An error estimator for separated representations of highly multidimensional models

By Amine Ammar, Francisco Chinesta, Pedro Díez, Antonio Huerta. 2010

134 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0baf88c1aa374e296e678e91c9cbcad2be1f31e7

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Citations in the corpus

Tensor-based methods for numerical homogenization from high-resolution images (2013, methodology): “The aim is here to provide a methodology for an adaptive approximation of the solution relying on the classical dual-primal error estimator [1, 2].”
Parametric solutions involving geometry: A step towards efficient shape optimization (2014, background): “The final number of terms n for convergence requires an error estimate, see, for instance, [24].”
Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models (2010, methodology): “In this section we are summarizing an error estimator procedure proposed in [8] and based on the use of primal and dual formulations.”
PGD-Based Computational Vademecum for Efficient Design, Optimization and Control (2013, background): “The enrichment is stopped as soon as the equation residual or any other error estimator based on a quantity of interest [8, 50] is small enough.”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “We refer to the review papers [46, 49], the literature [6, 45, 50, 77, 153, 154] and the references therein for details on PGD and its applications.”