Use Of Separated Representations Research Articles

Review of Reduced Order Models for Heat and Moisture Transfer in Building Physics with Emphasis in PGD Approaches

This paper presents a review of the use of model reduction techniques for building physics applications. The use of separated representations, the so called Proper Generalised Decomposition (PGD), is particularly investigated. This technique can be applied for efficient building physics modelling at different levels: the wall and multizone models, the whole-building (coupled envelope and air volumes) simulation and their practical applications. The PGD can be formulated as a space-time representation to provide fast and accurate solutions of 2- and 3-dimensional problems at the wall or the whole-building level. Furthermore, the PGD solution allows to treat efficiently parametric problems of practical building applications.

Efficient Stabilization of Advection Terms Involved in Separated Representations of Boltzmann and Fokker-Planck Equations

AbstractThe fine description of complex fluids can be carried out by describing the evolution of each individual constituent (e.g. each particle, each macromolecule, etc.). This procedure, despite its conceptual simplicity, involves many numerical issues, the most challenging one being that related to the computing time required to update the system configuration by describing all the interactions between the different individuals. Coarse grained approaches allow alleviating the just referred issue: the system is described by a distribution function providing the fraction of entities that at certain time and position have a particular conformation. Thus, mesoscale models involve many different coordinates, standard space and time, and different conformational coordinates whose number and nature depend on the particular system considered. Balance equation describing the evolution of such distribution function consists of an advection-diffusion partial differential equation defined in a high dimensional space. Standard mesh-based discretization techniques fail at solving high-dimensional models because of the curse of dimensionality. Recently the authors proposed an alternative route based on the use of separated representations. However, until now these approaches were unable to address the case of advection dominated models due to stabilization issues. In this paper this issue is revisited and efficient procedures for stabilizing the advection operators involved in the Boltzmann and Fokker-Planck equation within the PGD framework are proposed.

Arlequin based PGD domain decomposition

Problems defined in fully or partially separable domains can be solved by considering a space separated representation of the unknown fields. Thus three-dimensional problems can be solved from the solution of some one-dimensional problems in the case of fully separated representations involving the three space coordinates or as a sequence of 2D and 1D problems in the case of partially separated representations (plates, shells or extruded geometries). When the domains become more complex, sometimes they can be simplified by using appropriate mappings. When it is not possible or such a transformation becomes too complex, the use of domain decomposition could facilitate the use of separated representations. However, domain coupling in the context of space separated representations have never been analyzed. In this paper we propose a domain decomposition strategy based on the use of space separated representations and the Arlequin coupling strategy. First we consider separated representations of the physical space that will be then extended to address parametric solutions.

A Short Review on Model Order Reduction Based on Proper Generalized Decomposition

This paper revisits a new model reduction methodology based on the use of separated representations, the so called Proper Generalized Decomposition—PGD. Space and time separated representations generalize Proper Orthogonal Decompositions—POD—avoiding any a priori knowledge on the solution in contrast to the vast majority of POD based model reduction technologies as well as reduced bases approaches. Moreover, PGD allows to treat efficiently models defined in degenerated domains as well as the multidimensional models arising from multidimensional physics (quantum chemistry, kinetic theory descriptions,…) or from the standard ones when some sources of variability are introduced in the model as extra-coordinates.

SUPG-based stabilization using a separated representations approach

We have developed a new method for the construction of Streamline Upwind Petrov Galerkin (SUPG) stabilization techniques for the resolution of convection-diffusion equations based on the use of separated representations inside the Proper Generalized Decompositions (PGD) framework. The use of SUPG schemes produces a consistent stabilization adding a parameter to all the terms of the equation (not only the convective one). SUPG obtains an exact solution for problems in 1D, nevertheless, a generalization does not exist for elements of high order or for any system of convection-diffusion equations. We introduce in this paper a method that achieves stabilization in the context of Proper Generalzied Decomposition (PGD). This class of approximations use a representation of the solution by means of the sum of a finite number of terms of separable functions. Thus it is possible to use the technique of separation of variables in the context of problems of convection-diffusion that will lead to a sequence of problems in 1D where the parameter of stabilization is well known.

