skip to main content

This content will become publicly available on May 3, 2024

Title: Exponentially Convergent Multiscale Finite Element Method
We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of the MsFEM in the literature, the ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by the ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.  more » « less
Award ID(s):
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Communications on Applied Mathematics and Computation
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Summary

    In this paper, we propose a general concept for constructing multiscale basis functions within Generalized Multiscale Finite Element Method, which uses oversampling and stable decomposition. The oversampling refers to using larger regions in constructing multiscale basis functions and stable decomposition allows estimating the local errors. The analysis of multiscale methods involves decomposing the error by coarse regions, where each error contribution is estimated. In this estimate, we often use oversampling techniques to achieve a fast convergence. We demonstrate our concepts in the mixed, the Interior Penalty Discontinuous Galerkin, and Hybridized Discontinuous Galerkin discretizations. One of the important features of the proposed basis functions is that they can be used in online Generalized Multiscale Finite Element Method, where one constructs multiscale basis functions using residuals. In these problems, it is important to achieve a fast convergence, which can be guaranteed if we have a stable decomposition. In our numerical results, we present examples for both offline and online multiscale basis functions. Our numerical results show that one can achieve a fast convergence when using online basis functions. Moreover, we observe that coupling using Hybridized Discontinuous Galerkin provides a better accuracy compared with Interior Penalty Discontinuous Galerkin, which is due to using multiscale glueing functions.

    more » « less
  2. null (Ed.)
    An important problem in designing human-robot systems is the integration of human intent and performance in the robotic control loop, especially in complex tasks. Bimanual coordination is a complex human behavior that is critical in many fine motor tasks, including robot-assisted surgery. To fully leverage the capabilities of the robot as an intelligent and assistive agent, online recognition of bimanual coordination could be important. Robotic assistance for a suturing task, for example, will be fundamentally different during phases when the suture is wrapped around the instrument (i.e., making a c- loop), than when the ends of the suture are pulled apart. In this study, we develop an online recognition method of bimanual coordination modes (i.e., the directions and symmetries of right and left hand movements) using geometric descriptors of hand motion. We (1) develop this framework based on ideal trajectories obtained during virtual 2D bimanual path following tasks performed by human subjects operating Geomagic Touch haptic devices, (2) test the offline recognition accuracy of bi- manual direction and symmetry from human subject movement trials, and (3) evalaute how the framework can be used to characterize 3D trajectories of the da Vinci Surgical System’s surgeon-side manipulators during bimanual surgical training tasks. In the human subject trials, our geometric bimanual movement classification accuracy was 92.3% for movement direction (i.e., hands moving together, parallel, or away) and 86.0% for symmetry (e.g., mirror or point symmetry). We also show that this approach can be used for online classification of different bimanual coordination modes during needle transfer, making a C loop, and suture pulling gestures on the da Vinci system, with results matching the expected modes. Finally, we discuss how these online estimates are sensitive to task environment factors and surgeon expertise, and thus inspire future work that could leverage adaptive control strategies to enhance user skill during robot-assisted surgery. 
    more » « less
  3. Abstract We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction-diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction-diffusion problems. The reduced basis does not depend on the fractional power s for 0 < s min ≤ s ≤ s max < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments. 
    more » « less
  4. Taylor And Francis Online (Ed.)
    We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one dimension coincides with the finite element approximation of the solution. This result can be viewed as an extension of the Green function formula for the solution at the continuous level. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices and compare the right hand side load vectors for the two methods. Using evaluation of the Green function, a formula for the inverse of the stiffness matrix is extended to the case of non-uniformly distributed mesh points. We provide an error analysis based on the connection between the two methods and estimate the energy norm of the difference of the two solutions. Interesting extensions to the 2D case are provided. 
    more » « less
  5. Abstract

    The reduction of a large‐scale symmetric linear discrete ill‐posed problem with multiple right‐hand sides to a smaller problem with a symmetric block tridiagonal matrix can easily be carried out by the application of a small number of steps of the symmetric block Lanczos method. We show that the subdiagonal blocks of the reduced problem converge to zero fairly rapidly with increasing block number. This quick convergence indicates that there is little advantage in expressing the solutions of discrete ill‐posed problems in terms of eigenvectors of the coefficient matrix when compared with using a basis of block Lanczos vectors, which are simpler and cheaper to compute. Similarly, for nonsymmetric linear discrete ill‐posed problems with multiple right‐hand sides, we show that the solution subspace defined by a few steps of the block Golub–Kahan bidiagonalization method usually can be applied instead of the solution subspace determined by the singular value decomposition of the coefficient matrix without significant, if any, reduction of the quality of the computed solution.

    more » « less