scholarly journals Parametric Nonlinear Model Reduction Using K-Means Clustering for Miscible Flow Simulation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Norapon Sukuntee ◽  
Saifon Chaturantabut
Keyword(s):  
Clustering Algorithm ◽  
Interpolation Method ◽  
Flow Simulation ◽  
Order Reduction ◽  
Galerkin Projection ◽  
Parameter Domain ◽  
Miscible Flow ◽  
Multidimensional Parameter ◽  
Discrete Empirical Interpolation Method ◽  
Low Dimensional

This work considers the model order reduction approach for parametrized viscous fingering in a horizontal flow through a 2D porous media domain. A technique for constructing an optimal low-dimensional basis for a multidimensional parameter domain is introduced by combining K-means clustering with proper orthogonal decomposition (POD). In particular, we first randomly generate parameter vectors in multidimensional parameter domain of interest. Next, we perform the K-means clustering algorithm on these parameter vectors to find the centroids. POD basis is then generated from the solutions of the parametrized systems corresponding to these parameter centroids. The resulting POD basis is then used with Galerkin projection to construct reduced-order systems for various parameter vectors in the given domain together with applying the discrete empirical interpolation method (DEIM) to further reduce the computational complexity in nonlinear terms of the miscible flow model. The numerical results with varying different parameters are demonstrated to be efficient in decreasing simulation time while maintaining accuracy compared to the full-order model for various parameter values.

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.

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.

scholarly journals Interplay of Sensor Quantity, Placement and System Dimension in POD-Based Sparse Reconstruction of Fluid Flows

Fluids ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 109 ◽  
Author(s):  
Balaji Jayaraman ◽  
S M Abdullah Al Mamun ◽  
Chen Lu
Keyword(s):  
Interpolation Method ◽  
Dimensional Space ◽  
Fluid Flows ◽  
Training Data ◽  
Design Parameters ◽  
Sparse Reconstruction ◽  
Discrete Empirical Interpolation Method ◽  
Low Dimensional ◽  
Value Decomposition ◽  
Accurate Reconstruction

Sparse linear estimation of fluid flows using data-driven proper orthogonal decomposition (POD) basis is systematically explored in this work. Fluid flows are manifestations of nonlinear multiscale partial differential equations (PDE) dynamical systems with inherent scale separation that impact the system dimensionality. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches require the knowledge of the underlying low-dimensional space spanning the manifold in which the system resides. In this paper, we adopt an approach that learns basis from singular value decomposition (SVD) of training data to recover sparse information. This results in a set of four design parameters for sparse recovery, namely, the choice of basis, system dimension required for sufficiently accurate reconstruction, sensor budget and their placement. The choice of design parameters implicitly determines the choice of algorithm as either l 2 minimization reconstruction or sparsity promoting l 1 minimization reconstruction. In this work, we systematically explore the implications of these design parameters on reconstruction accuracy so that practical recommendations can be identified. We observe that greedy-smart sensor placement, particularly interpolation points from the discrete empirical interpolation method (DEIM), provide the best balance of computational complexity and accurate reconstruction.

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.

scholarly journals Application of multiphysics model order reduction to doppler/neutronic feedback

2019 ◽  
Vol 5 ◽  
pp. 17 ◽  
Author(s):  
Peter German ◽  
Jean C. Ragusa ◽  
Carlo Fiorina
Keyword(s):  
Interpolation Method ◽  
Input Parameter ◽  
Order Reduction ◽  
Latin Hypercube Sampling ◽  
Order Model ◽  
Continuity Equations ◽  
Discrete Empirical Interpolation Method ◽  
Empirical Interpolation Method ◽  
Efficient Coupling ◽  
State Models

In this paper, a proper orthogonal decomposition based reduced-order model is presented for parametrized multiphysics computations. Our application physics is Doppler feedback in a simplified model of the molten salt fast reactor concept. The reduced model is created using the method of snapshots where the offline training set is obtained by exercising a full-order model created with the OpenFOAM based multiphysics solver, GeN-Foam. The steady state models solve the multi-group diffusion k-eigenvalue equations with moving precursors together with the energy equation. A fixed velocity field is assumed throughout the computations, hence the momentum and continuity equations are not solved. The discrete empirical interpolation method is used for the efficient coupling of the ROM solvers, while the input parameter space is surveyed using the improved distributed latin hypercube sampling algorithm.

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