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An error estimator for separated representations of highly multidimensional models. (English) Zbl 1231.74503

Summary: Fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics. Most of these models consist of a partial differential equation defined in a highly multidimensional domain (e.g. Schrodinger equation, Fokker-Planck equations among many others). The main challenge related to these models is their associated curse of dimensionality. We proposed in some of our former works a new strategy able to circumvent the curse of dimensionality based on the use of separated representations (also known as finite sums decomposition). This technique proceeds by computing at each iteration a new sum that consists of a product of functions each one defined in one of the model coordinates. The issue related to error estimation has never been addressed. This paper presents a first attempt on the accuracy evaluation of such a kind of discretization techniques.

MSC:

74S99 Numerical and other methods in solid mechanics
76M99 Basic methods in fluid mechanics
65N15 Error bounds for boundary value problems involving PDEs
00A71 General theory of mathematical modeling
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References:

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