficiently accurate approximations of the parameter-to-solution map. From a mathematical perspective, creating a surrogate model corresponds to the approximation of (Banach space-valued) functions with infinitely many variables. In this lecture, we will discuss the construction of efficient surrogate models through sparse grid approximations, kernel approximation, and quasi-Monte Carlo methods. If time permits, we will also explore how this theory can be utilized to derive existence results in the approximation theory for certain classes of neural networks.
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