A localized reduced-order modeling approach for PDEs with bifurcating solutions☆
Introduction
Reduced-order modeling has become an indispensable tool for the simulation and control of systems governed by complex, nonlinear partial differential equations (PDEs), as well as for the quantification of uncertainties in outputs of interest that depend on the solutions of those systems. The works cited in this section attest to the broad success such modeling has had in many settings. Here, reduced-order models generally refer to inexpensive surrogates for expensive models that are built based on a relatively few solutions of the latter model and for which the expense incurred in the construction process is then amortized over many solutions of the surrogate.
The focus here is on the construction of efficient reduced-order models (ROMs) for PDEs whose solutions bifurcate as the values of parameters appearing in the PDE change. In such cases, using a single ROM results in surrogates that themselves are expensive enough so that their repeated use can incur high costs. Here, we instead build multiple local ROMs, each of which can serve to inexpensively determine a set of approximations of PDE solutions that do not span discontinuous or other large changes due to parameter changes. In our methodology, the construction of the multiple local ROMs is automated as is the determination of which local basis one should use for values of the parameters not used in the construction process. A more complete general discussion of ROMs and of our localized ROM approach is given in the rest of this section and details are about the latter are provided in subsequent sections.
We consider the problem of determining a function such that where denotes a point in a parameter domain , a function space, a given form that is linear in but generally nonlinear in , and a linear functional on . Note that either or both and could depend on some or all the components of the parameter vector1 . We view the problem (1) as a variational formulation of a nonlinear partial differential equation (PDE) or a system of such equations2 in which parameters appear. However, it is not necessary to do so, e.g., we could treat strong forms of PDEs as well; the notation of (1) allows us to keep the exposition relatively simple.
We are interested in situations in which the solution of (1) differs in character for parameter vectors in different subregions of the parameter domain . Such situations occur if undergoes bifurcations as changes from one subregion to another; in this case we refer to the subdivision of into subregions as a bifurcation diagram. We are particularly interested in situations that require solutions of (1) for a set of parameter vectors that span across two or more of the subregions of the bifurcation diagram. Such a situation arises in optimization problems for which an (iterative) optimization algorithm updates the parameter vector at each iteration so that it is possible that the updated parameter vector is in a different subregion than is the previous vector. Uncertainty quantification problems also give rise to the need to find solutions of (1) for possibly many parameter vectors that span across two or more subregions of the bifurcation diagram.
In this paper, we assume that we have a priori knowledge of the bifurcation diagram to allow a more accurate sampling of parameter space around the critical values for the bifurcation. We do so because we focus on the construction and application of a reduced-order model (ROM) for problems with bifurcating solutions. This assumption only affects the first step in the construction of a ROM. In a follow-up paper we treat the case for which no knowledge of the bifurcation diagram is presumed. In Section 4, we discuss what changes are needed so that our approach can handle the more general case.
In general, one cannot solve (1) for so that one instead chooses an approximating -dimensional subspace and then seeks an approximation satisfying the discretized system3 Note that if denotes a basis for , we have that for some set of coefficients .4
Solving the discretized nonlinear PDE (2) for is often an expensive proposition, especially if multiple solutions are needed. For example, if denotes a finite element subspace constructed with respect to a meshing of nominal grid size of a -dimensional spatial domain, we have that so that the discretized system (2) could be huge. For this reason, one is interested in building surrogates for the solution of (2) that are much less costly to evaluate so that obtaining approximations to the exact solution of (1) for many choices of the parameter vector now becomes feasible. Such surrogates are invariably constructed using a “few” solutions of the expensive, full-order discrete system (2) and can take on many forms such as interpolants and least-squares approximations. Here, we are interested in reduced-order models (ROMs) for which one constructs a relatively low-dimensional approximating subspace of dimension that still contains an acceptably accurate approximation to the solution of (1). That approximation is determined from the reduced discrete system that, if , is much cheaper to solve compared to (2). We note that ROM systems are generally dense in the sense that, e.g., matrices associated with (3) are not sparse.
As already stated, the goal is to the determine ROM approximations that are acceptably accurate approximations of the solutions of the continuous model (1). However, because the ROM approximation is constructed through the use of solutions of the spatially discretized problem (2), we have two sources of error in the ROM approximation, an error that can be estimated (in a chosen norm ) by We mostly focus on the last term in (4). However, one should keep in mind that an efficient overall computational methodology should try to balance the two error terms on the right-hand side of (4) because there is not much sense in having the error due to the ROM be much smaller than the spatial error. This observation may be used to relate to so as to provide guidance as to what should be the dimension of the ROM approximations space.
ROMs in the setting of bifurcating solutions are considered in the early papers [1], [2], [3], [4] in the setting of buckling bifurcations in solid mechanics. More recently, in [5] it is shown that a POD approach allows for considerable computational time savings for the analysis of bifurcations in some nonlinear dissipative systems. In [6], [7], a reduced basis (RB) method is used to track solution branches from bifurcation points arising in natural convection problems. A RB method is used in [8] to investigate Hopf bifurcations in natural convection problems and in [9] for symmetry breaking bifurcations in contraction–expansion channels. An investigation of symmetry breaking in an expansion channel can be found in [10]. In [11], [12], [13], reliable error estimation is used to determine the critical parameter points where bifurcations occur in the Navier–Stokes setting. A recent work on ROMs for bifurcating solutions in structural mechanics is [14]. Finally, we would like to mention that machine learning techniques based on sparse optimization have been applied to detect bifurcating branches of solutions in [15], [16] for a two-dimensional laterally heated cavity and Ginzburg–Landau model, respectively.
