A Multi-Fidelity Parametric Framework for Reduced-Order Modeling using Optimal Transport-based Interpolation: Applications to Diffused-Interface Two-Phase Flows
This work introduces a data-driven, non-intrusive reduced-order modeling (ROM) framework that leverages Optimal Transport (OT) for multi-fidelity and parametric problems in two-phase flows modelling. Building upon the success of displacement interpolation for data augmentation in handling nonlinear dynamics, we extend its application to more complex and practical scenarios. The framework is designed to correct a computationally inexpensive low-fidelity (LF) model to match an accurate high-fidelity (HF) one by capturing its temporal evolution via displacement interpolation while preserving the problem's physical consistency. The framework is further extended to address systems dependent on a physical parameter, for which we construct a surrogate model using a hierarchical, two-level interpolation strategy. First, it creates synthetic HF checkpoints via displacement interpolation in the parameter space. Second, the residual between these synthetic HF checkpoints and a true LF solution is interpolated in the time domain using the multi-fidelity OT-based methodology. This strategy provides a robust and efficient way to explore the parameter space and to obtain a refined description of the dynamical system. The potential of the method is discussed in the context of complex and computationally expensive diffuse-interface methods for two-phase flow simulations, which are characterized by moving interfaces and nonlinear evolution, and challenging to be dealt with traditional ROM techniques.
A new application of subspaces interpolation for the construction of nonlinear Parametric Reduced Order Models (PROMs) is proposed. This approach is based upon the Riemannian geometry of the manifold formed by the quotient of the set of full-rank N-by-q matrices by the orthogonal group of dimension q. By using a set of untrained parametrized Proper Orthogonal Decomposition (POD) subspaces of dimension q, the subspace for a new untrained parameter is obtained as the generalized Karcher barycenter which solution is sought after by solving a simple fixed point problem. Contrary to existing PROM a…
In this paper we present a novel framework for obtaining high-order numerical methods for scalar conservation laws in one-space dimension for both the homogeneous and non-homogeneous case. The numerical schemes for these two settings are somewhat different in the presence of shocks, however at their core they both rely heavily on the solution curve being represented parametrically. By utilizing high-order parametric interpolation techniques we succeed to obtain fifth order accuracy ( in space ) everywhere in the computation domain, including the shock location itself. In the presence of source…
Hermite parametric surface interpolation based on Argyris element
University of Ljubljana（卢布尔雅那大学）；University of Primorska（普里莫斯卡大学）；INdAM, Unità di Ricerca di Firenze c/o DiMaI “U. Dini”, University of Florence（意大利国家数学研究所佛罗伦萨研究单元迪马伊“乌·迪尼”系，佛罗伦萨大学）
In this paper we present a novel framework for obtaining high order numerical methods for 1-D scalar conservation laws with non-convex flux functions. When solving Riemann problems, the Oleinik entropy condition, [16], is satisfied when the resulting shocks and rarefactions correspond to correct portions of the appropriate (upper or lower) convex envelope of the flux function. We show that the standard equal-area principle fails to select these solutions in general, and therefore we introduce a generalized equal-area principle which always selects the weak solution corresponding to the correct…
When balanced truncation is used for model order reduction, one has to solve a pair of Lyapunov equations for two Gramians and uses them to construct a reduced-order model. Although advances in solving such equations have been made, it is still the most expensive step of this reduction method. Parametric model order reduction aims to determine reduced-order models for parameter-dependent systems. Popular techniques for parametric model order reduction rely on interpolation. Nevertheless, the interpolation of Gramians is rarely mentioned, most probably due to the fact that Gramians are symmetri…
We present a novel projection-based model reduction framework for parametric linear time-invariant systems that allows interpolating the transfer function at a given frequency point along parameter-dependent curves as opposed to the standard approach where transfer function interpolation is achieved for a discrete set of parameter and frequency samples. We accomplish this goal by using parameter-dependent projection spaces. Our main result shows that for holomorphic system matrices, the corresponding interpolatory projection spaces are also holomorphic. The coefficients of the power series rep…
In this paper, we present an interpolation framework for structure-preserving model order reduction of parametric bilinear dynamical systems. We introduce a general setting, covering a broad variety of different structures for parametric bilinear systems, and then provide conditions on projection spaces for the interpolation of structured subsystem transfer functions such that the system structure and parameter dependencies are preserved in the reduced-order model. Two benchmark examples with different parameter dependencies are used to demonstrate the theoretical analysis.
