An error estimator for separated representations of highly multidimensional models

• Published in: Computer methods in applied mechanics and engineering Vol. 199; no. 25; pp. 1872 - 1880
• Main Authors: Ammar, A., Chinesta, F., Diez, P., Huerta, A.
• Format: Journal Article Publication
• Language: English
• Published: Kidlington Elsevier B.V 01.01.2010; Elsevier
• Subjects: Anàlisi d'error (Matemàtica); Computer simulation; Curse of dimensionality; Discretization; Engineering Sciences; Enginyeria dels materials; Enginyeria mecànica; Error analysis (Mathematics); Error estimation; Errors; Exact sciences and technology; Finite sums decomposition; Fundamental areas of phenomenology (including applications); Física; High dimensional models; Materials; Mathematical analysis; Mathematical models; Mechanics; Mecànica estadística; Micromechanics; Micromecànica; Physics; Quantum statistical mechanics; Representations; Schroedinger equation; Separated representations; Solid mechanics; Statistical mechanics; Statistical physics, thermodynamics, and nonlinear dynamical systems; Structural and continuum mechanics; Àrees temàtiques de la UPC; Finite sums decomposition; Curse of dimensionality; High dimensional models; Error estimation; Separated representations; Modeling; Schrödinger equation; Fine structure; System with n degrees of freedom; Nanometer scale; Quantum mechanics; Fokker Planck equation; Multidimensional model; Statistical mechanics; Structural analysis
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2010.02.012

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.