## Seminar in Numerical Analysis: Steffen Börm (Universität Kiel)

**Hierarchical Tensor Approximation**

In the context of stochastic partial differential equation, we are frequently faced with equations in high-dimensional domains. In order to obtain efficient numerical methods for these equations, we have to take local regularity properties of the solution into account, e.g., by using locally refined finite element meshes. Extending standard meshing algorithms to higher dimensions poses a significant challenge.

We propose an alternative: the Galerkin trial space is constructed using a partition of unity. By multiplying local cut-off functions with polynomials, we can obtain discretizations of arbitrary order, and local grid refinement can be realized by reducing the supports of the cut-off functions. The main challenge lies in the construction of the corresponding system matrix, since even determining the sparsity pattern involves interactions between cut-off functions on different levels of the mesh hierarchy.

Our approach leads to a sparse system matrix, the basis functions are convenient tensor products of functions on lower-dimensional domains, and local regularity can be exploited by variable-order interpolation in order to obtain close to optimal complexity.

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iCal