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Symplectic Model Reduction of Hamiltonian Systems

Liqian Peng, Kamran Mohseni

arXiv:1407.6118v2math.NA

TL;DR

Large-scale Hamiltonian systems require reduced models that lower computational cost without losing symplectic structure. The paper introduces PSD with symplectic Galerkin projection, together with three PSD algorithms and SDEIM, and reports accurate, efficient, stable reduction for linear and nonlinear wave equations.

  • Problem

    Classical model reduction can fail to preserve the symplectic structure and stability needed for Hamiltonian systems, especially in long-time integration.

  • Method

    PSD fits empirical data with a symplectic subspace, while symplectic Galerkin projection constructs a reduced Hamiltonian system; three PSD algorithms and SDEIM support implementation.

  • Results

    The proposed technique demonstrated high accuracy, good efficiency, and stability preservation for linear and nonlinear wave equations.

  • Takeaways & Limitations

    PSD provides a structure-preserving alternative for reducing Hamiltonian systems, particularly when long-time integration is required.

  • Takeaways & Limitations

    The cotangent-lift momentum agrees with the Legendre-transform momentum only when q̇(t) lies in Range(Φ) or M = I_n, while complex SVD constructs only a near-optimal matrix in M_2.

Abstract

from arXiv · show

In this paper, a symplectic model reduction technique, proper symplectic decomposition (PSD) with symplectic Galerkin projection, is proposed to save the computational cost for the simplification of large-scale Hamiltonian systems while preserving the symplectic structure. As an analogy to the classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is designed to build a symplectic subspace to fit empirical data, while the symplectic Galerkin projection constructs a reduced Hamiltonian system on the symplectic subspace. For practical use, we introduce three algorithms for PSD, which are based upon: the cotangent lift, complex singular value decomposition, and nonlinear programming. The proposed technique has been proven to preserve system energy and stability. Moreover, PSD can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear Hamiltonian systems. Owing to these properties, the proposed technique is better suited than the classical POD-Galerkin approach for model reduction of Hamiltonian systems, especially when long-time integration is required. The stability, accuracy, and efficiency of the proposed technique are illustrated through numerical simulations of linear and nonlinear wave equations.

1. Introduction.

Model reduction addresses the computational burden of simulating high-dimensional dynamical systems, but classical POD-Galerkin reduction does not generally preserve stability or geometric structure. PSD instead constructs a symplectic subspace and reduced Hamiltonian system, with algorithms and SDEIM support for practical computation.

  • Model reduction approximates high-dimensional dynamical systems with lower-order models to reduce computational costs.Direct simulations can be too intensive for frequent use.
  • Classical POD-Galerkin reduction is widely used but is not guaranteed to produce a stable reduced system.Instability can accompany blowup of system energy and flow volume.
  • PSD constructs a symplectic subspace and reduced Hamiltonian system that preserve the underlying symplectic structure.The framework is designed to support energy and stability preservation during long-time integration.
  • Three PSD algorithms are based on the cotangent lift, complex SVD, and nonlinear programming.The paper presents these algorithms as practical ways to construct the symplectic matrix.

2. Hamiltonian System and Symplectic Integrator.

Hamiltonian systems evolve through structure-preserving dynamics, while symplectic integrators preserve the underlying symplectic structure during numerical time integration. This preservation conserves flow volume and system energy and supports long-time simulations.

  • A Hamiltonian system is defined by a smooth Hamiltonian function and its associated Hamiltonian vector field.The phase-space variable combines position and momentum coordinates.
  • Hamiltonian flow conserves the symplectic two-form, system Hamiltonian, and flow volume.
  • Symplectic integrators preserve the underlying symplectic structure during numerical integration.This preservation automatically conserves flow volume and system energy.
  • Symplectic Euler methods provide first-order integration schemes, while the implicit midpoint rule gives a second-order symplectic scheme.
  • Common numerical methods such as primitive Euler and classical Runge–Kutta are not symplectic integrators.

