Constraint Energy Minimization Research Articles

Convergence of the CEM-GMsFEM for compressible flow in highly heterogeneous media

This paper presents and analyses a Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving single-phase non-linear compressible flows in highly heterogeneous media. The construction of CEM-GMsFEM hinges on two crucial steps: First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. Then the basis functions are obtained by solving local energy minimization problems over the oversampling domains using the auxiliary space. The basis functions have exponential decay outside the corresponding local oversampling regions. The convergence of the proposed method is provided, and we show that this convergence only depends on the coarse grid size and is independent of the heterogeneities. An online enrichment guided by a posteriori error estimator is developed to enhance computational efficiency. Several numerical experiments on a three-dimensional case to confirm the theoretical findings are presented, illustrating the performance of the method and giving efficient and accurate numerical solutions.

Non-local multi-continuum method (NLMC) for Darcy–Forchheimer flow in fractured media

This work presents the application of the non-local multicontinuum method (NLMC) for the Darcy–Forchheimer model in fractured media. The mathematical model describes a nonlinear flow in fractured porous media with a high inertial effect and flow speed. The space approximation is constructed on the sufficiently fine grid using a finite volume method (FVM) with an embedded fracture model (EFM) to approximate lower dimensional fractures. A non-local model reduction approach is presented based on localization and constraint energy minimization. The multiscale basis functions are constructed in oversampled local domains to consider the flow effects from neighboring local domains. Numerical results are presented for a two-dimensional formulation with two test cases of heterogeneity. The influence of model nonlinearity on the multiscale method accuracy is investigated. The numerical results show that the non-local multicontinuum method provides highly accurate results for Darcy–Forchheimer flow in fractured media.

Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Convection Diffusion Equation

In this paper we present and analyze a constraint energy minimizing generalized multiscale finite element method for convection diffusion equations. To define the multiscale basis functions, we first build an auxiliary multiscale space by solving local spectral problems motivated by analysis. Then a constraint energy minimization performed in the oversampling domains is exploited to construct the multiscale space. The resulting multiscale basis functions have a good decay property even for high contrast diffusion and convection coefficients. Furthermore, if the number of oversampling layers is chosen properly, we can prove that the convergence rate is proportional to the coarse meshsize. Our analysis also indicates that the size of the oversampling domain weakly depends on the contrast of the heterogeneous coefficients. Several numerical experiments are presented illustrating the performance of our method.

Computational multiscale method for parabolic wave approximations in heterogeneous media

In this paper, we develop a computational multiscale method to solve the parabolic wave approximation with heterogeneous and variable media. Parabolic wave approximation is a technique to approximate the full wave equation. One benefit of the method is that one wave propagation direction can be taken as an evolution direction, and one then can discretize it using a classical scheme like backward Euler method. Consequently, one obtains a set of quasi-gas-dynamic (QGD) models with possibly different heterogeneous permeability fields. For coarse discretization, we employ constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) to perform spatial model reduction. The resulting system can be solved by combining the central difference in time evolution. Due to the variable media, we apply the technique of proper orthogonal decomposition (POD) to further the dimension of the problem and solve the corresponding model problem in the POD space instead of in the whole multiscale space spanned by all possible multiscale basis functions. We prove the stability of the full discretization scheme and give the convergence analysis of the proposed approximation scheme. Numerical results verify the effectiveness of the proposed method.

Constraint energy minimization generalized multiscale finite element method in mixed formulation for parabolic equations

In this paper, we develop the constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) in mixed formulation applied to parabolic equations with heterogeneous diffusion coefficients. The construction of the method is based on two multiscale spaces: pressure multiscale space and velocity multiscale space. The pressure space is constructed via a set of well-designed local spectral problems, which can be solved independently. Based on the computed pressure multiscale space, we will construct the velocity multiscale space by applying constrained energy minimization. The convergence of the proposed method is proved. In particular, we prove that the convergence of the method depends only on the coarse grid size, and is independent of the heterogeneities and contrast of the diffusion coefficient. Four typical types of permeability fields are exploited in the numerical simulations, and the results indicate that our proposed method works well and gives efficient and accurate numerical solutions.

