A localized reduced-order modeling approach for PDEs with bifurcating solutions

Abstract

Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE. 1 For finite-dimensional spaces, we use the nomenclature "points" and "vectors" interchangeably.2 In §3, we use the Navier-Stokes equations as a concrete setting to illustrate our methodology. 1 arXiv:1807.08851v1 [math.NA] 23 Jul 20183 In general, the forms N and F themselves are also discretized, e.g., because quadrature rules are used to approximate integrals appearing in their definition. However, here, we ignore such approximations, again to keep the exposition simple.4 Equation (1.2) represents a Galerkin type setting in which the trial function u N and test function v belong to the same space V N . We could easily generalize our discussion to the Petrov-Galerkin case for which these function would belong to different approximating spaces.

Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method

“…However, if one uses other discretization techniques or requires a very high accuracy, the computational cost can significantly increase. For instance, in [22], a very similar model has been computed with the FEM (in particular with the Taylor-Hood elements) exploiting 90,876 degrees of freedom to obtain the desired accuracy. Note that the 7372 degrees of freedom are associated with the mesh shown in Fig.…”

“…For instance, in [41] and in [40], the authors analyzed the effectiveness of similar techniques (with the SEM substituted by the FEM and without the deflation) to compute bifurcation diagrams for the Gross-Pitaevskii and the Von Kármán equations. In particular, the present work is strongly related to [22][23][24] and [21]. In fact, a possible future application of this work could be the mitral valve regurgitation [43].…”

“…Moreover, as described in [18], it is possible to increase the efficiency of the deflation observing that one does not need to assemble and solve the linear systems associated with the deflated problem. In order to describe a more efficient approach, let us assume that one wants to solve system (22) with the Newton method, that is characterized by the following iteration:…”

Mechanical dissimilarity of defects in welded joints via Grassmann manifold and machine learning

“…Clustering methods have been already used for the construction of local reduced order bases in parametric spaces. We refer the reader to [41,42], for instance, and for more recent examples to [43]. Local reduced order bases are known to be more accurate over a global reduced basis [41].…”

A Spectral Element Reduced Basis Method for Navier–Stokes Equations with Geometric Variations

“…The viscosity is kept constant to ν = 1. For these simulations, the Reynolds number (6) is between 5 and 10, with maximum velocity in the narrowing as characteristic velocity U and the height of the narrowing characteristic length L. For larger Reynolds numbers (about 30), a supercritical pitchfork bifurcation occurs giving rise to the so-called Coanda effect [8,9,22], which is not subject of the current study. Our model is similar to the model considered in [17,18], i.e.…”