3. Symplectic Projection.

Symplectic projection reduces a Hamiltonian system to a lower-dimensional system while preserving its symplectic structure and Hamiltonian form. The resulting reduced systems support energy and stability preservation under stated conditions.

  • Symplectic projection: Symplectic projection uses empirical data to construct a reduced Hamiltonian system with significantly fewer dimensions.
  • Definitions of symplectic projection: A symplectic matrix A satisfies A^T J_2n A = J_2k and spans a 2k-dimensional symplectic subspace with k ≤ n.The condition also requires rank(A) = 2k.
  • Definitions of symplectic projection: The symplectic Galerkin projection maps a 2n-dimensional Hamiltonian system to a 2k-dimensional system using a symplectic matrix and reduced Hamiltonian H~ = H ◦ A.
  • Scope: The framework considers linear transformations between symplectic vector spaces, although the original and reduced systems may be nonlinear.
  • Linear Hamiltonian systems: For linear systems, the projected operator K~ = J_2k L~ remains Hamiltonian because L~ = A^T L A is symmetric.
  • Energy preservation: The reduced system is energy preserving when the initial condition lies in Range(A), yielding ΔH(t) = 0 for all t.An extended symplectic basis can include an initial condition outside Range(A).
  • Stability preservation: Both original and reduced systems remain bounded under the theorem’s bounded-neighborhood energy condition.A strict local minimum or maximum of H in Range(A) is a stable equilibrium for both systems.
  • Comparison with Galerkin projection: Unlike ordinary Galerkin projection, symplectic projection preserves the reduced symplectic structure and supports energy and stability preservation with symplectic integration.

4. Proper Symplectic Decomposition (PSD).

PSD constructs low-dimensional symplectic subspaces from snapshot data and projects Hamiltonian systems onto them. Three algorithms trade offline cost against projection accuracy while preserving the symplectic constraints.

  • Symplectic projection: PSD projects the snapshot matrix Mx onto a low-dimensional symplectic subspace using a symplectic matrix A.The projection is Mz = A+Mx, with the reconstructed approximation given by AA+Mx.
  • Cotangent lift: The cotangent-lift algorithm computes a POD basis from an extended position-momentum snapshot matrix and forms A1 = diag(Φ, Φ).For γ = 1, this construction is optimal within the block-diagonal class M1(2n, 2k).
  • Cotangent lift: The cotangent lift uses both q(t) and p(t), whereas the tangent lift constructs its basis from q(t) alone, producing generally different reduced subspaces.The distinction changes the empirical data fitted by the basis.
  • Complex SVD: The complex-SVD algorithm represents trajectories as q(t) + ιp(t) and constructs a symplectic basis with nonzero off-diagonal blocks.Its associated truncated SVD provides the dominant singular directions of the transformed snapshot matrix.
  • Algorithmic trade-offs: The complex SVD is near optimal because it fits [Mx, −J2nMx] rather than Mx directly, while NLP can reduce projection error at higher offline cost.The cotangent lift and complex SVD require only SVD computations; NLP optimizes within a larger symplectic set using a pre-specified basis.

5. Symplectic Model Reduction of Nonlinear Hamiltonian Systems.

This section combines symplectic Galerkin projection with DEIM-style approximations to construct reduced nonlinear Hamiltonian systems while lowering online computational cost. The resulting SDEIM system preserves the symplectic reduction framework, although the DEIM approximation is not necessarily Hamiltonian.