Constraint energy minimizing generalized multiscale discontinuous Galerkin method

Numerical simulation of flow problems and wave propagation in heterogeneous media has important applications in many engineering areas. However, numerical solutions on the fine grid are often prohibitively expensive, and multiscale model reduction techniques are introduced to efficiently solve for an accurate approximation on the coarse grid. In this paper, we propose an energy minimization based multiscale model reduction approach in the discontinuous Galerkin discretization setting. The main idea of the method is to extract the non-decaying component in the high conductivity regions by identifying dominant modes with small eigenvalues of local spectral problems, and define multiscale basis functions in coarse oversampled regions by constraint energy minimization problems. The multiscale basis functions are in general discontinuous on the coarse grid and coupled by interior penalty discontinuous Galerkin formulation. The minimal degree of freedom in representing high-contrast features is achieved through the design of local spectral problems, which provides the most compressed local multiscale space. We analyze the method for solving Darcy flow problem and show that the convergence is linear in coarse mesh size and independent of the contrast, provided that the oversampling size is appropriately chosen. Numerical results are presented to show the performance of the method for simulation on flow problem and wave propagation in high-contrast heterogeneous media.

Multiscale stabilization for convection–diffusion equations with heterogeneous velocity and diffusion coefficients

We present a new stabilization technique for multiscale convection–diffusion problems. Stabilization for these problems has been a challenging task, especially for the case with high Péclet numbers. Our method is based on a constraint energy minimization idea and the discontinuous Petrov–Galerkin formulation. In particular, the test functions are constructed by minimizing an appropriate energy subject to certain orthogonality conditions, and are related to the trial space. The resulting test functions have a localization property, and can therefore be computed locally. We will prove the stability, and present several numerical results. Our numerical results confirm that our test space gives a good stability, in the sense that the solution error is close to the best approximation error.

An analysis of the NLMC upscaling method for high contrast problems

In this paper we propose simple multiscale basis functions with constraint energy minimization to solve elliptic problems with high contrast medium. Our methodology is based on the recently developed non-local multicontinuum method (NLMC). The main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non-local effects. In our method, each coarse block is decomposed into various regions according to the contrast ratio, and we require that the contrast ratio should be relatively small within each region. The basis functions are constructed by solving a local problem defined on the oversampling domains and they have mean value one on the chosen region and zero mean otherwise. Numerical analysis shows that the resulting basis functions can be localizable and have a decay property. The convergence of the multiscale solution is also proved. Finally, some numerical experiments are carried out to illustrate the performances of the proposed method. They show that the proposed method can solve problem with high contrast medium efficiently. In particular, if the oversampling size is large enough, then we can achieve the desired error.

Nonlocal multicontinuum (NLMC) upscaling of mixed dimensional coupled flow problem for embedded and discrete fracture models

In this work, we present an upscaled model for mixed dimensional coupled flow problem in fractured porous media. We consider both embedded and discrete fracture models (EFM and DFM) as fine scale models which contain coupled system of equations. For fine grid discretization, we use a conservative finite-volume approximation. We construct an upscaled model using the non-local multicontinuum (NLMC) method for the coupled system. The proposed upscaled model is based on a set of simplified multiscale basis functions for the auxiliary space and a constraint energy minimization principle for the construction of multiscale basis functions. Using the constructed NLMC-multiscale basis functions, we obtain an accurate coarse grid upscaled model. We present numerical results for both fine-grid models and upscaled coarse-grid models using our NLMC method. We consider model problems with (1) discrete fracture fine grid model with low and high permeable fractures; (2) embedded fine grid model for two types of geometries with differnet fracture networks and (3) embedded fracture fine grid model with heterogeneous permeability. The simulations using the upscaled model provide very accurate solutions with significant reduction in the dimension of the problem.

