scholarly journals Efficient Wildland Fire Simulation via Nonlinear Model Order Reduction

Fluids ◽  
2021 ◽  
Vol 6 (8) ◽  
pp. 280
Author(s):  
Felix Black ◽  
Philipp Schulze ◽  
Benjamin Unger
Keyword(s):  
Model Reduction ◽  
Nonlinear Model ◽  
Interpolation Method ◽  
Wildland Fire ◽  
Classical Model ◽  
Order Reduction ◽  
Discrete Empirical Interpolation Method ◽  
Empirical Interpolation Method ◽  
Novel Method ◽  
Parameter Dependent

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 transport-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 numerical experiments demonstrate that the ROMs obtained by the novel method outperform those obtained by a classical approach using the proper orthogonal decomposition and the discrete empirical interpolation method in terms of run time and accuracy.

scholarly journals The use of POD–DEIM model order reduction for the simulation of nonlinear hygrothermal problems

2020 ◽  
Vol 172 ◽  
pp. 04002
Author(s):  
Tianfeng Hou ◽  
Karl Meerbergen ◽  
Staf Roels ◽  
Hans Janssen
Keyword(s):  
Interpolation Method ◽  
Computational Cost ◽  
Order Reduction ◽  
Illustrative Case ◽  
Hygrothermal Simulation ◽  
Discrete Empirical Interpolation Method ◽  
Empirical Interpolation Method ◽  
Illustrative Case Study ◽  
Volume Method ◽  
Proper Orthogonal

In this paper, the discrete empirical interpolation method (DEIM) and the proper orthogonal decomposition (POD) method are combined to construct a reduced order model to lessen the computational expense of hygrothermal simulation. To investigate the performance of the POD-DEIM model, HAMSTAD benchmark 2 is selected as the illustrative case study. To evaluate the accuracy of the POD-DEIM model as a function of the number of construction modes and interpolation points, the results of the POD-DEIM model are compared with a POD and a Finite Volume Method (FVM). Also, as the number of construction modes/interpolation points cannot entirely represent the computational cost of different models, the accuracies of the different models are compared as function of the calculation time, to provide a fair comparison of their computational performances. Further, the use of POD-DEIM to simulate a problem different from the training snapshot simulation is investigated. The outcomes show that with a sufficient number of construction modes and interpolation points the POD-DEIM model can provide an accurate result, and is capable of reducing the computational cost relative to the POD and FVM.

A Modified Discrete Empirical Interpolation Method for Reducing Non-Linear Structural Finite Element Models

2013 ◽  
Author(s):  
Paolo Tiso ◽  
Rob Dedden ◽  
Daniel Rixen
Keyword(s):  
Finite Element ◽  
Interpolation Method ◽  
Computational Cost ◽  
Order Reduction ◽  
Galerkin Projection ◽  
Alternative Formulation ◽  
Discrete Empirical Interpolation Method ◽  
Empirical Interpolation Method ◽  
The Cost ◽  
Discrete Empirical Interpolation

Model Order Reduction (MOR) in nonlinear structural analysis problems in usually carried out by a Galerkin projection of the primary variables on a sensibly smaller space. However, the cost of computing the nonlinear terms is still of the order of the full system. The Discrete Empirical Interpolation Method (DEIM) is an effective algorithm to reduce the computational cost of the nonlinear terms of the discretized governing equations. However, its efficiency is diminished when applied to a Finite Element (FE) framework. We present here an alternative formulation of the DEIM that suits FE discretized problems and preserves the efficiency and the accuracy of the original DEIM method.

REDUCED-ORDER MODELS FOR THE IMPLIED VARIANCE UNDER LOCAL VOLATILITY

2014 ◽  
Vol 17 (08) ◽  
pp. 1450053
Author(s):  
EKKEHARD W. SACHS ◽  
MARINA SCHNEIDER
Keyword(s):  
Implied Volatility ◽  
Interpolation Method ◽  
Order Reduction ◽  
Parabolic Partial Differential Equation ◽  
Local Volatility ◽  
Reduced Order ◽  
Reduction Techniques ◽  
Discrete Empirical Interpolation Method ◽  
Pros And Cons ◽  
Empirical Interpolation Method

Implied volatility is a key value in financial mathematics. We discuss some of the pros and cons of the standard ways to compute this quantity, i.e. numerical inversion of the well-known Black–Scholes formula or asymptotic expansion approximations, and propose a new way to directly calculate the implied variance in a local volatility framework based on the solution of a quasilinear degenerate parabolic partial differential equation. Since the numerical solution of this equation may lead to large nonlinear systems of equations and thus high computation times compared to the classical approaches, we apply model order reduction techniques to achieve computational efficiency. Our method of choice for the derivation of a reduced-order model (ROM) will be proper orthogonal decomposition (POD). This strategy is additionally combined with the discrete empirical interpolation method (DEIM) to deal with the nonlinear terms. Numerical results prove the quality of our approach compared to other methods.

scholarly journals Model Order Reduction for Pattern Formation in FitzHugh-Nagumo Equation

2016 ◽  
Author(s):  
Bulent Karasozen ◽  
Murat Uzunca ◽  
Tugba Kucukseyhan
Keyword(s):  
Interpolation Method ◽  
Computational Cost ◽  
Order Reduction ◽  
Euler Method ◽  
Order Model ◽  
Backward Euler ◽  
Discrete Empirical Interpolation Method ◽  
Dg Method ◽  
Empirical Interpolation Method ◽  
Proper Orthogonal

We developed a reduced order model (ROM) using the proper orthogonal decomposition (POD) to compute efficiently the labyrinth and spot like patterns of the FitzHugh-Nagumo (FNH) equation. The FHN equation is discretized in space by the discontinuous Galerkin (dG) method and in time by the backward Euler method. Applying POD-DEIM (discrete empirical interpolation method) to the full order model (FOM) for different values of the parameter in the bistable nonlinearity, we show that using few POD and DEIM modes, the patterns can be computed accurately. Due to the local nature of the dG discretization, the PODDEIM requires less number of connected nodes than continuous finite element for the nonlinear terms, which leads to a significant reduction of the computational cost for dG POD-DEIM.

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