Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction

Joshua Barnett; Charbel Farhat

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Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction

Joshua Barnett, Charbel Farhat

arXiv:2204.02462v1Apr 5, 2022cs.CE

TL;DR

Traditional affine PMOR can face a Kolmogorov barrier for highly nonlinear, convection-dominated problems. The paper develops a data-driven quadratic approximation manifold and corresponding hyperreduced quadratic PROMs, which achieve comparable accuracy with substantially lower dimensions and computational costs in the Ahmed-body example.

Problem
Traditional affine PMOR faces a Kolmogorov barrier because highly nonlinear PDE solution manifolds can have slowly decaying n-widths, requiring large reduced dimensions.
Method
The paper develops a data-driven polynomial approximation, fully specializes it to a quadratic manifold, and tailors ECSW hyperreduction for the resulting quadratic PROM.
Results
The quadratic PROM achieves the same if not better accuracy as a traditional hyperreduced PROM while using an order of magnitude smaller dimension and reduced mesh; its online execution is more than 32 times faster.
Takeaways & Limitations
For the Ahmed-body turbulent wake, quadratic PMOR provides a substantially smaller and faster alternative to traditional affine PMOR at comparable accuracy.
Takeaways & Limitations
A piece-wise quadratic extension is identified as ongoing work rather than a completed part of the present method.

Abstract

from arXiv · show

A quadratic approximation manifold is presented for performing nonlinear, projection-based, model order reduction (PMOR). It constitutes a departure from the traditional affine subspace approximation that is aimed at mitigating the Kolmogorov barrier for nonlinear PMOR, particularly for convection-dominated transport problems. It builds on the data-driven approach underlying the traditional construction of projection-based reduced-order models (PROMs); is application-independent; is linearization-free; and therefore is robust for highly nonlinear problems. Most importantly, this approximation leads to quadratic PROMs that deliver the same accuracy as their traditional counterparts using however a much smaller dimension -- typically, $n_2 \sim \sqrt n_1$, where $n_2$ and $n_1$ denote the dimensions of the quadratic and traditional PROMs, respectively. The computational advantages of the proposed high-order approach to nonlinear PMOR over the traditional approach are highlighted for the detached-eddy simulation-based prediction of the Ahmed body turbulent wake flow, which is a popular CFD benchmark problem in the automotive industry. For a fixed accuracy level, these advantages include: a reduction of the total offline computational cost by a factor greater than five; a reduction of its online wall clock time by a factor greater than 32; and a reduction of the wall clock time of the underlying high-dimensional model by a factor greater than two orders of magnitude.

1. Introduction

Traditional PMOR uses data-driven affine subspace approximations, but nonlinear convection-dominated problems can encounter a Kolmogorov barrier that motivates nonlinear alternatives such as the proposed quadratic manifold.

1. Introduction: PMOR constructs reduced-order bases from solution snapshots sampled from the high-dimensional model and compressed, for example, by truncated SVD.Galerkin projection uses identical right and left bases, whereas Petrov-Galerkin projection permits a distinct test basis.
1. Introduction: Petrov-Galerkin PMOR is particularly appropriate for first-order hyperbolic problems such as convection-dominated turbulent flows, where it can improve numerical stability.LSPG represents the stated state of the art for nonlinear convection-dominated turbulent flow problems in the paper’s introduction.
1. Introduction: The Kolmogorov barrier arises because the n-width decays much more slowly for highly nonlinear PDEs than for linear PDEs, limiting accurate low-dimensional affine approximations.This issue can be exacerbated by high-dimensional parameter spaces, where snapshot sampling and the resulting reduced basis may become unaffordable.
1. Introduction: The paper proposes a data-driven polynomial approximation, fully develops the quadratic case, and adapts ECSW hyperreduction for the resulting quadratic PMOR.The approach is motivated by highly nonlinear, convection-dominated, very high-dimensional computational models.
1. Introduction: Unlike an earlier quadratic method limited to special structural-dynamics systems, the proposed development targets highly nonlinear unsteady convection-dominated problems without relying on that method’s linearization and modal-analysis setting.The earlier approach constructs its approximation a priori around equilibrium using eigenmodes and their derivatives.

