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Energy-minimizing domain decomposition

Muhammad Hassan, Benjamin Stamm, Lambert Theisen

arXiv:2609.31182v1math.NAcs.NA

TL;DR

Domain decomposition methods are well developed for some problem types, but available approaches are fragmented across linear, nonlinear, source, and eigenvalue problems. This paper introduces an energy-minimizing domain decomposition framework with parallel local solves and a global optimization step. Numerical tests report strong performance across model problems, while nonlinear eigenvalue minimizers are not generally guaranteed unique or to represent the ground state.

  • Problem

    Existing domain decomposition approaches are fragmented across linear, nonlinear, source, and eigenvalue problems, with no single energy-based framework covering them all.

    Source: §1 · p. 2

  • Method

    EMDD applies domain decomposition to energy minimization, using parallel local optimization followed by global minimization over local updates and prior iterates.

    Source: §1 · p. 2

  • Results

    Across a wide set of model problems, EMDD is among the best-performing methods compared with model-specific domain decomposition methods.

    Source: §6 · p. 24

  • Takeaways & Limitations

    The authors conclude that energy-based domain decomposition is a promising robust framework for energy-minimization problems.

    Source: §6 · p. 24

    For nonlinear eigenvalue problems, minimizers are not known to be unique or to correspond to the ground state without further assumptions.

    Source: §4.4 · p. 17

Abstract

from arXiv · show

We propose a novel domain decomposition framework based on energy minimization, applicable to a wide class of linear and nonlinear partial differential equations. This energy-minimizing domain decomposition (EMDD) method addresses problems whose solutions are minimizers of an energy functional, either unconstrained or constrained to a manifold, and thus covers source and eigenvalue problems in a single formulation. Besides introducing the general framework, we derive the algorithmic realization of the method for four classes of model problems, namely linear and nonlinear source and eigenvalue problems, and derive the computational complexity of the method. Numerical experiments demonstrate the robustness and efficiency of the method across all four model problems and show that it compares favourably against existing model-specific domain decomposition methods.

1. Introduction.

The introduction frames domain decomposition as mature for linear elliptic problems but fragmented across nonlinear, source, and eigenvalue settings. It proposes an energy-based framework and a two-level optimization method to address these classes together.

1. Introduction.

  • Linear source methods commonly use Schwarz variants as Krylov preconditioners, while nonlinear methods differ in where decomposition and linearization occur.Approaches include nonlinear subdomain iterations, Newton-Krylov-Schwarz, and preconditioned nonlinear systems.Source: §1 · p. 1
  • Eigenvalue domain decomposition is commonly indirect, with rare methods posing local eigenvalue or minimization problems on subdomains.The usual approach preconditions repeated linear solves within an outer eigensolver.Source: §1 · p. 2
  • EMDD formulates linear, nonlinear, source, and eigenvalue problems through energy minimization, extending one framework across classes usually treated separately.The energy functional also combines prior iterates with local updates without explicit transmission conditions.Source: §1 · p. 2
  • Local subproblems are solved in parallel, then a global optimization minimizes energy over local updates and past iterates.The previous global iterate is included in local approximation spaces, so no local boundary conditions are imposed.Source: §1 · p. 2

2. Problem Setting and Numerical Method.

The method applies domain decomposition to minimization over a space or constraint manifold. Its two-level algorithm alternates parallel local optimization with a global optimization, with fixed-point guarantees conditional on assumptions.

2. Problem Setting and Numerical Method.

  • Each iteration solves local optimization problems in parallel and then minimizes energy globally over local updates and past iterates.The local spaces include the previous global iterate, and no local boundary conditions are imposed.Source: §2 · p. 3
  • For an unconstrained problem, the local and second-level steps are Galerkin approximations that inherit the global minimization structure.The second-level problem is posed over a small-dimensional space when local subspaces are localized.Source: §2 · p. 4
  • A unique global minimizer is an EMDD fixed point; conversely, a fixed point is critical under local solvability, transversality, and global spanning assumptions.The section assumes global minimizers exist and are unique, possibly up to trivial gauge invariance.Source: §2 · pp. 3–5
  • For unconstrained problems, unique global minimization ensures local unique solvability; for linear eigenproblems this holds up to sign, but nonlinear cases remain unclear.The authors report no observed non-uniqueness in their nonlinear eigenvalue experiments, discounting trivial gauge freedoms.Source: §2 · p. 6

3. Model problems, discretization, and complexity.

The model-problem setting uses a bounded Lipschitz domain in dimensions one through three and a Sobolev space. The listed nonlinearities are subject to growth bounds.

