Low-rank tensor methods for partial differential equations

Markus Bachmayr

doi:10.1017/S0962492922000125

Low-rank tensor methods for partial differential equations

2023-05-11

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Markus Bachmayr *

DOI：10.1017/S0962492922000125

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Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

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41A46

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65D40

65F55

65J10

65M12

65N12

65N25

65Y20

Low-rank tensor methods for partial differential equations

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