Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions

Chen, Yifan; Hou, Thomas Y.; Wang, Yixuan

doi:10.1137/20m1352922

Cited by 13 publications

(4 citation statements)

References 32 publications

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“…where n is the total number of scales and h(n) and g(n) are the filter coefficients. Formulas ( 7) and ( 8) describe the relationship between adjacent two-scale spatial basis functions [21], and these two equations are called two-scale equations.…”

Section: D Point Cloud Reconstruction Of Inverted Arches Of Soft Rock...mentioning

confidence: 99%

Research on Deformation Monitoring of Invert Uplifts in Soft Rock Tunnels Based on 3D Laser Scanning

Zhang,

Niu,

Liu

2023

Applied Sciences

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The soft surrounding rock in tunnels has the characteristics of low strength, easy softening after soaking, and poor self-stability, which makes the inverted arch structure in soft rock tunnels prone to uplift deformations. Therefore, a deformation monitoring method for inverted arch uplifts of soft rock tunnels based on 3D laser scanning technology is studied to improve deformation monitoring. A Leica Scan Station2 3D laser scanner is used to collect 3D point cloud data of soft rock tunnel inverts. Using the automatic matching method of public landmarks and the improved Rodrigues parameter method, the collected 3D point cloud data are spliced and through the Mallat algorithm, the 3D point cloud data are reconstructed and processed after splicing. The whole least square method is used to fit the reconstructed 3D point cloud data. Through the principal component analysis method, the normal vector of the fitting plane is estimated and the best datum plane of the soft rock tunnel invert is found. By calculating and extracting the geometric parameters of the slice point cloud, the monitoring of inverted arch uplift deformations in soft rock tunnels is completed. The experiment shows that this method can effectively collect 3D point cloud data of soft rock tunnel inverts, and complete point cloud stitching and reconstruction. This method can effectively monitor the uplift deformation of inverted arches at different grouting depths.

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Section: D Point Cloud Reconstruction Of Inverted Arches Of Soft Rock...mentioning

confidence: 99%

Research on Deformation Monitoring of Invert Uplifts in Soft Rock Tunnels Based on 3D Laser Scanning

Zhang,

Niu,

Liu

2023

Applied Sciences

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“…There is a vast literature on numerical upscaling of multiscale PDEs. For our context, i.e., elliptic PDEs with rough coefficients, rigorous theoretical results include generalized finite element methods [1,2], harmonic coordinates [28], LOD [23,15,18,10,14,22], Gamblets related approaches [29,30,25,26,17,27], and generalizations of multiscale finite element methods [16,8,20,12,6,7]. Among them, the ones most related to this paper are LOD and Gamblets; the connection has been explained in subsection 1.1.3.…”

Section: Numerical Upscalingmentioning

confidence: 99%

Multiscale Elliptic PDE Upscaling and Function Approximation via Subsampled Data

Chen¹,

Hou²

2022

Multiscale Model. Simul.

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There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled information used for approximation (in the latter). As such, both problems can be thought of as recovering a target function based on some coarse data that are either artificially chosen by an upscaling algorithm or determined by some physical measurement process. The purpose of this paper is then to study, under such a setup and for a specific elliptic problem, how the lengthscale of the coarse data, which we refer to as the subsampled lengthscale, influences the accuracy of recovery, given limited computational budgets. Our analysis and experiments identify that reducing the subsampling lengthscale may improve the accuracy, implying a guiding criterion for coarse-graining or data acquisition in this computationally constrained scenario, especially leading to direct insights for the implementation of the Gamblets method in the numerical homogenization literature. Moreover, reducing the lengthscale to zero may lead to a blow-up of approximation error if the target function does not have enough regularity, suggesting the need for a stronger prior assumption on the target function to be approximated. We introduce a singular weight function to deal with it, both theoretically and numerically. This work sheds light on the interplay of the lengthscale of coarse data, the computational costs, the regularity of the target function, and the accuracy of approximations and numerical simulations.

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“…9). The local 15 (global 16 ) results are more accurate than required by about 14 Since we assume that u 0 = g D = g N = 0 for Example 3 and 4 (thus u b = 0), we used Algorithm 2 with the improved relative global error bound 2 Proposition 5.4) to generate the results shown in Fig. 10.…”

Section: Global Gfem Approximation With Random Local Basis Generationmentioning

confidence: 99%

“…A technically similar approach is proposed in [28] for the solution of Laplacian eigenvalue problems. Furthermore, optimal interface spaces for (parametrized) elliptic problems are introduced in [57], generalized to geometry changes in [56], and also proposed in [14]. Concerning (real-world) applications the optimal local approximation spaces are employed, for instance, for the construction of digital twins [34,38] and in the context of data assimilation [59].…”

Section: Introductionmentioning

confidence: 99%

Optimal local approximation spaces for parabolic problems

Schleuß¹,

Smetana²

2020

Preprint

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We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods. The diffusion coefficient can be arbitrarily rough in space and time. To construct local approximation spaces we consider a compact transfer operator that acts on the space of local solutions and covers the full time dimension. The optimal local spaces are then given by the left singular vectors of the transfer operator. To proof compactness of the latter we combine a suitable parabolic Caccioppoli inequality with the compactness theorem of Aubin-Lions. In contrast to the elliptic setting [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406] we need an additional regularity result to combine the two results. Furthermore, we employ the generalized finite element method to couple local spaces and construct an approximation of the global solution. Since our approach yields reduced space-time bases, the computation of the global approximation does not require a time stepping method and is thus computationally efficient. Moreover, we derive rigorous local and global a priori error bounds. In detail, we bound the global approximation error in a graph norm by the local errors in the L 2 (H 1 )-norm, noting that the space the transfer operator maps to is equipped with this norm. Numerical experiments demonstrate an exponential decay of the singular values of the transfer operator and the local and global approximation errors for problems with high contrast or multiscale structure regarding space and time.

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Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions

Cited by 13 publications

References 32 publications

Research on Deformation Monitoring of Invert Uplifts in Soft Rock Tunnels Based on 3D Laser Scanning

Research on Deformation Monitoring of Invert Uplifts in Soft Rock Tunnels Based on 3D Laser Scanning

Multiscale Elliptic PDE Upscaling and Function Approximation via Subsampled Data

Optimal local approximation spaces for parabolic problems

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