2. Nonlinear projection-based model order reduction

Nonlinear PMOR projects a high-dimensional semi-discrete model onto a low-dimensional basis, while LSPG and ECSW address projection choice and nonlinear computational cost. The Ahmed-body example illustrates the large dimension gap that can arise for traditional affine reduction.

2. Nonlinear projection-based model order reduction: The nonlinear HDM is written as M u_dot(t) + f(u(t)) − g(t) = 0, with u(t) the solution, M a constant mass-like matrix, f a nonlinear flux or force, and g external or boundary forcing.The formulation also encompasses nonlinear second-order dynamical systems with arbitrarily complex dissipative forces and can represent configuration-dependent mass-like systems after transformation.
2.1. Traditional nonlinear Galerkin and Petrov-Galerkin projection-based reduced-order models: Traditional PMOR approximates the solution as u_tilde = Vq + u_ref, where V is a reduced basis, q contains generalized coordinates, and u_ref is a reference solution.Substitution yields a nonlinear residual equation that is time-discretized into reduced algebraic systems.
2.1. Traditional nonlinear Galerkin and Petrov-Galerkin projection-based reduced-order models: The reduced basis V is constructed by compressing a snapshot matrix with thin SVD, with its dimension selected using a user-defined singular-value tolerance.The retained basis vectors are the leading left singular vectors of the snapshot matrix.
2.1. Traditional nonlinear Galerkin and Petrov-Galerkin projection-based reduced-order models: Petrov-Galerkin PMOR enforces residual orthogonality to a left basis W, and the paper adopts LSPG for highly nonlinear, unsteady, convection-dominated problems.When W differs from V, the projection is Petrov-Galerkin; choosing W = V gives Galerkin projection.
2.2. Hyperreduction using ECSW: ECSW hyperreduction approximates projected quantities through a cubature rule on a reduced mesh, removing online complexity that would otherwise scale with the HDM dimension N.The reduced mesh is a subset of the mesh entities defining the underlying discretization.
2.3. Kolmogorov n-width barrier to reducibility: For the Ahmed-body flow, traditional global affine LSPG requires n ≈600 for sufficient accuracy, whereas a 100-subdomain piece-wise affine model achieves comparable accuracy with average n ≈11.The example uses an HDM with dimension greater than N = 17×10^6 and is parameterized by time only.

3. Data-driven quadratic approximation manifold

The paper develops a data-driven quadratic approximation manifold that extends traditional affine PMOR to improve accuracy and reduce PROM dimension and computational cost. The construction sets a linear basis from snapshot data, fits a quadratic coefficient matrix row-by-row with regularized least squares, and supports application-independent nonlinear PMOR.

3.5. Numerical algorithm and required computational resources: The quadratic approach has offline construction cost similar to or lower than POD, while offering substantially higher accuracy at a fixed target dimension.The computational cost includes constructing V and the quadratic row matrices, with the latter described as a small fraction of the former in favorable cases.
3.1. Higher-order polynomial approximation manifolds: The method constructs a quadratic manifold by setting G_1=V from snapshot-based POD data and fitting H in H[q⊗q]+Vq+u_ref.The quadratic term uses a vectorized Kronecker product of the generalized coordinates.
3.2. Construction of the quadratic approximation: The two-step procedure first constructs V and then determines H by minimizing snapshot approximation errors, preserving V's full-rank and orthogonality properties.Separating the steps also lowers computational complexity relative to simultaneous determination.
3. Data-driven quadratic approximation manifold: The data-driven construction is agnostic to the nonlinear HDM and can therefore be applied to PMOR for any nonlinear HDM without requiring a specific model form.The polynomial structure is predetermined, allowing H to be computed row-by-row.
3.3. Computation of the coefficient matrix of the quadratic term of the approximation: H is computed row-by-row through N independent regularized least-squares problems, enabling smaller solves and embarrassingly parallel implementation.Tikhonov regularization is introduced because the optimization problems can be susceptible to over-fitting.