An error estimator for separated representations of highly multidimensional models

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.

Non-incremental strategies for simulating thermomechanical models with uncertainty

Description of complex materials involves numerous computational challenges. Thus, the analysis of transient models needs for intensive computation. In some of our former works we proposed a technique based on the separated representation of the unknown field able to circumvent the curse of dimensionality in the treatment of highly multidimensional models. In this work, we are addressing the challenge related to the transient behaviour. For this purpose we propose a separated representation of transient models leading to a non-incremental strategy, allowing to impressive CPU time savings. The use of separated representations allows computing the solution of transient parametric models (the parameters are assumed as additional coordinates). The parameters include thermal coefficients but also thermal sources and/or initial conditions. The resulting curse of dimensionality is circumvented by using the separated representation that implies a complexity that scales linearly with the dimension of the space.

Recent advances on the use of separated representations

AbstractSeparated representations based on finite sum decompositions constitute an appealing strategy for reducing the computer resources and the calculation costs by reducing drastically the number of degrees of freedom that the functional approximations involve (the number of degrees of freedom scale linearly with the dimension of the space in which the model is defined instead of the exponential growing characteristic of mesh‐based discretization strategies). In our knowledge the use of separated representations is the only possibility for circumventing the terrific curse of dimensionality related to some highly multidimensional models involving hundreds of dimensions, as we proved in some of our former works. Its application is not restricted to multidimensional models, obviously separated representation can also be applied in standard 2D or 3D models, allowing for high resolution computations. Because its early life numerous issues persist, many of them attracting the curiosity of many research groups within the computational mechanics community. In this paper we are focusing in two issues never until now addressed: (i) the imposition of non‐homogenous essential boundary conditions and (ii) the consideration of complex geometries. Copyright © 2009 John Wiley & Sons, Ltd.

Coupling finite elements and reduced approximation bases

Models encountered in computational physics and engineering, usually involve too many degrees of freedom, too many simulation time-steps, too many iterations (e.g. non-linear models, optimization or inverse identification…), or simply excessive simulation time (for example when simulation in real time is envisaged). In some of our former works different reduction techniques were developed, some of them based on the use of an adaptive proper orthogonal decomposition and the other ones based on the use of separated representations. In this paper we are analyzing the coupling between reduced basis and standard finite element descriptions.

Coupling finite elements and reduced approximation bases

Models encountered in computational physics and engineering, usually involve too many degrees of freedom, too many simulation time-steps, too many iterations (e.g. non-linear models, optimization or inverse identification…), or simply excessive simulation time (for example when simulation in real time is envisaged). In some of our former works different reduction techniques were developed, some of them based on the use of an adaptive proper orthogonal decomposition and the other ones based on the use of separated representations. In this paper we are analyzing the coupling between reduced basis and standard finite element descriptions.

Fast adaptive algorithms in the non-standard form for multidimensional problems

We present a fast, adaptive multiresolution algorithm for applying integral operators with a wide class of radially symmetric kernels in dimensions one, two and three. This algorithm is made efficient by the use of separated representations of the kernel. We discuss operators of the class ( − Δ + μ 2 I ) − α , where μ ⩾ 0 and 0 < α < 3 / 2 , and illustrate the algorithm for the Poisson and Schrödinger equations in dimension three. The same algorithm may be used for all operators with radially symmetric kernels approximated as a weighted sum of Gaussians, making it applicable across multiple fields by reusing a single implementation. This fast algorithm provides controllable accuracy at a reasonable cost, comparable to that of the Fast Multipole Method (FMM). It differs from the FMM by the type of approximation used to represent kernels and has the advantage of being easily extendable to higher dimensions.

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

Kinetic theory models described within the Fokker–Planck formalism involve high-dimensional spaces (including physical and conformation spaces and time). One appealing strategy for treating this kind of problems lies in the use of separated representations and tensor product approximations basis. This technique that was introduced in a former work [A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J. Non-Newtonian Fluid Mech. 139 (2006) 153–176] for treating steady state kinetic theory models is extended here for treating transient models.