Constraint Energy Minimizing Generalized Multiscale Finite Element Method

In this paper, we propose Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM). The main goal of this paper is to design multiscale basis functions within GMsFEM framework such that the convergence of method is independent of the contrast and linearly decreases with respect to mesh size if oversampling size is appropriately chosen. We would like to show a mesh-dependent convergence with a minimal number of basis functions. Our construction starts with an auxiliary multiscale space by solving local spectral problems. In auxiliary multiscale space, we select the basis functions that correspond to small (contrast-dependent) eigenvalues. These basis functions represent the channels (high-contrast features that connect the boundaries of the coarse block). Using the auxiliary space, we propose a constraint energy minimization to construct multiscale spaces. The minimization is performed in the oversampling domain, which is larger than the target coarse block. The constraints allow handling non-decaying components of the local minimizers. If the auxiliary space is correctly chosen, we show that the convergence rate is independent of the contrast (because the basis representing the channels are included in the auxiliary space) and is proportional to the coarse-mesh size (because the constraints handle non-decaying components of the local minimizers). The oversampling size weakly depends on the contrast as our analysis shows. The convergence theorem requires that channels are not aligned with the coarse edges, which hold in many applications, where the channels are oblique with respect to the coarse-mesh geometry. The numerical results confirm our theoretical results. In particular, we show that if the oversampling domain size is not sufficiently large, the errors are large. To remove the contrast-dependence of the oversampling size, we propose a modified construction for basis functions and present numerical results and the analysis.

Impact Evaluation of Feature Reduction Techniques on Classification of Hyper Spectral Imagery

Different classification techniques are being designed and under usage for classification of hyper spectral images. The usage of these classifiers differ for different type of hyper spectral data and application domain. The performance of these classifiers are influenced by feature preprocessing stage. In this research work we have investigated the impact of feature preprocessing using Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA) on classification stage. The experiment is carried out using three sets of hyperspectral data. Classifications were carried out using three different classification techniques namely Maximum Likelihood Estimation (MLE), Constraint Energy Minimization (CEM) and Spectral Angle Mapper (SAM) on the preprocessed hyperspectral data. It is observed that the impact of PCA and LDA on the classification techniques are in two aspects (a) The preprocessing techniques facilitates to achieve high classification accuracy even with less number of training features and (b) Preprocessing expedites the classification process for large data sets. Also it can be concluded that PCA outperforms LDA in case of noisy data.

Automated TIMI frame counting using 3-d modeling

Three dimensional coronary modeling and reconstruction can assist in the quantitative analysis of coronary flow velocity from 2-d coronary images. In this paper a novel method to assess coronary flow velocity is proposed. First, 3-d models of the coronary arteries are estimated from bi-plane X-ray images using epipolar constraint energy minimization for the selected fiducial points like bifurcations, and subsequently 3-d B-spline energy minimization for the arterial segments. A 4-d model is assembled from a set of 3-d models representing different phases of the cardiac cycle. The 4-d model is fitted to the 2-d image sequences containing basal or hyperemic blood flow information. Then, by counting the frames in analogy with TIMI frame counting, an index of the mean coronary flow velocity can be estimated. Our experimental results show that the algorithm correlates with r=0.98 (P<0.0001, 95% CI 0.92–0.99) to the clinical measurements of the TFC.

