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Residual-driven online Generalized Multiscale Finite Element Methods
Eric T. Chung, Yalchin Efendiev, Wing Tat Leung
TL;DR
High-contrast multiscale problems motivate reduced local models, but local offline bases may miss distant effects. The paper constructs adaptive online bases from residuals using an offline space with specified properties. It reports that sufficient offline bases permit contrast-independent error reduction, while inadequate offline spaces yield little decay.
Problem
Local multiscale bases may miss distant effects in high-contrast problems, while fine-scale discretization can require large, ill-conditioned systems.
Source: §2.1 · p. 3Method
The method constructs local online basis functions from the current residual and adaptively enriches selected regions using an offline GMsFEM space.
Source: §1 · p. 3, §6 · p. 18Results
With sufficient initial offline basis functions, online enrichment can reduce error independently of contrast; too few initial bases yield little error decay.
Source: §1 · pp. 2–3Takeaways & Limitations
Including eigenvectors associated with small eigenvalues in the offline space supports contrast-independent error reduction from online enrichment.
Source: §1 · p. 2The contrast-independent error-reduction guarantee requires an offline space satisfying ONERP; the online basis also has no sparsity pattern in the snapshot space.
Source: §4 · p. 9, §5.3 · p. 16
Abstract
from arXiv · showhide
The construction of local reduced-order models via multiscale basis functions has been an area of active research. In this paper, we propose online multiscale basis functions which are constructed using the offline space and the current residual. Online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators. We derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error. This error decrease is independent of physical parameters, such as the contrast and multiple scales in the problem. The offline spaces are constructed using Generalized Multiscale Finite Element Methods (GMsFEM). We show that if one chooses a sufficient number of offline basis functions, one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast. We note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects. Using the residual information, we can incorporate the distant information provided the offline approximation satisfies certain properties. In the paper, theoretical and numerical results are presented. Our numerical results show that if the offline space is sufficiently large (in terms of the dimension) such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues, then the error reduction by adding online multiscale basis function is independent of the contrast. We discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces.
1 Introduction
Local multiscale methods can miss distant effects, motivating residual-based online basis functions that adaptively enrich selected regions. The paper establishes that contrast-independent error reduction requires a sufficiently rich offline space, and reports improved results with adaptive and reduced-dimensional online enrichment.
1 Introduction
- Local multiscale reduction targets efficient coarse-grid computation, while global reduction may require costly offline construction and lacks local adaptivity.The paper focuses on local methods with local online computations.Source: §1 · pp. 1–2
- GMsFEM enriches coarse spaces with additional basis functions and separates computation into offline and online stages.Offline spaces are reused to construct multiscale basis functions for input parameters.Source: §1 · p. 2
- Online basis functions use residuals to capture distant effects that local offline basis functions can miss.They are added adaptively in selected regions using error indicators.Source: §1 · p. 2
- The offline space must satisfy ONERP to guarantee that online enrichment reduces error independently of contrast; the first excluded eigenvalue controls the decay.Including eigenvectors associated with small eigenvalues is necessary, particularly for high-contrast problems.Source: §1 · p. 2
- Adaptive online enrichment substantially improves GMsFEM, while too few offline basis functions yield little error decay.Online basis computation can use a reduced-dimensional space of consecutive offline basis functions and still achieve substantial error reduction.Source: §1 · pp. 2–3
2 GMsFEM for high contrast flow
GMsFEM addresses high-contrast, multiscale flow problems by constructing local basis functions on coarse neighborhoods. Its offline space is reduced from a snapshot space through spectral selection, while online basis functions are added locally and adaptively from residuals.
2.1 Overview
- High-contrast, multiscale coefficients make fine-scale discretizations large and ill-conditioned, motivating efficient schemes with fewer degrees of freedom.The considered flow problem has homogeneous Dirichlet boundary conditions and a given source function.Source: §2.1 · p. 3
- GMsFEM builds nodal basis functions supported on coarse neighborhoods and combines offline bases with locally computed, adaptive online bases.Multiple basis functions and different numbers of bases can be used at different coarse nodes.Source: §2.1 · pp. 4–5
2.2 Construction of offline basis functions
- A snapshot space contains components of the fine-scale solution on a neighborhood; spectral reduction extracts dominant modes as offline basis functions.Snapshots may be restrictions of the conforming space or κ-harmonic extensions, yielding spectral or harmonic basis functions.Source: §2.2 · p. 5
- The offline construction selects the first li eigenfunctions of the local spectral problem to form neighborhood basis functions.The local offline space is spanned by the selected offline basis functions.Source: §2.2 · pp. 5–6
3 Offline Adaptive GMsFEM
Offline adaptive GMsFEM enriches regions with large residuals using subsequent local eigenfunctions. Its convergence depends on the eigenvalue structure, and local bases may miss global behavior that online bases are designed to capture.
