Abstract
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.
Highlights
Some models encountered in science and engineering are sometimes defined in multidimensional spaces that exhibit the terrific curse of dimensionality when usual mesh-based discretization techniques are applied.In those models the difficulty is quite natural and their solution needs for new 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
One possibility lies in the use of sparse grids [1] but its use is restricted to models defined in spaces of moderate dimensions. Another technique able to circumvent, or at least alleviate, the curse of dimensionality consists of using a separated representation of the unknown field as we proposed in our former works [2, 3] and applied in numerous contexts: (i) quantum chemistry [4]; (ii) Brownian dynamics [5]; (iii) kinetic theory description of polymers solutions and melts [6]; (iv) kinetic theory descriptions of rods suspensions [7]; among many others
Summary
Some models encountered in science and engineering are sometimes defined in multidimensional spaces (as the ones involved in quantum mechanics or kinetic theory descriptions of materials, including complex fluids) that exhibit the terrific curse of dimensionality when usual mesh-based discretization techniques are applied. In the context of computational mechanics a similar decomposition was proposed, that was called radial approximation and that was applied for separating the space and time coordinates in thermomechanical models [9] This kind of approximation only needs a technique able to construct, in a way completely transparent for the user, the separated representation of the unknown field involved in a partial differential equation (PDE). From the coefficients j just computed the approximation basis can be enriched by adding the new function For this purpose we solve the non-linear Galerkin variational formulation related to Equation (2). In the enrichment step, function R1s+1(x1) is updated by assuming known all the others functions (given at the previous iteration of the non-linear solver R2s(x2), .
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
