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Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods

Eric Chung, Yalchin Efendiev, Thomas Y. Hou

arXiv:1604.08312v1math.NA

TL;DR

Existing numerical homogenization approaches can miss important local solution features when microscale problems require more degrees of freedom or include perforations. The paper presents GMsFEM as an adaptive multiscale model-reduction framework with offline and online stages, local enrichment, and online basis functions; numerical results report fast convergence, low velocity error with few basis functions, and nearly perfect parallel scaling.

  • Problem

    Numerical homogenization approaches may fail when the local solution space needs more degrees of freedom, while perforated domains introduce excluded portions and complex geometry that challenge standard macroscopic models.

  • Method

    GMsFEM performs adaptive local model reduction by constructing reusable multiscale basis functions offline and enriching the coarse space iteratively using local error indicators and online basis functions.

  • Results

    The numerical results show fast convergence with increased offline bases, velocity error below 5% with three mixed GMsFEM basis functions per node, and nearly perfect scaling using up to 1000 processors.

  • Takeaways & Limitations

    The framework supports local reduced-order models that can also construct global model-reduction snapshots and be updated adaptively during online computations.

  • Takeaways & Limitations

    The overview is not comprehensive, and the paper does not discuss applying the method to nonlinear problems.

Abstract

from arXiv · show

In this paper, we discuss a general multiscale model reduction framework based on multiscale finite element methods. We give a brief overview of related multiscale methods. Due to page limitations, the overview focuses on a few related methods and is not intended to be comprehensive. We present a general adaptive multiscale model reduction framework, the Generalized Multiscale Finite Element Method. Besides the method's basic outline, we discuss some important ingredients needed for the method's success. We also discuss several applications. The proposed method allows performing local model reduction in the presence of high contrast and no scale separation.

1 Introduction

The paper introduces GMsFEM as an adaptive local multiscale model reduction framework for challenging problems with multiple scales, high contrast, and no scale separation. It systematically selects and enriches local degrees of freedom through snapshots, spectral reduction, error indicators, and related computational ingredients.

  • 1 Introduction: GMsFEM targets multiscale problems with high contrast and no scale separation by adaptively adding local degrees of freedom as needed.The framework is intended for local model reduction across varied challenging applications.
  • 1 Introduction: Unlike numerical homogenization, the approach determines necessary local degrees of freedom as needed rather than prespecifying effective local features.Numerical homogenization may fail to represent important local solution-space features unless they are identified beforehand.
  • 1 Introduction: GMsFEM generalizes MsFEM through local snapshot spaces and spectral decompositions, enabling adaptive enrichment and rigorous error estimation.The method constructs local multiscale spaces and adds degrees of freedom based on error estimators.
  • 1 Introduction: The framework reduces snapshot spaces to dominant modes, while adding extra degrees of freedom when scale interactions cannot be represented by reduced-dimensional spaces.Appropriate snapshots and local spectral problems yield reduced spaces under scale separation and constructive enrichment without it.
  • 1 Introduction: Adaptivity assigns different numbers of basis functions to regions according to local heterogeneities, high contrast, and estimated errors.Regions with scale separation are expected to require fewer degrees of freedom.
  • 1 Introduction: The framework is tested across applications, while many encountered multiscale problems are described as benefiting from local adaptive model reduction.The paper presents applications rather than a comprehensive overview of all multiscale methods.

2 A brief introduction to numerical homogenization

Numerical homogenization computes coarse-scale effective coefficients from local fine-scale problems, aiming to preserve average flux responses. Oversampling reduces boundary-induced resonance errors, while the approach assumes a reduced-dimensional local solution structure.

  • Local upscaling: Numerical homogenization computes upscaled conductivity in each coarse block by solving local problems with boundary conditions representing local heterogeneities.The example uses Dirichlet conditions N_l = x_l on each coarse-block boundary.
  • Local upscaling: The upscaled coefficients are defined so the average flux response matches that of the fine-scale local problem under prescribed boundary conditions.
  • Convergence: Under periodic coefficient structure κ_ij(x) = κ_ij(x/ϵ), convergence of numerical homogenization can be studied, and the resulting coefficients are identified as correct homogenized coefficients.
  • Oversampling: Oversampling solves local problems on a slightly larger domain and reduces errors caused by oscillatory boundary conditions and artificial boundary layers.The residual error scales as ϵ/H instead of the larger boundary-induced error described for the standard construction.
  • Scope: Numerical homogenization assumes a reduced-dimensional local solution space, such as three local fields per coarse block in three dimensions.

3 Multiscale Finite Element Method

MsFEM combines localized multiscale basis functions with a global formulation on a coarse grid. Its basis functions capture fine-scale solution features, while oversampling and limited global information address boundary and localization effects.