We propose a model order reduction approach for non-intrusive surrogate modeling of parametric dynamical systems. The reduced model over the whole parameter space is built by combining surrogates in frequency only, built at few selected values of the parameters. This, in particular, requires matching the respective poles by solving an optimization problem. If the frequency surrogates are constructed by a suitable rational interpolation strategy, frequency and parameters can both be sampled in an adaptive fashion. This, in general, yields frequency surrogates with different numbers of poles, a …
In the paper, the planar polynomial geometric interpolation of data points is revisited. Simple sufficient geometric conditions that imply the existence of the interpolant are derived in general. They require data points to be convex in a certain discrete sense. Since the geometric interpolation is based precisely on the known data only, one may consider it as the parametric counterpart to the polynomial function interpolation. The established result confirms the Höllig-Koch conjecture on the existence and the approximation order in the planar case for parametric polynomial curves of any degre…
In this paper, a non-intrusive reduced-order model (ROM) for parametric reactor kinetics simulations is presented. Time-dependent ROMs are notoriously data intensive and difficult to implement when nonlinear multiphysics phenomena are considered. These challenges are exacerbated when parametric dependencies are included. The proper orthogonal decomposition mode coefficient interpolation (POD-MCI) ROM presented in this work can be constructed directly from lower-dimensional quantities of interest (QoIs) and is independent of the underlying model. This greatly alleviates the data requirement of …
In this paper, we discuss a novel model reduction framework for generalized linear systems. The transfer functions of these systems are assumed to have a special structure, e.g., coming from second-order linear systems and time-delay systems, and they may also have parameter dependencies. Firstly, we investigate the connection between classic interpolation-based model reduction methods with the reachability and observability subspaces of linear structured parametric systems. We show that if enough interpolation points are taken, the projection matrices of interpolation-based model reduction en…
The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and have not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This in…
This work introduces a novel approach for data-driven model reduction of time-dependent parametric partial differential equations. Using a multi-step procedure consisting of proper orthogonal decomposition, dynamic mode decomposition and manifold interpolation, the proposed approach allows to accurately recover field solutions from a few large-scale simulations. Numerical experiments for the Rayleigh-Bénard cavity problem show the effectiveness of such multi-step procedure in two parametric regimes, i.e.~medium and high Grashof number. The latter regime is particularly challenging as it nears …
A parametric, hybrid reduced order model approach based on the Proper Orthogonal Decomposition with both Galerkin projection and interpolation based on Radial Basis Functions method is presented. This method is tested against a case of turbulent non-isothermal mixing in a T-junction pipe, a common ow arrangement found in nuclear reactor cooling systems. The reduced order model is derived from the 3D unsteady, incompressible Navier-Stokes equations weakly coupled with the energy equation. For high Reynolds numbers, the eddy viscosity and eddy diffusivity are incorporated into the reduced order …
An adaptive scheme to generate reduced-order models for parametric nonlinear dynamical systems is proposed. It aims to automatize the POD-Greedy algorithm combined with empirical interpolation. At each iteration, it is able to adaptively determine the number of the reduced basis vectors and the number of the interpolation basis vectors for basis construction. The proposed technique is able to derive a suitable match between the reduced basis and the interpolation basis vectors, making the generation of a stable, compact and reliable reduced-order model possible. This is achieved by adaptively …
Dynamic Mode Decomposition (DMD) is a model-order reduction approach, whereby spatial modes of fixed temporal frequencies are extracted from numerical or experimental data sets. The DMD low-rank or reduced operator is typically obtained by singular value decomposition of the temporal data sets. For parameter-dependent models, as found in many multi-query applications such as uncertainty quantification or design optimization, the only parametric DMD technique developed was a stacked approach, with data sets at multiples parameter values were aggregated together, increasing the computational wor…
This work investigates the use of sparse polynomial interpolation as a model order reduction method for the incompressible Navier-Stokes equations. Numerical results are presented underscoring the validity of sparse polynomial approximations and comparing with established reduced basis techniques. Two numerical models serve to access the accuracy of the reduced order models (ROMs), in particular parametric nonlinearities arising from curved geometries are investigated in detail. Besides the accuracy of the ROMs, other important features of the method are covered, such as offline-online splitti…
