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Distributed finite element solution using model order reduction

Tom Gustafsson, Antti Hannukainen, Vili Kohonen

TL;DR

The paper tackles scalable distributed finite element solution of elliptic boundary value problems by integrating localized model order reduction in a cloud setting. It partitions the domain into overlapping subdomains, constructs local reduced bases via a weighted low-rank approximation of the harmonic extension operator \\mathcal{Z}_i, and projects the global FE system onto the reduced space with minimal memory and communication. A global error bound combines the standard FE error with a reduction term controlled by the tolerance \\epsilon, yielding practical accuracy guarantees while keeping the reduced system small. Numerical experiments demonstrate the method's scalability on up to 85 million DOFs on a laptop by leveraging cloud resources, achieving substantial DOF reduction and improved conditioning, and enabling large-scale 3D simulations in realistic geometries.

Abstract

We extend a localized model order reduction method for the distributed finite element solution of elliptic boundary value problems in the cloud. We give a computationally efficient technique to compute the required inner product matrices and optimal reduced bases. A memory-efficient methodology is proposed to project the global finite element linear system onto the reduced basis. Our numerical results demonstrate the technique using non-trivial tetrahedral meshes and subdomain interfaces with up to 85 million degrees-of-freedom on a laptop computer by distributing the bulk of the model order reduction to the cloud.

Distributed finite element solution using model order reduction

TL;DR

The paper tackles scalable distributed finite element solution of elliptic boundary value problems by integrating localized model order reduction in a cloud setting. It partitions the domain into overlapping subdomains, constructs local reduced bases via a weighted low-rank approximation of the harmonic extension operator \\mathcal{Z}_i, and projects the global FE system onto the reduced space with minimal memory and communication. A global error bound combines the standard FE error with a reduction term controlled by the tolerance \\epsilon, yielding practical accuracy guarantees while keeping the reduced system small. Numerical experiments demonstrate the method's scalability on up to 85 million DOFs on a laptop by leveraging cloud resources, achieving substantial DOF reduction and improved conditioning, and enabling large-scale 3D simulations in realistic geometries.

Abstract

We extend a localized model order reduction method for the distributed finite element solution of elliptic boundary value problems in the cloud. We give a computationally efficient technique to compute the required inner product matrices and optimal reduced bases. A memory-efficient methodology is proposed to project the global finite element linear system onto the reduced basis. Our numerical results demonstrate the technique using non-trivial tetrahedral meshes and subdomain interfaces with up to 85 million degrees-of-freedom on a laptop computer by distributing the bulk of the model order reduction to the cloud.
Paper Structure (17 sections, 7 theorems, 64 equations, 9 figures, 3 tables)

This paper contains 17 sections, 7 theorems, 64 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. Domain decomposition
  4. Low-rank approximation of $\mathcal{Z}_i$
  5. Error analysis
  6. Estimate on a single subdomain
  7. Full error estimate
  8. Parallel implementation
  9. Numerical experiments
  10. Scaling tests
  11. Subproblem scaling
  12. Nonconstant $a$
  13. Engineering geometry
  14. Conclusions and future work
  15. Proof to \ref{['lem:boundarytol2norm']}
  16. ...and 2 more sections

Key Result

Lemma 4.1

Let $\bm A$ be a square symmetric and positive definite matrix with the block structure and the Cholesky factor Then $\bm A / \bm A_{II} = \bm R_{BB}^T \bm R_{BB}$ where $\bm A / \bm A_{II} = \bm A_{BB} - \bm A_{BI}\bm A_{II}^{-1}\bm A_{IB}$ is the Schur's complement.

Figures (9)

  • Figure 1: An example with two subdomains. (Top left.) A polygonal domain is split into overlapping subdomains $\omega_1$ (orange and white) and $\omega_2$ (blue and white). The subdomain $\omega_0$ (white), consists of elements that belong to both subdomains -- in general, to more than one subdomain. Such a decomposition can be obtained by feeding the mesh edge connectivity to a graph partitioning software, e.g., METIS karypis_metis_1997 or Scotch pellegrini_scotch_1996, and finding elements that have at least one node belonging to a specific subgraph. (Top right.) The subdomain $\omega_1$ is extended by the distance $r$ to define $\omega_{1}^+$ which then consists of the orange and the grey elements. Larger $r$ will decrease the size of the global reduced linear system while increasing the size of the local basis reduction problem. (Bottom.) The index sets $I$ and $B$ are used in the definition of a discrete lifting operator from $\partial \omega_1^+$ to $\omega_1$ in \ref{['sec:lowrank']}.
  • Figure 2: An example of the stitching operator with two finite element functions (red and black) over two subdomains in one dimension.
  • Figure 3: A high level overview of the data flow in our distributed implementation of step (2). Decomposing the mesh into overlapping subdomains is done by the master node in step (1). The overlapping meshes are then uploaded to a cloud storage bucket where the worker nodes are able to access the files. The worker nodes are virtual machines launched as a batch job. The figure depicts an example with three worker nodes/subdomains. The worker nodes write their output data to a separate cloud storage bucket which can be accessed by the master node in order to proceed with step (3).
  • Figure 4: Convergence of the method on a log-log scale. The $x$-axis depicts the mesh parameter $h$, and the $y$-axis shows the errors $\| \nabla (\phi - \tilde{u}) \|_{0, \Omega}$, the reduction errors $\| \nabla (u - \tilde{u}) \|_{0,\Omega}$ and the theoretical FEM convergence rate. The reduced method follows the conventional rate since the reduction error stays well below the FEM error. The reduction errors were not computed for the largest cases due to limitations in scaling the conventional FEM.
  • Figure 5: Case 7 reduced solution and error. The error is negligible and concentrates on the subdomain interfaces.
  • ...and 4 more figures

Theorems & Definitions (13)

  • Definition 3.4: Stitching operator $\mathcal{S}_i$
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 5.1
  • Lemma 5.2
  • proof
  • Remark 1
  • Theorem 5.4
  • proof
  • ...and 3 more