  • Basic formulation: MsFEM uses multiscale basis functions and a global numerical formulation that couples them on a coarse grid.The basis functions are designed to capture fine-scale features of the solution.
  • Basic formulation: The coarse-grid solution is represented using localized basis functions supported in coarse neighborhoods and coefficients determined by the global formulation.
  • Basic formulation: In the discrete setting, fine-grid column vectors define a coarse space, and the multiscale solution is the finite element projection into that space.
  • Basis construction: Standard multiscale basis functions match finite element basis functions on coarse-block boundaries but are oscillatory inside the blocks.
  • Basis construction: MsFEM can exploit scale separation by using smaller local domains with re-scaled stiffness-matrix integrals.
  • Basis construction: Oversampling enlarges the local domain to reduce boundary errors, but the resulting method can be nonconforming unless oversampling functions are restricted with linear basis functions.
  • Global information: Limited global information can be encoded through global fields that capture essential heterogeneities and help construct multiple multiscale basis functions.Finding these global fields can be difficult.

4 Generalized Multiscale Finite Element Method. Basic Concepts

GMsFEM adaptively enriches local coarse spaces by combining snapshot construction with offline dimension reduction and online coarse solves. The framework supports reusable parameter-dependent bases and problem-relevant snapshot spaces.

  • Motivation: GMsFEM addresses complex heterogeneities by using multiple basis functions when a single coarse-region basis cannot represent the local solution space.
  • Framework: The method separates computation into offline and online stages, with offline spaces reused to construct multiscale bases for different input parameters.This reuse provides computational savings in the online stage.
  • Framework: Offline computations generate the coarse grid, construct snapshot spaces, and reduce them to small-dimensional offline spaces.
  • Framework: Online computations construct parameter-dependent multiscale basis functions, solve coarse-grid problems for force terms and boundary conditions, and optionally use iterative solvers.
  • Snapshot spaces: Snapshot spaces are chosen for each coarse region and can improve convergence, impose application-relevant restrictions, and reduce offline-space construction cost.Oversampling in snapshot spaces can further improve convergence.
  • Snapshot spaces: Snapshots may use local fine-grid basis functions, harmonic extensions of boundary data, oversampled harmonic extensions, or random boundary conditions.

4.3 Offline spaces

GMsFEM constructs offline spaces by spectrally reducing local snapshot spaces and selecting dominant modes. The resulting approximation is controlled by eigenvalue thresholds, auxiliary bilinear forms, and oversampling choices.

  • Offline-space construction: The offline space is formed by applying a local spectral decomposition to snapshot spaces and selecting eigenvectors associated with the smallest eigenvalues.
  • Spectral problem: The local spectral problem is derived from an energy-error decomposition over coarse subdomains and is used to select local offline basis functions.
  • Spectral problem: The auxiliary bilinear form s_ω and eigenvalue threshold δ determine the generalized eigenvalue problem and the basis-selection criterion.
  • Spectral problem: Oversampling is sought to produce fast spectral decay, while energy-minimizing snapshots may be needed to obtain parameter- and mesh-independent energy bounds.
  • Offline-space construction: The convergence rate is proportional to 1/Λ*, where Λ* is the smallest eigenvalue whose eigenvector is excluded from the offline space.
  • Conforming construction: Selected eigenfunctions are multiplied by partition-of-unity functions to construct conforming basis functions, while mixed or discontinuous Galerkin discretizations can avoid this multiplication.
  • Global solve: The global offline space is the union of local offline spaces, and the coarse solution is obtained by solving the resulting reduced problem.
  • Implementation: In implementation, local spectral problems become matrix generalized eigenvalue problems whose selected eigenvectors form offline matrices for online-space construction.

4.4 A numerical example

The GMsFEM converges as the number of multiscale basis functions increases for a high-contrast permeability field, whereas polynomial-basis FEM does not converge until reaching the fine-scale threshold.

  • GMsFEM converges as the number of multiscale basis functions per coarse node increases.The experiment uses a 100 × 100 fine grid, a 10 × 10 coarse grid, and harmonic snapshot functions in an oversampled domain.
  • Polynomial-basis FEM does not converge until a very large number of basis functions crosses the fine-scale threshold.
  • One GMsFEM basis function per node performs poorly because of the high contrast.
  • The permeability field κ(x) used in the numerical example is shown in Figure 4.

5 Adaptivity in GMsFEM

GMsFEM adaptivity uses local residual-based error indicators to identify regions for enrichment, adding eigenfunctions iteratively until the approximation meets a tolerance. Numerical results show faster convergence for adaptive than uniform enrichment, while online enrichment can be ineffective with too few offline basis functions.

  • Error indicators: The error indicators can use either the L2 norm or a coefficient-weighted H−1 norm of the local residual.The paper studies the weighted H−1-based indicator in detail.
  • Error indicators: The H−1 residual norm provides a computable indicator of the energy-norm error.
  • Adaptive enrichment: The number of added basis functions depends on the decay of the local eigenvalues.The next eigenfunctions are used in the selected coarse neighborhoods.
  • Motivation: Adaptivity identifies coarse-grid regions with large errors, while local basis functions alone lack global information needed to capture distant effects.The paper therefore introduces online basis functions as a further ingredient for rapid adaptive convergence.
  • Adaptive enrichment: Adaptive enrichment computes local residuals, marks coarse neighborhoods with large errors, and adds basis functions there iteratively.The process starts from a small initial space and repeats until the error falls below a prescribed tolerance.
  • Numerical results: Adaptive GMsFEM converges faster than GMsFEM with a uniform number of basis functions in the reported H−1-adaptivity experiment.The comparison uses the permeability field in Figure 4 and the forcing term in Figure 5.

6 Residual-based online procedure

The residual-based online procedure enriches the GMsFEM space where current local residuals indicate error, with convergence controlled by the offline space and local spectral information.

  • Residual-based enrichment: Online basis functions are computed from local residuals to accelerate convergence of the current multiscale solution.The residual norm measures the potential reduction in energy error.
  • Convergence: Convergence requires a sufficient number of initial offline basis functions for error decay independent of coefficient contrast.The decay rate depends on the smallest relevant eigenvalue among coarse blocks.
  • Residual-based enrichment: The enriched space adds online basis functions to the offline space in selected coarse neighborhoods.Non-overlapping neighborhoods can be enriched simultaneously.
  • Numerical result: With only one offline basis function, online enrichment does not improve the error substantially because the smallest eigenvalue is 0.0024.Increasing the number of offline basis functions makes convergence very fast; one online iteration reduces the energy error below 1%.

7 Selected global discretizations and energy minimizing oversampling

GMsFEM adapts its local construction to the chosen global discretization, while energy-minimizing oversampling improves snapshot construction for efficient online simulations.

  • Discretization-dependent construction: For each discretization, snapshot spaces and local spectral decompositions are defined consistently with the global formulation.The construction therefore changes across mixed and discontinuous Galerkin settings.
  • Mixed and discontinuous discretizations: Discontinuous Galerkin GMsFEM supports basis functions within individual coarse elements, matching the discretization’s local support.DG coupling uses the classical interior-penalty formulation.
  • Mixed and discontinuous discretizations: Mixed GMsFEM constructs velocity basis functions on pairs of coarse elements sharing an edge and preserves global H(div)-conformity.The associated pressure space is coarse-blockwise constant, while velocity is approximated in a multiscale space.
  • Spectral reduction: Offline spaces retain eigenfunctions with small eigenvalues because these modes are important for accurate multiscale approximation.The global offline space is assembled from local offline spaces.
  • Energy-minimizing oversampling: Energy-minimizing oversampling computes snapshot vectors through local constrained minimization problems on oversampled regions.This construction is motivated by the need for efficient online simulations.
  • Numerical result: With 3 basis functions per node, the mixed GMsFEM velocity error is less than 5%, compared with 20200 fine-grid degrees of freedom.The reported errors are measured in the weighted velocity norm.

8 Discussions on the sparsity in the snapshot space

The paper exploits sparsity in local snapshot representations to reduce computational effort, using different strategies for parameter-dependent problems and Helmholtz equations.

  • Sparse snapshot representations: When solutions use only a portion of the snapshot space, many expansion coefficients are zero and sparsity techniques can save computational effort.The central requirement is a snapshot space in which the solution is sparse.
  • Sparse approaches: The Local-Sparse Snapshot Subspace Approach determines an online sparse space locally through spectral sparse decomposition.It is motivated by parameter-dependent multiscale problems.
  • Sparse approaches: The Sparse Snapshot Subspace Approach determines the online space globally through a global solve using plane-wave snapshots for Helmholtz problems.It directly applies l1 minimization to the underlying PDE.
  • Sparsity conditions: Sparsity is expected when online parameters are close to selected offline values or when wave solutions involve only a few propagation directions.These conditions motivate the two application-specific constructions.

9 Space-time GMsFEM

Space-time GMsFEM constructs time-dependent local reduced spaces for heterogeneous parabolic problems, reducing the dimension and potentially the computational cost compared with space-only bases.

  • Space-time formulation: The space-time construction modifies spatial GMsFEM to account for parabolic cell problems, space-time-cell degrees of freedom, and local space-time boundary conditions.Time-dependent multiscale bases are constructed on the coarse grid.
  • Time decomposition: The method decomposes the full time interval into sequential coarse time-interval problems when the space-time offline space is a direct sum of interval-local spaces.The solution on the whole interval is assembled from the interval solutions.
  • Motivation: Space-time multiscale bases represent the solution over coarse time intervals rather than rebuilding spatial bases at every fine time step.Space-only bases require separate multiscale basis functions at each fine time instant.
  • Motivation: Space-time model reduction can provide substantial CPU savings because space-only representations have a much larger dimension.The reduced-system size is tied to the coefficients used over each space-time interval.
  • Snapshot and offline spaces: Space-time snapshot spaces contain fine-scale solution components on local space-time regions, and spectral problems extract dominant offline modes.Snapshots can be generated from random boundary conditions on oversampled space-time regions.

10 GMsFEM in perforated domains

GMsFEM handles perforated-domain problems by retaining geometry in local snapshot spaces and adaptively reducing them into offline basis functions. The resulting coarse spaces approximate fine-scale solutions while allowing substantially fewer degrees of freedom.

  • 10 GMsFEM in perforated domains: Perforated domains exclude portions of the computational domain, making their geometries challenging for conventional upscaling and numerical homogenization.Variable perforation sizes and geometries generate multiscale solution features.
  • 10 GMsFEM in perforated domains: Snapshot spaces retain perforation geometry, after which local spectral decomposition identifies the multiscale basis functions.For Stokes problems, local problems are solved in both node-based and edge-based coarse neighborhoods.
  • 10 GMsFEM in perforated domains: The offline velocity space is formed from selected local eigenvectors, while the coarse pressure space uses piecewise constant functions on the coarse mesh.The number of retained eigenvectors per neighborhood controls the reduced velocity dimension.
  • 10 GMsFEM in perforated domains: Partition-of-unity multiplication can destroy constant divergence and fail to honor perforation boundary conditions, motivating local optimization corrections.These issues affect stability and boundary-condition conformity of the resulting basis functions.

11 Selected Applications

The paper applies GMsFEM to two-phase flow and transport and to fractured-media gas transport. These examples use multiscale spaces that incorporate mobility or fracture information while reducing the cost of repeated coarse solves.

  • 11 Selected Applications: The two-phase-flow setup uses zero initial saturation, localized source terms, and relative permeabilities κrw(S) = S^2 and κro(S) = (1−S)^2.Water and oil viscosities are set to µw = 1 and µo = 5.
  • 11 Selected Applications: The mixed GMsFEM solves the two-phase flow equation on a coarse grid, while finite volume solves saturation on the fine grid.Multiscale velocity bases are computed at time zero with unit mobility and reused to obtain fine-scale velocities for saturation updates.
  • 11 Selected Applications: The applications include Stokes flow in a perforated domain, with multiscale velocity solutions evaluated against a fine-scale reference.The perforated region is circular and the computational domain is coarsely discretized using uniform triangulation.
  • 11 Selected Applications: For fractured-media gas transport, zero-thickness fractures are represented as fine-scale edge elements within a matrix–fracture decomposition.Snapshot spaces use harmonic extensions and basis construction accounts for the fracture distribution.
  • 11 Selected Applications: The framework is also reported for elastic and acoustic wave propagation, inverse problems, and uncertainty quantification.These additional uses are cited as extensions of the GMsFEM framework.

12 Discussions

The discussion positions GMsFEM as an adaptable local reduction framework that can support global reduction and several extensions. Its scope is bounded by the paper’s noncomprehensive overview and omission of nonlinear applications.

  • 12 Discussions: The paper’s overview of multiscale methods is selective rather than comprehensive, and nonlinear applications are not discussed.Nonlinear eigenvalue problems, hybridization, and nonlinear interpolation are deferred to other work.
  • 12 Discussions: Local reduced-order models can construct snapshots for global model reduction, including POD-based reduction and adaptive online snapshot updates.This creates a global-local workflow combining local adaptivity with global reduced representations.
  • 12 Discussions: The adaptive framework can incorporate optimized multiscale basis functions produced by oversampling, optimization, or singular value decomposition.The paper emphasizes a general strategy adaptable across applications and discretizations.
  • 12 Discussions: The framework is discussed as applicable to stochastic problems, stabilization of convection-dominated problems, and other multiscale settings.Suggested extensions include Monte Carlo or separation-of-variables techniques and multiscale test spaces.
  • 12 Discussions: Using up to 1000 processors, 3D perforated-problem simulations reportedly achieve almost perfect parallel scaling.Multiscale basis computations are described as embarrassingly parallel.
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