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The global behavior of dynamical systems can be studied by analyzing theeigenvalues and corresponding eigenfunctions of linear operators associatedwith the system. Two important operators which are frequently used to gaininsight into the system's behavior are the Perron-Frobenius operator and theKoopman operator. Due to the curse of dimensionality, computing theeigenfunctions of high-dimensional systems is in general infeasible. We willpropose a tensor-based reformulation of two numerical methods for computingfinite-dimensional approximations of the aforementioned infinite-dimensionaloperators, namely Ulam's method and Extended Dynamic Mode Decomposition (EDMD).The aim of the tensor formulation is to approximate the eigenfunctions bylow-rank tensors, potentially resulting in a significant reduction of the timeand memory required to solve the resulting eigenvalue problems, provided thatsuch a low-rank tensor decomposition exists. Typically, not all variables of ahigh-dimensional dynamical system contribute equally to the system's behavior,often the dynamics can be decomposed into slow and fast processes, which isalso reflected in the eigenfunctions. Thus, the weak coupling between differentvariables might be approximated by low-rank tensor cores. We will illustratethe efficiency of the tensor-based formulation of Ulam's method and EDMD usingsimple stochastic differential equations.

Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator