A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids

By Amine Ammar, B. Mokdad, Francisco Chinesta, Roland Keunings. 2006

612 citations recorded by the source corpus.

Kinetic theory models involving the Fokker-Planck equation can be accurately discretized using a mesh support (finite elements, finite differences, finite volumes, spectral techniques, etc.). However, these techniques involve a high number of approximation functions. In the finite element framework, widely used in complex flow simulations, each approximation function is related to a node that defines the associated degree of freedom. When the model involves high dimensional spaces (including physical and conformation spaces and time), standard discretization techniques fail due to an excessive computation time required to perform accurate numerical simulations. One appealing strategy that allows circumventing this limitation is based on the use of reduced approximation basis within an adaptive procedure making use of an efficient separation of variables. (c) 2006 Elsevier B.V. All rights reserved.

Subfields: Applied Mathematics, Computational Mechanics, Statistics and Probability

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “context of various applications, including Fokker-Planck equations [4] and stochastic partial differential equations [203].”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, methodology): “The first attempt to implement a tensor-structured solution method for a multispring problem traces back to [1], where a greedy algorithm giving the approximate solution as a sum of rank-1 tensors (the so-called canonical tensor format) was presented.”
PGD-Based Computational Vademecum for Efficient Design, Optimization and Control (2013, methodology): “In our earlier papers [1] and [3], we used Newton’s method.”
Self-consistent clustering analysis: An efficient multi-scale scheme for inelastic heterogeneous materials (2016, methodology): “Analytical micromechanical methods20 [14, 15, 16, 17], the Voronoi cell finite element method [18], fast Fourier transforms [19], spectral methods21 [20], the generalized method of cells [21], the transformation field analysis (TFA) [22], the nonuniform trans-22 formation field analysis (NTFA) [23, ”
Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations (2009, neutral): “[3, 66, 68, 92] for space-time separation, [26] for space-space separation, [2] for multi-dimensional separation).”
The LATIN multiscale computational method and the Proper Generalized Decomposition (2010, methodology): “This technique has already been used in the context of stochastic problems [19] and multi-dimensional problems [1, 2, 6].”
A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids - Part II: Transient simulation using space-time separated representations (2007, methodology): “13 and compared to the steady state simulation done in [3].”
An error estimator for separated representations of highly multidimensional models (2010, methodology): “We proposed in some of our former works [2,3] a separated representation strategy able to circumvent the curse of dimensionality that those models involve.”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, methodology): “We consider the case of 4 two-dimensional finitely extensible nonlinear elastic (FENE) springs in the shear flow regime according to [2],”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, background): “Alternatively, following the work in [4, 13], a greedy approach may be applied to perform updates only on the new separation term while freezing all the previous ones.”
Proper Generalized Decomposition and layer-wise approach for the modeling of composite plate structures (2013, methodology): “Over the past years, the Proper Generalized Decomposition (PGD) has shown interesting features in the reduction model framework (Ammar et al., 2006).”
Non incremental strategies based on separated representations: applications in computational rheology (2010, methodology): “Thus, the efficient application of the technique that we proposed in [3] requires the separability of all the fields involved in the model.”
Convergence of a greedy algorithm for high-dimensional convex nonlinear problems (2010, methodology): “Let us first describe a method which has been proposed by Nouy[10] and Chinesta[1], that is the fixed-point procedure and which we use in our final numerical implementation (see Section 6.2.2).”
Alleviating mesh constraints : Model reduction, parallel time integration and high resolution homogenization (2008, methodology): “In the following paragraphs we summarize the main elements of this technique that we have successfully applied for solving some multidimensional models usually encountered in the kinetic theory models of complex fluids in steady [2] and transient [3] regimes.”
Parametric solutions involving geometry: A step towards efficient shape optimization (2014, methodology): “Newton’s method is a straightforward alternative, see, for instance, [8, 9].”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, positive): “In this test, we consider the 3D Hookean model for a polymer chain [53, 54, 55, 56, 57] with 4 springs in the shear flow regime, which yields us the 12-dimensional problem, with the matrix A = I4×4 ⊗ 0 β 0 0 0 0 0 0 0 − 1 2 D, D = 1 2  2 −1 0 0 −1 2 −1 0 0 −1 2 −1 0 0 −1 2 ⊗ I3×3, β = 0.”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, methodology): “Some variants of this methodology have been proposed for the a priori construction of such separated representations of functions in tensor product spaces [27,28,41,5,1,21,29, 39].”