  • Motivation: POD-Galerkin reduction can remain computationally expensive when the nonlinear term cannot be analytically simplified.Evaluating the nonlinear field in the original coordinates and projecting it back can leave the reduced solve dependent on the full dimension.
  • DEIM: DEIM approximates the nonlinear vector field using a collateral basis and selected spatial indices, with coefficients determined by interpolation.The approximation uses τ(x) = (P^T Ψ)^−1P^T fN(x) when the interpolation matrix is nonsingular.
  • DEIM: The DEIM-reduced system precomputes W offline and evaluates the nonlinear term only at m selected spatial indices online.With this construction, the online complexity can be independent of the original dimension n.
  • DEIM: SVD constructs state and collateral POD bases from empirical ensembles, while a greedy residual procedure selects interpolation indices.The procedure selects the first index from the largest basis-function magnitude and later indices from the largest residual magnitudes.
  • SDEIM: For Hamiltonian systems, symplectic projection produces a reduced Hamiltonian system, and DEIM yields the SDEIM approximation of its nonlinear term.The symplectic basis matrix A is constructed from empirical data, including cotangent-lift and complex-SVD approaches.
  • SDEIM: SDEIM has O(1) online complexity when k and m′ are fixed, but its DEIM approximation is not necessarily Hamiltonian.When the approximation is good, the authors expect the reduced dynamics not to exhibit a large energy variation.

6. Numerical Examples.

The numerical examples compare symplectic and non-symplectic reductions for linear and nonlinear wave equations. Symplectic reductions preserve energy and boundedness, while POD/DEIM can become unstable; SDEIM additionally improves nonlinear evaluation efficiency.

  • Linear wave equation: The experiments compare PSD algorithms, tangent lift, and POD using L2 errors between benchmark and reduced generalized-coordinate solutions.The PSD algorithms construct symplectic matrices from weighted data, whereas tangent lift uses configuration-space data.
  • Linear wave equation: With 20 modes, POD and cotangent lift are accurate at t = 2.5, but POD significantly deviates from the full model at t = 5.The linear-wave data ensemble contains 101 uniformly spaced snapshots.
  • Linear wave equation: POD systems with 20 or 40 modes blow up, whereas PSD errors grow slowly and geometric algorithms preserve the system energy.The 40-mode POD system blows up faster than the 20-mode system.
  • Linear wave equation: For k ∈ {10, 20, ..., 80}, the POD total error remains bounded on [0, 50] only at k = 10, while geometric-method errors decrease systematically with more modes.NLP is most accurate, but improves on cotangent lift by at most 0.028% in relative percentage error.
  • Linear wave equation: The linear-wave original system is stable despite an unstable mode because its initial-condition projection coefficient onto that mode vanishes.The resulting trajectory is stable and bounded for all t.
  • Linear wave equation: Symplectic projection yields a stable reduced origin under the stated Hamiltonian minimum condition, and energy conservation implies bounded original and reduced trajectories.The simulations observe the required basis condition for cotangent lift and complex SVD.
  • Sine-Gordon equation: For sine-Gordon, POD and DEIM become unbounded for k ≥ 80, while DEIM and SDEIM reduce POD or PSD running times by three orders of magnitude.Direct POD-Galerkin or symplectic projection does not provide speedups for the nonlinear vector term sin(u).

7. Conclusion.

The paper develops symplectic model reduction with PSD algorithms that construct low-dimensional symplectic subspaces and preserve Hamiltonian structure. Combined with SDEIM, the approach supports accurate, stable, and efficient reduction for nonlinear systems.

  • The proposed method combines symplectic projection with PSD algorithms to construct reduced Hamiltonian systems on low-dimensional symplectic subspaces.The PSD algorithms include the cotangent lift, complex SVD, and nonlinear programming.
  • Symplectic model reduction preserves the system's symplectic structure, energy, and stability.
  • PSD combined with SDEIM reduces the complexity of nonlinear vector terms and can provide significant speedups for general nonlinear problems.
  • For the sine-Gordon equation, POD and DEIM reduced systems blow up for k ≥80 over t ∈[0,150], making their error norm infinite.The reported error calculations for POD and DEIM were limited to k = 40 and k = 60 because of this blowup.
  • The method was demonstrated on linear and nonlinear wave equations with high accuracy, good efficiency, and stability preservation.
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