The usual approach to model reduction for parametric partial differential equations (PDEs) is to construct a linear space $V_n$ which approximates well the solution manifold $\mathcal{M}$ consisting of all solutions $u(y)$ with $y$ the vector of parameters. This linear reduced model $V_n$ is then used for various tasks such as building an online forward solver for the PDE or estimating parameters from data observations. It is well understood in other problems of numerical computation that nonlinear methods such as adaptive approximation, $n$-term approximation, and certain tree-based methods may provide improved numerical efficiency. For model reduction, a nonlinear method would replace the linear space $V_n$ by a nonlinear space $Σ_n$. This idea has already been suggested in recent papers on model reduction where the parameter domain is decomposed into a finite number of cells and a linear space of low dimension is assigned to each cell.
Up to this point, little is known in terms of performance guarantees for such a nonlinear strategy. Moreover, most numerical experiments for nonlinear model reduction use a parameter dimension of only one or two. In this work, a step is made towards a more cohesive theory for nonlinear model reduction. Framing these methods in the general setting of library approximation allows us to give a first comparison of their performance with those of standard linear approximation for any general compact set. We then turn to the study these methods for solution manifolds of parametrized elliptic PDEs. We study a very specific example of library approximation where the parameter domain is split into a finite number $N$ of rectangular cells and where different reduced affine spaces of dimension $m$ are assigned to each cell. The performance of this nonlinear procedure is analyzed from the viewpoint of accuracy of approximation versus $m$ and $N$.
We extend the index-aware model-order reduction method to systems of nonlinear differential-algebraic equations with a special nonlinear term f(Ex), where E is a singular matrix. Such nonlinear differential-algebraic equations arise, for example, in the spatial discretization of the gas flow in pipeline networks. In practice, mathematical models of real-life processes pose challenges when used in numerical simulations, due to complexity and system size. Model-order reduction aims to eliminate this problem by generating reduced-order models that have lower computational cost to simulate, yet ac…
In this paper, we develop a nonlinear reduction framework based on our recently introduced extended group finite element method. By interpolating nonlinearities onto approximation spaces defined with the help of finite elements, the extended group finite element formulation achieves a noticeable reduction in the computational overhead associated with nonlinear finite element problems. However, the problem's size still leads to long solution times in most applications. Aiming to make real-time and/or many-query applications viable, we apply model order reduction and complexity reduction techniq…
Numerical solutions of a 2D fluid problem coupled to a nonlinear non-local reaction-advection-diffusion problem for cell crawling migration in a discoidal domain
Université Paul Sabatier（图卢兹第三大学）；Université Paris Descartes（巴黎笛卡尔大学）
In many applications, for instance when describing dynamics of fluids or gases, hyperbolic conservation laws arise naturally in the modeling of conserved quantities of a system, like mass or energy. These types of equations exhibit highly nonlinear behaviors like shock formation or shock interaction. In the case of parametrized hyperbolic equations, where, for instance, varying transport velocities are considered, these nonlinearities and strong transport effects result in a highly nonlinear solution manifold. This solution manifold cannot be approximated properly by linear subspaces. To this …
We propose a projection-based monolithic model order reduction (MOR) procedure for a class of problems in nonlinear mechanics with internal variables. The work is is motivated by applications to thermo-hydro-mechanical (THM) systems for radioactive waste disposal. THM equations model the behaviour of temperature, pore water pressure and solid displacement in the neighborhood of geological repositories, which contain radioactive waste and are responsible for a significant thermal flux towards the Earth's surface. We develop an adaptive sampling strategy based on the POD-Greedy method, and we de…
Classical reduced models are low-rank approximations using a fixed basis designed to achieve dimensionality reduction of large-scale systems. In this work, we introduce reduced deep networks, a generalization of classical reduced models formulated as deep neural networks. We prove depth separation results showing that reduced deep networks approximate solutions of parametrized hyperbolic partial differential equations with approximation error $ε$ with $\mathcal{O}(|\log(ε)|)$ degrees of freedom, even in the nonlinear setting where solutions exhibit shock waves. We also show that classical redu…
We propose a new hyper-reduction method for a recently introduced nonlinear model reduction framework based on dynamically transformed basis functions and especially well-suited for advection-dominated systems. Furthermore, we discuss applying this new method to a wildland fire model whose dynamics feature traveling combustion waves and local ignition and is thus challenging for classical model reduction schemes based on linear subspaces. The new hyper-reduction framework allows us to construct parameter-dependent reduced-order models (ROMs) with efficient offline/online decomposition. The num…
TGPT-PINN: Nonlinear model reduction with transformed GPT-PINNs
University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）；Shanghai Jiao Tong University（上海交通大学）；University of Utah（犹他大学）；Scientific Computing and Imaging Institute（科学计算与成像研究所）
We propose a sparse grids based adaptive noise reduction strategy for electrostatic particle-in-cell (PIC) simulations. Our approach is based on the key idea of relying on sparse grids instead of a regular grid in order to increase the number of particles per cell for the same total number of particles, as first introduced in Ricketson and Cerfon (Plasma Phys. and Control. Fusion, 59(2), 024002). Adopting a new filtering perspective for this idea, we construct the algorithm so that it can be easily integrated into high performance large-scale PIC code bases. Unlike the physical and Fourier dom…
In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such nonlinearities in high dimensional settings occur, e.g., when stochastic reaction diffusion equations are discretized in space. We provide a brief discussion around existence, uniqueness and stability of solutions. (Almost) stability then is the basis for new concepts of Gramians that we introduce and study in this work. With the help of these Gramians, dominant subsp…
Foundations of a new projection-based model reduction approach for convection dominated nonlinear fluid flows are summarized. In this method the evolution of the flow is approximated in the Lagrangian frame of reference. Global basis functions are used to approximate both the state and the position of the Lagrangian computational domain. It is demonstrated that in this framework, certain wave-like solutions exhibit low-rank structure and thus, can be efficiently compressed using relatively few global basis. The proposed approach is successfully demonstrated for the reduction of several simple …
We present a population control method with sampling and regulation steps for Monte Carlo particles involved in the numerical simulation of a transport equation. We recall in the first section the difficulties related to the variance reduction methods in the general framework of transport equations; we continue with a brief presentation of the mathematical tools invoked when solving the radiative transport equations and we focus on the importance of the emission and control of existing Monte Carlo particles. The next part discusses several novel methods based on the cell-based population contr…
MULTIBAT: Unified workflow for fast electrochemical 3D simulations of lithium-ion cells combining virtual stochastic microstructures, electrochemical degradation models and model order reduction
Ulm University（乌尔姆大学）；German Aerospace Center (DLR)（德国航空航天中心）；Helmholtz Institute for Electrochemical Energy Storage (HIU)（亥姆霍兹电化学储能研究所）；University of Münster（明斯特大学）；Fraunhofer Institute for Industrial Mathematics ITWM（弗劳恩霍夫工业数学研究所）；Bulgarian Academy of Science（保加利亚科学院）
Clustering by projection has been proposed as a way to preserve network structure in linear multi-agent systems. Here, we extend this approach to a class of nonlinear network systems. Additionally, we generalize our clustering method which restores the network structure in an arbitrary reduced-order model obtained by projection. We demonstrate this method on a number of examples.