In most implementations of ROMs, including the papers cited above related to bifurcations, a single global basis is used to determine the ROM approximation at any chosen parameter point by solving (3). However, in a setting in which consists of subregions for which the corresponding solutions of (1) have different character, such as is the case in the bifurcation setting, it may be the case that , although small compared to , may be large enough so that solving the dense ROM system (3) many times becomes a costly proposition. Thus, it seems prudent to construct several local bases, each of which is used for parameters belonging to a different subregion of the bifurcation diagram and also possibly to bridge across the boundary between those subregions. Thus, our goal is to construct, say, such local bases5 of dimension , each spanning a local subspace . We then construct local reduced-order models that provide acceptably accurate approximations to the solution of (1) for parameters belonging to different disjoint parts of the bifurcation diagram. Of course, if one is to employ several local bases, then one must also determine when one needs to switch from one basis to another.
The use of local ROM bases has been considered in previous works. In [17] snapshots are obtained by sampling the full-order solutions at various time instants; snapshots corresponding to different parameter values are not considered. In [18], [19], [20] local bases are determined by projection based clustering instead of Euclidean closeness. In [21], [22], k-means clustering and nearest neighbor classifier with respect to parameters or a low-dimensional representation of the current state are used. In [23] k-means clustering is used to generate local bases for use in conjunction with the empirical interpolation method. In [24] local bases determined using k-means clustering and logistic regression classifiers are used for aero-icing problems. In [25] time-based snapshot clustering as well as k-means and a bisection process clusterings of parameter based snapshots are considered and applied to a model problem in cardiac electrophysiology. In [26], [27], [28] localized bases are used to resolve fine-scale phenomena in a multiscale setting. In [29] an hp-ROM approach is used to localize ROMs for improved accuracy. In [30] local projection spaces are used in multi-scale turbulence models.
The plan for the rest of the paper is as follows. In Section 2, we discuss in detail the various ingredients that, together, constitute our recipe for the construction and application of a ROM that is well suited for bifurcation problems. Specifically, we discuss how to
- 1.
select sample points in the parameter domain ;
- 2.
compute the corresponding snapshots , i.e., solutions of the full-order discretized problem (2) for each of the parameter points ;
- 3.
cluster the snapshots so that each cluster corresponds to parameters in a different part of the parameter domain that could be a subregion of the bifurcation diagram or that could bridge across the boundary of two such subregions;
- 4.
construct the local bases corresponding to each cluster;
- 5.
detect which cluster a new parameter choice belongs to so that one can use the corresponding local basis to determine a ROM approximate solution.
When these steps are completed, we can solve (5) for the ROM approximation using the appropriate local basis. For the sake of completeness we would like to shortly describe the approach proposed in [15]. The authors deal with time dependent problems with parameter bifurcation. First, they discretize the parameter domain, compute time-dependent snapshots and POD basis for each parameter independently. Then, they collect all the POD basis and select the proper basis, in the online stage, by means of sparse optimization methods. In this way, they are able to detect a subset of the POD basis for the parameter of interest.
In [16], similarly, the authors present a sparse sensing framework to detect bifurcation regimes, through Dynamic Mode Decomposition (DMD, see e.g. [31]) which is a data-driven method that allows to find the best linear fit dynamical systems that approximates the original one. As in [15], they collect all the modes, for any parameter, together and then select the DMD basis by sparse optimization algorithms. The advantage is that with a DMD approach one can learn the dynamics and study eigenvalues and eigenvectors of the linearized system and detect the bifurcation regimes.
In our work, we first collect steady-state snapshots, cluster them and compute POD basis for each cluster. Then, we select the proper basis to use in the online stage through different criteria with respect to [15], as explained in Section 2.5.
In Section 3, to illustrate the implementation and employment of the new algorithm defined in Section 2 in a concrete setting, we consider two problems for the Navier–Stokes equations. Concluding remarks are provided in Section 4.
Section snippets
Detailed description of the new localized-basis method
In this section we provide a more detailed description of each of the five steps listed in the recipe given in Section 1.
Application to the incompressible Navier–Stokes equations
We use a variational formulation of the incompressible Navier–Stokes equations as the concrete setting of (1) to illustrate our methodology. We first define the Navier–Stokes problem and then consider examples for both continuous and discontinuous transitions through bifurcation points. For the first case, we consider two types of spatial discretizations. Together, the examples are meant to illustrate the robustness of our approach with respect to both solution behaviors and spatial
Concluding remarks
In the paper, a k-means clustering of snapshots is used as the starting point for constructing a ROM that uses localized bases to treat PDE problems having bifurcating solutions. A recipe for detecting which local basis to use for any given parameter point not used to determine the snapshots is also given. Careful attention given to account for the differences between bifurcations that cause continuous or discontinuous changes in the solution. At this point, for the sake of simplicity, the new
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AA supported by the US Department of Energy Office of Science grant DE-SC0009324. MG supported by the US Air Force Office of Scientific Research grant FA9550-15-1-0001 and US Department of Energy Office of Science grant DE-SC0009324. MH and GR supported by European Union Funding for Research and Innovation through the European Research Council project H2020 ERC CoG 2015 AROMA-CFD project 681447, P.I. Prof. G. Rozza). AQ supported by US National Science Foundation grant DMS-1620384.