The paper is concerned with classic kernel interpolation methods, in addition to approximation methods that are augmented by gradient measurements. To apply kernel interpolation using radial basis functions (RBFs) in a stable way, we propose a type of quasi-optimal interpolation points, searching from a large set of \textit{candidate} points, using a procedure similar to designing Fekete points or power function maximizing points that use pivot from a Cholesky decomposition. The proposed quasi-optimal points result in a smaller condition number, and thus mitigates the instability of the interp…
Oak Ridge National Laboratory（橡树岭国家实验室）；Politecnico di Milano（米兰理工大学）
We propose a new approach to generate a reliable reduced model for a parametric elliptic problem, in the presence of noisy data. The reference model reduction procedure is the directional HiPOD method, which combines Hierarchical Model reduction with a standard Proper Orthogonal Decomposition, according to an offline/online paradigm. In this paper we show that directional HiPOD looses in terms of accuracy when problem data are affected by noise. This is due to the interpolation driving the online phase, since it replicates, by definition, the noise trend. To overcome this limit, we replace int…
One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This work will be featured as a chapter in the upcoming Handbook on Model Order Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the numerical treatment of the most important matrix manifolds that arise in the context of…
The interpolation on Grassmann manifolds in the framework of parametric evolution partial differential equations is presented. Interpolation points on the Grassmann manifold are the subspaces spanned by the POD bases of the available solutions corresponding to the chosen parameter values. The well-known Neville-Aitken's algorithm is extended to Grassmann manifold, where interpolation is performed in a recursive way via the geodesic barycenter of two points. The performances of the proposed method are illustrated through three independent CFD applications, namely: the Von Karman vortex shedding…
We present a technique for the approximation of a class of Hilbert space-valued maps which arise within the framework of Model Order Reduction for parametric partial differential equations, whose solution map has a meromorphic structure. Our MOR stategy consists in constructing an explicit rational approximation based on few snapshots of the solution, in an interpolatory fashion. Under some restrictions on the structure of the original problem, we describe a priori convergence results for our technique, hereafter called minimal rational interpolation, which show its ability to identify the mai…
We present a novel direct integral pseudospectral (PS) method (a direct IPS method) for solving a class of continuous-time infinite-horizon optimal control problems (IHOCs). The method transforms the IHOCs into finite-horizon optimal control problems (FHOCs) in their integral forms by means of certain parametric mappings, which are then approximated by finite-dimensional nonlinear programming problems (NLPs) through rational collocations based on Gegenbauer polynomials and Gegenbauer-Gauss-Radau (GGR) points. The paper also analyzes the interplay between the parametric maps, barycentric ration…
In this article, we develop a reduced basis method for efficiently solving the coupled Stokes/Darcy equations with parametric internal geometry. To accommodate possible changes in topology, we define the Stokes and Darcy domains implicitly via a phase-field indicator function. In our reduced order model, we approximate the parameter-dependent phase-field function with a discrete empirical interpolation method (DEIM) that enables affine decomposition of the associated linear and bilinear forms. In addition, we introduce a modification of DEIM that leads to non-negativity preserving approximatio…
The need for accelerating the repeated solving of certain parametrized systems motivates the development of more efficient reduced order methods. The classical reduced basis method is popular due to an offline-online decomposition and a mathematically rigorous {\em a posterior} error estimator which guides a greedy algorithm offline. For nonlinear and nonaffine problems, hyper reduction techniques have been introduced to make this decomposition efficient. However, they may be tricky to implement and often degrade the online computation efficiency. To avoid this degradation, reduced residual re…
We present a general (i.e., independent of the underlying model) interpolation technique based on optimal transportation of Gaussian models for parametric advection-dominated problems. The approach relies on a scalar testing function to identify the coherent structure we wish to track; a maximum likelihood estimator to identify a Gaussian model of the coherent structure; and a nonlinear interpolation strategy that relies on optimal transportation maps between Gaussian distributions. We show that well-known self-similar solutions can be recast in the frame of optimal transportation by appropria…