3. Model problems, discretization, and complexity.

  • The setting uses a bounded Lipschitz domain Ω ⊂ ℝ^d for d ∈ {1, 2, 3}, sets V = H_0^1(Ω), and assumes a symmetric, continuous, coercive bilinear form a and bounded linear functional f.Source: §3.1 · p. 7
  • The source and eigenvalue nonlinearities are subject to growth bounds involving q_src and q_eig, respectively.The supplied conditions bound their magnitudes by a constant times 1 plus the argument raised to the corresponding growth exponent.Source: §3.1 · p. 7

1. Coercive linear source problems.

The linear source problem is characterized by its associated Euler-Lagrange equation, and coercivity yields a unique minimizing solution.

1. Coercive linear source problems.

  • The Euler-Lagrange equation associated with the coercive linear source minimization problem is given as its stationarity condition.The passage identifies it as the equation associated with problem (3.3).Source: §1 · p. 7
  • The Lax-Milgram lemma gives a unique minimizer of problem (3.3), and that minimizer solves the Euler-Lagrange equations.Source: §1 · p. 7

2. Monotone semi-linear source problems.

The Euler–Lagrange equation for the semilinear source energy has a unique minimizer under the stated sufficient conditions.

2. Monotone semi-linear source problems.

  • Under the sufficient conditions (3.1), the semilinear source energy has a unique minimizer solving its Euler–Lagrange equation.The supplied passage identifies this result with the Browder–Minty theorem.Source: §2 · p. 7

3. Linear eigenvalue problems.

The linear eigenvalue problem is formulated as minimizing energy on a normalized manifold; its minimizers are the ground-state eigenfunctions.

3. Linear eigenvalue problems.

  • The coercivity condition on the bilinear form may be weakened to a Gårding inequality.Source: §3 · p. 8
  • Minimizing the energy in (3.5) over the normalized manifold yields the ground-state eigenfunctions of the associated linear eigenvalue problem.The lowest eigenvalue is assumed to be non-degenerate.Source: §3 · p. 8

4. Nonlinear eigenvalue problems.

The section treats nonlinear eigenvalue minimization, Galerkin discretization, and the resulting computational costs under stated assumptions.

4. Nonlinear eigenvalue problems.

  • Under assumptions (3.2), the nonlinear eigenvalue energy has a unique minimizer up to sign.The discretized nonlinear eigenvalue problem may not be uniquely solvable, excluding gauge freedoms.Source: §4 · p. 8

3.2. General framework for Galerkin discretization.

  • The model problems are discretized by replacing the function space with an N-dimensional approximation subspace while preserving their structural setting.The approximation space is decomposed into subspaces, and QR orthogonalization of the coarse-space basis is advised for numerical stability near convergence.Source: §3.2 · pp. 8, 10

3.3. General framework for complexity analysis.

  • The complexity analysis uses function-evaluation, bilinear-form, overlap, and parameter assumptions, including bounded overlap and q ≤ min(M, Nloc).These assumptions are described as natural for finite-element discretizations of local PDEs, with bounded overlap standard in domain decomposition.Source: §3.3 · pp. 10–11

3.3.1. The vectors f i and f 0.

  • After the first iteration, updating the f vectors costs O(N), while assembling f0 costs O(MNloc) when the required quantities are precomputed.At the first iteration, assembling all local vectors costs O(N + MNloc).Source: §3.3.1 · p. 11

3.3.2. The matrices Ai and A0.

  • Local bilinear-form matrix-vector products scale as O(Ni), and the matrices Si and S0 have construction and application costs matching Ai and A0.Source: §3.3.2 · p. 12, §3.3.3 · p. 13

3.3.3. The matrices Si and S0.

  • The construction and application costs of Si and S0 match those of Ai and A0, respectively.Source: §3.3.3 · p. 13

3.3.4. The nonlinear operators Ni(u), N0(u), Hi(u) and H0(u).

  • For nonlinear operators, local matrix-vector products scale as O(Ni), while assembly and matrix-vector products for N0 scale as O(q2N) and O(MNloc), respectively.The corresponding Hi and H0 costs match those for Ni and N0.Source: §3.3.4 · p. 13

4. Application to the model problems.

The model-problem applications realize EMDD for source and eigenvalue tasks, with linear cases having stated per-iteration costs and nonlinear cases also using iterative or quadratic approximations.