4. Impact on LSPG and ECSW

The quadratic approximation changes LSPG by adding a derivative-dependent term and changes ECSW training because snapshot coordinates must be identified through a nonlinear solve. Although setup costs increase, the smaller quadratic reduced dimension can yield greater hyperreduction efficiency.

Impact on LSPG: The quadratic LSPG left reduced-order basis contains the traditional terms plus the additional term W_m+1,ℓ+1.This term arises from differentiating the quadratic approximation with respect to the generalized coordinates.
Impact on LSPG: Hyperreduction removes the dependence on N from the quadratic and traditional PROM construction costs, making both complexities reasonable.Without hyperreduction, the quadratic construction requires O(2N^3n) operations versus O(N^3n) for the traditional PROM.
Impact on ECSW: Quadratic ECSW training is more expensive because generalized coordinates for solution snapshots require solving a nonlinear problem rather than projecting onto a right reduced-order basis.The nonlinear coordinate identification can use an additional Gauss–Newton procedure, increasing setup cost for the same number of training snapshots.
Impact on ECSW: For convection-dominated nonlinear problems, quadratic PROMs can achieve similar accuracy with smaller dimension, enabling ECSW to use a smaller reduced mesh and improve computational efficiency.The Ahmed body turbulent-wake results support this expected hyperreduction advantage.

5. Application

The Ahmed body application compares traditional and quadratic hyperreduced LSPG models, showing that the HQPROM preserves accuracy with substantially lower reduced dimension and computational cost.

5. Application: The HDM simulation requires 15.1 h wall-clock time on 240 cores, providing the reference execution cost for the reduced models.This benchmark uses 10 cluster nodes and 240 total cores.
5.3. Nonlinear LSPG hyperreduced reduced-order models and accuracy results: The flow fields computed by the HDM, HPROM, and HQPROM are largely indistinguishable at the end of the simulation interval.Figure 1 compares iso-vorticity contours colored by local Mach number at t = 2 × 10^-1 s.
5.3. Nonlinear LSPG hyperreduced reduced-order models and accuracy results: The HQPROM delivers excellent accuracy across flow fields, lift and drag histories, and local velocity histories, slightly outperforming the HPROM for the considered QoIs.The comparison uses the HDM, HPROM, and HQPROM; the HPROM has n = 627, while the HQPROM has n = 39.
5.4. Performance results: The HQPROM reduces total offline computational cost by a factor of five relative to the HPROM, largely because ECSW constructs its reduced mesh much faster.The offline total includes basis and quadratic-manifold construction, ECSW reduced-mesh construction, I/O, and communication.
5.4. Performance results: The HQPROM is more than 32 times faster online than the HPROM and achieves a wall-clock speed-up factor of 131 relative to the HDM.The HPROM provides a wall-clock speed-up factor of 4, whereas CPU-time speed-ups reach two orders for HPROM and three orders for HQPROM.

6. Conclusions

The quadratic approximation manifold constructs QPROMs that match or exceed traditional hyperreduced PROM accuracy with much smaller reduced dimensions and hyperreduction meshes. Its piece-wise quadratic extension is proposed as future work for potentially further mitigating the Kolmogorov barrier.

An order of magnitude smaller dimension and hyperreduction mesh achieve the same if not better accuracy than traditional hyperreduced PROMs.These results concern the Ahmed body turbulent wake flow problem using detached-eddy simulation.
For the same training tolerance, ECSW computes a QPROM reduced mesh two orders of magnitude faster than a traditional PROM reduced mesh.
The conclusions are based on an Ahmed body turbulent wake solution rich enough to expose the Kolmogorov barrier without parameterization beyond time.
A piece-wise quadratic formulation using most-appropriate local right ROBs is identified as a possible future extension that may address the Kolmogorov barrier more effectively.Its development is ongoing and was not reported as part of the present work.