3 Offline Adaptive GMsFEM
- The offline adaptive algorithm computes local residuals and adds subsequent eigenfunctions in coarse neighborhoods selected by residual magnitude.It uses bulk marking to select neighborhoods for enrichment and repeats until an error tolerance is reached.Source: §3 · pp. 6–7
- Offline enrichment converges rapidly when local eigenvalues grow quickly, with a convergence rate independent of coefficient contrast.The residual norm provides a computable indicator of energy-norm error, and the first excluded eigenvalue enters the error bound.Source: §3 · pp. 6–7
- Local offline bases can miss global solution behavior, so online basis functions are needed for rapid convergence in such cases.The paper attributes this limitation to the absence of global information in local basis functions.Source: §3 · p. 7
4 Residual based online adaptive GMsFEM
The method adds online multiscale basis functions during adaptive enrichment, using local residuals from the current solution. Error reduction depends on the offline space satisfying a property that can ensure convergence independent of physical parameters.
4 Residual based online adaptive GMsFEM
- Online basis functions are computed during iteration from local residuals of the current multiscale solution, and the enriched space can contain both offline and online bases.The construction requires offline basis functions, whose number affects whether the solution sequence converges rapidly.Source: §4 · p. 8
- The method selects each online basis function to maximize the local residual, whose norm measures the amount of energy-error reduction.The basis is normalized in the energy form, and the maximization is over the local space.Source: §4 · p. 8
- Adding online basis functions on non-overlapping coarse neighborhoods in one iteration can improve convergence; the rate depends on Cerr and Λ(I)_min.The paper gives a convergence rate for adding one basis per iteration and considers multiple non-overlapping neighborhoods to enhance convergence and efficiency.Source: §4 · p. 9
- The offline space must contain enough basis functions for Λ(I)_min to be sufficiently large, a condition that may be difficult when channels make it very small.The paper presents ONERP as a condition for guaranteeing error decrease independent of physical parameters.Source: §4 · pp. 9–10
- When the offline space satisfies ONERP, online enrichment decreases error independently of physical parameters such as contrast and scales.The paper states that sufficiently large offline spaces can satisfy the property; channels may make the relevant minimum eigenvalue very small.Source: §4 · pp. 9–10
5 Numerical result
The numerical studies show that online enrichment depends on a sufficiently rich offline space: ONERP is associated with fast, contrast-independent error reduction, while inadequate spaces can slow or even prevent reduction. Adaptive residual selection targets error levels, and reduced online solves become effective in later iterations when expanded with neighboring snapshots.
5.1 Comparison of using different number of initial basis
- 130-fold error decay with five initial basis functions versus 90-fold with four was observed over three iterations.The comparison is reported for the last three iterations following the initial one.Source: §5.1 · p. 12, Table 5 · p. 13, Figure 5 · p. 14
- Without ONERP, higher contrast can slow error decay and sometimes online additions do not reduce error; with ONERP, convergence is fast and contrast-independent.The reported comparison includes 1–2 initial basis functions in the slower-decay cases; the paper also reports cases with no error decrease when ONERP is absent.Source: §5.1 · pp. 11–12
5.2 Adaptive online enrichment
- Residual-tolerance enrichment brings energy errors to the same order as the chosen tolerance, while cumulative-residual selection uses fewer basis functions for a similar error.The tested tolerances were 10^-3, 10^-4, and 10^-5; the cumulative-residual comparison used θ = 0.7 and tol = 10^-4.Source: §5.2 · pp. 15–16, Table 9 · p. 16
5.3 Inexpensive online basis construction
- Reduced-dimensional online spaces work well only in the first iteration when using the next N0 snapshots, but 40 next snapshots with neighboring regions give convergence comparable to the full local snapshot space.The later-iteration issue arises because only one extra snapshot is added each time in the first approach.Source: §5.3 · p. 16, Figure 6 · p. 17, Figure 7 · p. 17
- Selecting a basis-function class by testing residual reduction in each region can improve GMsFEM accuracy.The example considers eigenvectors and polynomial basis functions, choosing the class that gives the largest residual reduction.Source: §5.3 · pp. 17–18
6 Conclusions
The paper develops residual-driven online multiscale basis functions within GMsFEM, adding them adaptively in selected regions. Contrast-independent error reduction requires an offline space satisfying ONERP.
6 Conclusions
- Online basis functions solve local problems using the residual at the current solution iterate, incorporating global effects missing from local GMsFEM bases.The method adds online functions in selected regions through adaptivity, without resorting to global solves.Source: §6 · p. 18
- The offline space must satisfy ONERP for added online basis functions to reduce error independently of contrast and small scales.The first several GMsFEM basis functions are needed to guarantee contrast-independent error reduction.Source: §6 · p. 18
- Numerical examples show rapid error decay with ONERP-satisfying offline spaces, and the paper proposes strategies to reduce online basis computation costs.Source: §6 · p. 18