4.1. Model Problem 1: Linear Source Problems of the form (3.3).

  • For linear source problems, one EMDD iteration costs O(N + NkryMNloc), with only one global O(N) computation per iteration.Source: §4.1 · p. 14

4.2. Example Class 2: Semilinear Source Problems of the form (3.4).

  • For semilinear source problems, one EMDD iteration costs O(Nnl(q2 + M)N), while qEMDD uses a second-order Taylor approximation to reduce the nonlinear bottleneck.The quadratic approximation may increase outer iterations, though the authors report not observing this in their numerical tests.Source: §4.2 · p. 15

4.3. Example Class 3: Linear Eigenvalue Problems of the form (3.5).

  • For linear eigenvalue problems, one EMDD iteration costs O(N + NevpMNloc); a spectral gap at the lowest eigenvalue is conditional on non-degeneracy and a sufficiently accurate initialization.Source: §4.3 · p. 16

4.4. Example Class 4: Nonlinear Eigenvalue Problems of the form (3.6).

  • For nonlinear eigenvalue problems, local and second-level constrained minimizations can be solved by Riemannian optimization or their Euler–Lagrange equations by SCF-type methods.The constrained minimizers need not be unique or correspond to the ground state without further assumptions.Source: §4.4 · p. 17

5. Numerical Experiments.

Across four model classes, the experiments compare EMDD with domain-decomposition alternatives under varied problem, mesh, overlap, and subdomain settings. Results include faster convergence for selected EMDD variants, while mesh refinement and problem type affect iteration behavior.

5. Numerical Experiments.

  • The experiments use a serial Gridap implementation in Julia and generally construct subdomains with an irregular METIS partition.The code is archived in.Source: §5 · p. 18

5.1. Linear Source Problems.

  • EMDD q=2 is consistently faster than q=1, though EMDD needs more iterations on variable diffusion than on Poisson.The other compared methods behave similarly on the diffusion and Poisson problems.Source: §5.1 · pp. 18–19, Figure 1 · p. 19
  • Fixed overlap layers lead to more iterations as the mesh is refined; keeping δ/H≈0.1 nearly levels counts, and increasing overlap generally lowers them.The reported table gives parallel local-solve iterations for the Poisson problem and varies mesh refinement and overlap.Source: Table 1 · p. 19, §5.1 · pp. 19–20

5.2. Semilinear Source Problems.

  • EMDD q=2 is consistently faster than q=1; the quadratic local model converges similarly to exact EMDD at a fraction of its cost.Newton–PCG exhibits second-order convergence when enough inner PCG iterations are performed; Anderson-accelerated nonlinear RAS has monotonicity difficulties at β=1.Source: Figure 2 · p. 20, §5.2 · p. 21
  • On the graded L-shaped diffusion–reaction problem, the history vector improves convergence, and the quadratic-model variant is nearly equivalent to exact EMDD.The test uses a graded triangular P1 mesh, two overlap layers, and m∈{2,4,8} subdomains.Source: Figure 3 · p. 21, §5.2 · p. 21

5.3. Linear Eigenvalue Problems.

  • EMDD q=1 performs similarly to LOPSD+AS, while q=2 is comparable to LOBPCG+AS; JD-GMRES(AS, ν) with ν=2 or 4 outperforms them per outer iteration.The JD-GMRES variants use approximate linear-system inversion and are considerably more expensive per outer iteration.Source: Figure 4 · p. 22, §5.3 · p. 22

5.4. Nonlinear Eigenvalue Problems.

  • EMDD q=2 needs the fewest outer iterations for the Gross–Pitaevskii ground state, while projected qEMDD closely approximates full EMDD at significantly lower cost.All compared methods converge, and the CG variants of GFDN need fewer iterations than plain GFDN.Source: Figure 5 · p. 23, §5.4 · p. 23

6. Outlook.

The outlook presents additive, two-level EMDD as an initial energy-based domain decomposition method, with competitive tests and more sophisticated variants and theoretical questions left for future work.

6. Outlook.

  • Across model problems, additive two-level EMDD is among the best-performing methods compared with model-specific domain decomposition methods.The authors view energy-based domain decomposition as a promising robust framework for energy-minimization problems.Source: §6 · p. 24
  • The authors did not explore more sophisticated variants, including multiplicative, restricted, and coarse-space-enriched EMDD.The multiplicative variant replaces parallel local solves with sequential ones; coarse-space enrichment adds a problem-adapted space to the second-level minimization.Source: §6 · p. 24
  • Future work includes testing variants and richer finite-element basis functions, studying more general constrained optimization, and deriving explicit convergence rates, particularly for eigenvalue problems.Examples of broader constrained problems include Stiefel or Grassmann manifold optimization and nonlinear eigenvalue problems with eigenvalue nonlinearity.Source: §6 · p. 24

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