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The authors focus on solving the high-dimensional Kolmogorov backward equation, which arises from the discretization of an infinite-dimensional Markov chain into a large finite dimension.
They propose a tensor network approach to approximate the Markov operator, which is the key to solving the Kolmogorov backward equation. The Markov operator is obtained under a functional hierarchical tensor (FHT) ansatz using a hierarchical sketching algorithm.
When the terminal condition admits an FHT ansatz, the proposed operator-based approach allows efficient computation of the PDE solution through tensor network contraction. It also provides an efficient way to solve the Kolmogorov forward equation when the initial distribution is in an FHT ansatz.
The authors apply the proposed approach successfully to two challenging time-dependent Ginzburg-Landau models with hundreds of variables, demonstrating its effectiveness in overcoming the curse of dimensionality.
The tensor network structure and hierarchical sketching algorithm are crucial components that enable the efficient representation and computation of the high-dimensional Markov operator.
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by Xun Tang,Lea... at arxiv.org 04-16-2024
https://arxiv.org/pdf/2404.08823.pdfDeeper Inquiries