Low-rank tensor methods for partial differential equations. (English) Zbl 07736652
Summary: 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.
MSC:
| 65-XX | Numerical analysis |
| 41A46 | Approximation by arbitrary nonlinear expressions; widths and entropy |
| 41A63 | Multidimensional problems |
| 65D40 | Numerical approximation of high-dimensional functions; sparse grids |
| 65F55 | Numerical methods for low-rank matrix approximation; matrix compression |
| 65J10 | Numerical solutions to equations with linear operators |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65Y20 | Complexity and performance of numerical algorithms |
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