AMG by element agglomeration and constrained energy minimization interpolation

AbstractThis paper studies AMG (algebraic multigrid) methods that utilize energy minimization construction of the interpolation matrices locally, in the setting of element agglomeration AMG. The coarsening in element agglomeration AMG is done by agglomerating fine‐grid elements, with coarse element matrices defined by a local Galerkin procedure applied to the matrix assembled from the individual fine‐grid element matrices. This local Galerkin procedure involves only the coarse basis restricted to the agglomerated element. To construct the coarse basis, one exploits previously proposed constraint energy minimization procedures now applied to the local matrix. The constraints are that a given set of vectors should be interpolated exactly, not only globally, but also locally on every agglomerated element. The paper provides algorithmic details, as well as a convergence result based on a ‘local‐to‐global’ energy bound of the resulting multiple‐vector fitting AMG interpolation mappings. A particular implementation of the method is illustrated with a set of numerical experiments. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Blind Chip-Level Equalizer for the Downlink of Cyclic-Prefix CDMA Systems

The authors present a chip-level blind frequency domain equalizer (FEQ) for the forward-link channel of a cyclic-prefix code division multiple access system. The FEQ coefficients are obtained without the need of training symbols or knowledge of channel state information. The coefficients are instead acquired by solving a constraint energy minimization problem involving the subspace spanned by the active and passive spreading codewords. They also prove that the random scrambling code sequences is required for the operation of the proposed equalization algorithm. Results from computer simulations are provided to verify the performance of the proposed FEQ

Efficient Search on Energy Minima for Structure Prediction of Nucleic Acid Motifs

Structure prediction of non-canonical motifs such as mismatches, extra unmatched nucleotides or internal and hairpin loop structures in nucleic acids is of great importance for understanding the function and design of nucleic acid structures. Systematic conformational analysis of such motifs typically involves the generation of many possible combinations of backbone dihedral torsion angles for a given motif and subsequent energy minimization (EM) and evaluation. Such approach is limited due to the number of dihedral angle combinations that grows very rapidly with the size of the motif. Two conformational search approaches have been developed that allow both an effective crossing of barriers during con-formational searches and the computational demand grows much less with system size then search methods that explore all combinations of backbone dihedral torsion angles. In the first search protocol single torsion angles are flipped into favorable states using constraint EM and subsequent relaxation without constraints. The approach is repeated in an iterative manner along the backbone of the structural motif until no further energy improvement is obtained. In case of two test systems, a DNA-trinucleotide loop (sequence: GCA) and a RNA tetraloop (sequence: UUCG), the approach successfully identified low energy states close to experiment for two out of five start structures. In the second method randomly selected combinations of up to six backbone torsion angles are simultaneously flipped into preset ranges by a short constraint EM followed by unconstraint EM and acceptance according to a Metropolis acceptance criterion. This combined stochastic/EM search was even more effective than the single torsion flip approach and selected low energy states for the two test cases in between two and four cases out of five start structures.

Three-dimensional quantitative structure–activity relationship for several bioactive peptides searched by a convex hull–comparative molecular field analysis approach

Three-dimensional (3D) convex hulls are computed for theoretically generated structures of a group of 18 bioactive tachykinin peptides. The number of peptides treated as a training set is 14, whereas that treated as a test set is four. The frequency of atoms of the same atomic type lying at the vertices of all the hulls computed for all the structures in a structural set is counted. Vertex atoms with non-zero frequency counted are collected together as a set of commonly exposed groups. These commonly exposed atoms are then treated as a set of correspondences for aligning all the other 13 structures in a structural set against a common template, which is the structure of the most potent peptide in the set using the FIT module of the sybyl 6.6 program. Each aligned structural set is then analyzed by the comparative molecular field analysis (CoMFA) module using the C.3 probe having a charge of +1.0. The corresponding cross-validated r 2 values range from −0.99 to 0.57 for a number of 73 structural sets studied. The comparative molecular similarity indices analysis (CoMSIA) module within the sybyl 6.6 package is also used to analyze some of these aligned structural sets. Although the CoMSIA results are in accord with those of CoMFA, it is also found that the CoMFA results of several structural sets can be improved somewhat for conformations of the structures in the sets that are adjusted by constraint energy minimization and then constraint molecular dynamics simulation runs using distance constraints derived from some commonly exposed groups determined for them. This result further implies that the convex hull–CoMFA is a feasible approach to screen the bioactive conformations for molecules of this class.