Related papers: High-order empirical interpolation methods for rea…
We present a model reduction approach for the real-time solution of time-dependent nonlinear partial differential equations (PDEs) with parametric dependencies. The approach integrates several ingredients to develop efficient and accurate…
We present a model reduction approach that extends the original empirical interpolation method to enable accurate and efficient reduced basis approximation of parametrized nonlinear partial differential equations (PDEs). In the presence of…
This work proposes novel techniques for the efficient numerical simulation of parameterized, unsteady partial differential equations. Projection-based reduced order models (ROMs) such as the reduced basis method employ a (Petrov-)Galerkin…
For nonlinear reduced-order models, especially for those with non-polynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. As a result, the reduced-order model loses its…
We provide first the functional analysis background required for reduced order modeling and present the underlying concepts of reduced basis model reduction. The projection-based model reduction framework under affinity assumptions,…
In this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the…
Reduced-order modeling is an efficient approach for solving parameterized discrete partial differential equations when the solution is needed at many parameter values. An offline step approximates the solution space and an online step…
In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based…
We investigate new developments of the combined Reduced-Basis and Empirical Interpolation Methods (RB-EIM) for parametrized nonlinear parabolic problems. In many situations, the cost of the EIM in the offline stage turns out to be…
A novel algorithmic discussion of the methodological and numerical differences of competing parametric model reduction techniques for nonlinear problems are presented. First, the Galerkin reduced basis (RB) formulation is presented which…
At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly,…
To speed-up the solution to parametrized differential problems, reduced order models (ROMs) have been developed over the years, including projection-based ROMs such as the reduced-basis (RB) method, deep learning-based ROMs, as well as…
In this paper we extend the hierarchical model reduction framework based on reduced basis techniques for the application to nonlinear partial differential equations. The major new ingredient to accomplish this goal is the introduction of…
In this paper, we introduce the neural empirical interpolation method (NEIM), a neural network-based alternative to the discrete empirical interpolation method for reducing the time complexity of computing the nonlinear term in a reduced…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
We introduce a method for the fast numerical approximation of linear, second-order parabolic partial differential equations (PDEs for short) with time-independent coefficients based on model order reduction techniques and the Laplace…
In this paper, we combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational…
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…
Nonlinear parametric inverse problems appear in many applications and are typically very expensive to solve, especially if they involve many measurements. These problems pose huge computational challenges as evaluating the objective…
A novel refinement measure for non-intrusive surrogate modelling of partial differential equations (PDEs) with uncertain parameters is proposed. Our approach uses an empirical interpolation procedure, where the proposed refinement measure…