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Matrix-Free Moment-Matching Method for Reduced-Order Modeling of Quadratic-Bilinear Descriptor Systems with Multiple Inputs

Zeyuan Gao, Cheol W. Lee, Oleg Zikanov

arXiv:2610.00830v1math.NAcs.NA

TL;DR

Large-scale systems can make repeated simulations under changing inputs computationally expensive, while multi-input QB descriptor systems require reduction methods that accommodate their differential and algebraic inputs. This paper develops a matrix-free moment-matching method with separate projection spaces, and its reduced models reproduce tested flow responses closely.

  • Problem

    Repeated high-fidelity simulations can be costly, and multi-input QB descriptor systems may have inputs acting on both differential and algebraic equations.

    Source: §1 · pp. 1–2

  • Method

    The method converts descriptor systems to ODEs, constructs separate projection spaces for individual and mixed input transfer functions, and computes reduced quantities without explicitly assembling full-order matrices or tensors.

    Source: §1 · p. 2, §2.2 · pp. 5, 7

  • Results

    ROM and FOM transient responses are almost indistinguishable at six monitoring points across three heat-transfer test cases, and temperature relative L2 error falls below 0.5% near steady state.

    Source: §4.2 · p. 18

  • Takeaways & Limitations

    The matrix-free formulation avoids explicit construction and storage of large operators, making the method particularly suitable for large-scale CFD problems.

    Source: §1 · p. 2, §5 · p. 20

    The conversion neglects O(r^3) terms because they do not contribute to the first- and second-order transfer functions considered.

    Source: §2.1 · p. 4

Abstract

from arXiv · show

This paper develops a new moment-matching order reduction method for multi-input quadratic-bilinear descriptor systems. The proposed approach accommodates multiple inputs acting on both the differential and algebraic equations and constructs separate projection spaces associated with different input channels and input combinations. The method uses a matrix-free algorithm, allowing the reduced-order matrices and tensors to be computed without explicitly assembling or storing the full-order system matrices and tensors. This feature makes the proposed method particularly suitable for large-scale computational fluid dynamics problems. Numerical experiments carried out for two-dimensional flow problems demonstrate that the resulting reduced-order models accurately reproduce the transient responses of the full-order models under multiple time-varying inputs.

1 Introduction

Repeated high-fidelity simulations across input conditions can be costly, motivating reduced-order models for large quadratic-bilinear systems. This paper develops a matrix-free moment-matching method for multi-input descriptor systems, with separate projection spaces for input channels and combinations.

1 Introduction

  • Reduced-order models replace high-dimensional full-order models with lower-dimensional models that are faster to simulate while accurately representing solution behavior.Repeated simulations under different input conditions arise in design, optimization, control, and uncertainty quantification.Source: §1 · p. 1
  • The method builds separate Krylov subspaces for each input’s first- and second-order transfer functions and for mixed-input second-order terms, avoiding unnecessary coupling between transfer-function spaces.The construction addresses multi-input QB descriptor systems, including inputs acting on both differential and algebraic equations.Source: §1 · p. 2
  • The matrix-free algorithm computes reduced-order quantities from full-order operators without explicitly constructing or storing the full-order matrices and tensors.This is particularly relevant to CFD, where explicitly representing the quadratic term requires a third-order tensor whose construction and storage can be impractical.Source: §1 · p. 2
  • The paper evaluates the method on CFD tests to demonstrate its accuracy and effectiveness.The introduction identifies conversion of descriptor systems to equivalent ODE systems and ROM development as the next methodological steps.Source: §1 · pp. 2–3

2 Model Development

The model-development procedure first transforms the descriptor system into an equivalent ODE on its differential variables, then constructs reduced models using input-specific projection spaces. The resulting construction matches first- and second-order transfer functions for combinations of input channels.

2.1 Conversion of DAEs to ODE systems

  • The DAE conversion defines y1 = x1 + Γ1u and y2 = x2, with Γ1 = A12(A21A12)−1B2; this requires A21A12 to be nonsingular.For CFD, boundary values are eliminated by incorporating their contributions as equivalent source terms in adjacent cells. The O(r3) term can be neglected because it does not contribute to the first- and second-order transfer functions considered in this study.Source: §2.1 · pp. 3–4

2.2 ROM development

  • The construction matches first- and second-order transfer functions at the selected interpolation points for all combinations of input channels.The method forms reduced-order operators using the input- or input-pair-specific projection bases.Source: §2.2 · pp. 5–6
  • The proposed construction uses independent spaces for each input’s first- and second-order transfer functions and separate spaces for mixed-input second-order terms, avoiding unnecessary coupling between transfer-function spaces.Theorem 2 states that the one-sided construction preserves the interpolation conditions of the single-space Krylov construction.Source: §2.2 · pp. 5–6
  • To handle the transformed descriptor system, the method treats u and its time derivative as independent inputs, yielding 2m inputs when the original input has dimension m.The source term has no u-dot contribution, and the transformed equation has no u-dot-u-dot term, so the corresponding first- and second-order vectors vanish.Source: §2.2 · p. 7

3 Computational Implementation

The ROM construction builds Krylov subspaces and computes reduced matrices and tensors through full-order solves and operator evaluations, without assembling the full-order matrices and tensors. For finite-volume CFD, a volume-weighted inner product makes the discrete projection self-adjoint under stated boundary conditions and invertibility assumptions.

3.1 Procedure of ROM construction

  • The Krylov-subspace procedure solves for vectors at interpolation points and across individual input channels and input combinations, then forms G-orthonormal bases.The basis matrices Wp and Wpq are orthonormalized using the G-inner product.Source: §3.1 · p. 9
  • The matrix-free algorithm avoids assembling or storing full-order matrices and tensors, instead repeatedly solving the linearized system with different source terms.Such solves can use native CFD solvers and discretization schemes.Source: §3.1 · pp. 9–10
  • Algorithm 2 constructs ROM matrices and tensors by solving linear systems, evaluating discretized operators, and computing inner products.Its first two steps and operator evaluations can use native CFD discretization and solution procedures, avoiding direct calculation of full-order matrices and tensors.Source: §3.1 · p. 10

3.2 Inner product for ROM matrices calculation of CFD problems

  • The discrete operator P = G(DG)^−1D is self-adjoint in the volume-weighted inner product when DG is invertible and the stated boundary conditions hold.For these conditions, G is chosen diagonal with each entry equal to its cell volume for the inner products used in Algorithm 2.Source: §3.2 · pp. 10–11

4 Numerical Experiment

The numerical experiments test the proposed ROM on two-dimensional CFD problems by changing inputs from prescribed operating points and comparing transient responses with the FOM. The T-junction tests two inlet inputs and their cross terms; the coupled flow-and-heat example tests momentum and energy sources.

4.1 T-junction flow problem

  • The 20th-order T-junction ROM was tested against the FOM on a 37,500-cell mesh under top-inlet-only and simultaneous left- and top-inlet variations.Four interpolation points were used; a third comparison omitted cross terms by adding the separate inlet responses.Source: §4.1 · p. 13
  • Without cross terms, relative error stayed higher and steady-state predictions diverged at points c and e; the full ROM captured two-inlet coupling.For the two-input case with cross terms, the ROM and FOM also converged to nearly identical steady states at six monitoring points.Source: §4.1 · pp. 13–14, Figure 3 · p. 14, Figure 4 · p. 14
  • Approximately 1% of the maximum velocity magnitude was the maximum absolute error for simultaneous variations in both inlet velocities.The largest errors occurred near the upper wall in the flow-interaction region, where coupling and local flow gradients were strongest.Source: Figure 5 · p. 15, §4.1 · p. 15

4.2 Fluid flow problem coupled with heat transfer

  • The coupled flow-and-heat ROM was 28th order and tested against the FOM on a 100,000-cell mesh with momentum-source, energy-source, and combined variations.The additional Krylov subspace associated with the energy-source time derivative accounts for the higher order than in the previous example.Source: §4.2 · pp. 17–18
  • Temperature relative L2 error fell below 1% rapidly and below 0.5% near steady state across all three source-variation cases.Velocity relative errors remained small; the second case had a slightly higher error because temperature-induced velocity changes were smaller in magnitude.Source: Figure 9 · p. 18, §4.2 · p. 18
  • For coupled flow and heat transfer, ROM and FOM transient responses were almost indistinguishable at all six monitoring points across three source-variation cases.The cases varied the momentum source, energy source, or both simultaneously.Source: §4.2 · p. 18, Figure 10 · p. 19

5 Conclusion

The paper presents a matrix-free moment-matching method for multi-input quadratic-bilinear descriptor systems, with numerically stable compact models matching transfer functions of a traditional Krylov-based construction. Numerical experiments show good agreement between ROM and FOM transient responses under multiple time-varying inputs.

5 Conclusion

  • The matrix-free derivation avoids explicitly constructing and storing full-order matrices and tensors, which the paper identifies as intractable for large-scale CFD problems.The method is evaluated for multiple inputs, including inputs acting on algebraic equations.Source: §5 · p. 20
  • The proposed compact models are described as numerically stable and match the transfer functions of the traditional Krylov-based construction based on a single large model.Source: §5 · p. 20
  • Numerical experiments with multiple time-varying inputs show that ROM transient responses agree well with FOM responses.The paper notes that its experiments use two-dimensional examples and identifies three-dimensional turbulent flow as future work.Source: §5 · p. 20

A Transformation of a General QB Descriptor System

The transformation uses nonsingular matrices to partition the algebraic variables and operators, then rewrites the quadratic-bilinear descriptor system as DAEs.

A Transformation of a General QB Descriptor System

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