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arxiv: 2604.10089 · v1 · submitted 2026-04-11 · 🧮 math.OC

A Bifidelity Proximal Quasi-Newton Method for Dense Rigid Body Suspension Collision Resolution

Nicholas Rummel , Tyler Jensen , Stephen Becker , Eduardo Corona This is my paper

Pith reviewed 2026-05-10 16:36 UTC · model grok-4.3

classification 🧮 math.OC
keywords bifidelity approximationproximal quasi-Newtonlinear complementarity problemrigid body collision resolutiondense suspensionsStokesian particlesnumerical optimizationsimulation speedup
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The pith

Bifidelity proximal quasi-Newton methods solve LCPs for rigid body collisions in only three to four matrix-vector products.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Direct numerical simulation of dense rigid body suspensions requires solving a linear complementarity problem at each time step, where every matrix-vector product involves an expensive partial differential equation solve. The paper introduces a monofidelity proximal quasi-Newton method and a bifidelity variant that converge after just three or four such products. These deliver speedups of about 1.5 times and more than 2 times respectively over a competitive baseline in contact resolution. The bifidelity version shows robust convergence independent of problem size, reducing total runtime for a 216-particle simulation from eight days to five.

Core claim

The linear complementarity problem can be solved efficiently, often in only three to four matrix-vector products, by developing a custom monofidelity proximal quasi-Newton (Mono-PQN) method and a bi-fidelity variant (Bi-PQN). This approach is validated on representative systems of dense Stokesian Janus particles and achieves approximately 1.5 times and more than 2 times speedup respectively, with the latter displaying robust, problem-size-independent convergence.

What carries the argument

The bifidelity proximal quasi-Newton (Bi-PQN) method, which approximates the costly PDE solves in matrix-vector products with a low-fidelity model to accelerate LCP convergence while preserving accuracy.

Load-bearing premise

The bifidelity approximation preserves sufficient accuracy in the LCP solution for collision resolution without introducing errors that affect the overall simulation dynamics.

What would settle it

Running the same initial particle configurations with both the Bi-PQN solver and a high-accuracy reference LCP solver, then comparing resulting trajectories, velocities, and collision counts over many time steps for systematic deviations.

Figures

Figures reproduced from arXiv: 2604.10089 by Eduardo Corona, Nicholas Rummel, Stephen Becker, Tyler Jensen.

Figure 1
Figure 1. Figure 1: The true mobility matrix M = P∞Mstk ∞ P ∗ ∞ is approximated by the discretized oper￾ators PMstkP ⊤. It is prohibitively expensive to explicitly form Mstk, thus only MVPs are avail￾able. is seldom explicitly formed. Given the Stokes PDE formulation in Eqs. (1) and (2), it is useful to conceptualize it as a composition of three operators M = P∞Mstk ∞ P ∗ ∞ ≈ PMstk P ⊤. The linear operator P ∗ ∞ maps forces a… view at source ↗
Figure 2
Figure 2. Figure 2: An iteration of a forward-backward splitting algorithm: First, the unconstrained step [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: Heatmap of log10 of the maximum (over 50 realizations) absolute error between low and high fidelity LCP matrices. Right: Heatmap displaying average number of low fidelity MVPs computed for the cost of one high fidelity MVP. With these plots, values of p and ϵgmres can be identified that are promising candidates for low fidelity operators. • Choosing p: Ideally, p should be chosen to maximize a metric… view at source ↗
Figure 4
Figure 4. Figure 4: For each online experiment defined by lattice size on the x-axis, we compare box plots [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The percentage of the example LCPs generated in Section [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
read the original abstract

Direct numerical simulation of dense rigid body suspensions poses significant computational challenges. A popular approach to resolve collisions necessitates solving a linear complementary problem (LCP) per time step. Each matrix vector product (MVP) inside the LCP requires solving an expensive partial differential equation. In this work, we show the LCP can be solved efficiently, often in only three to four MVPs. Specifically, we develop a custom monofidelity proximal quasi-Newton (Mono-PQN) method and a bi-fidelity variant (Bi-PQN). Our approach is validated through an application to representative systems of dense Stokesian Janus particles. Notably, in contact resolution our Mono-PQN and Bi-PQN achieve $\approx 1.5 \times$ and $> 2 \times$ speed up respectively against a competitive baseline, with the latter method displaying robust, problem-size-independent convergence. For our largest simulation involving $216$ particles, our Bi-PQN cut total simulation runtime to five days, as compared to the eight days required by the prior state-of-the-art method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a monofidelity proximal quasi-Newton (Mono-PQN) solver and a bifidelity variant (Bi-PQN) for the linear complementarity problems that arise when resolving collisions in direct numerical simulations of dense rigid-body suspensions. Each matrix-vector product inside the LCP normally requires an expensive PDE solve; the proposed methods are reported to converge in only three to four MVPs, yielding approximately 1.5× and >2× speed-ups respectively over a competitive baseline, with Bi-PQN exhibiting robust, problem-size-independent convergence. The claims are illustrated on dense Stokesian Janus-particle systems, including a 216-particle run that reduces total wall-clock time from eight to five days.

Significance. If the bifidelity approximation preserves the accuracy of the contact forces and resolved collisions, the work would provide a practical route to substantially faster large-scale suspension simulations. The reported reduction in runtime for a 216-particle system and the size-independent convergence behavior are concrete indicators of potential impact in computational fluid dynamics and soft-matter modeling.

major comments (2)
  1. [Abstract] Abstract and numerical validation: the performance claims (3–4 MVPs, 1.5× and >2× speed-ups, five-day versus eight-day runtimes) are presented without any reported LCP residual norms, contact-force error statistics, or side-by-side particle-trajectory comparisons against a full-fidelity baseline. In dense Stokesian suspensions, O(10^{-3}) force perturbations can accumulate into qualitatively different jamming or sedimentation behavior, so the speedup assertions are load-bearing only if solution accuracy is explicitly verified.
  2. [Method description] The bifidelity surrogate construction is described as replacing selected expensive PDE-based MVPs with low-fidelity approximations, yet no analysis or numerical test is supplied showing that the resulting LCP solution remains sufficiently close (in the appropriate norm) to the monofidelity solution for the subsequent rigid-body dynamics to be statistically indistinguishable.
minor comments (2)
  1. [Abstract] The baseline solver against which the 1.5× and >2× speed-ups are measured is not named in the abstract; a brief statement of its identity and iteration count would improve clarity.
  2. [Introduction] Notation for the proximal quasi-Newton update and the fidelity-switching criterion could be introduced with a short equation block in the introduction to aid readers unfamiliar with the proximal-Newton literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The points raised highlight important aspects of validation that we will address to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract] Abstract and numerical validation: the performance claims (3–4 MVPs, 1.5× and >2× speed-ups, five-day versus eight-day runtimes) are presented without any reported LCP residual norms, contact-force error statistics, or side-by-side particle-trajectory comparisons against a full-fidelity baseline. In dense Stokesian suspensions, O(10^{-3}) force perturbations can accumulate into qualitatively different jamming or sedimentation behavior, so the speedup assertions are load-bearing only if solution accuracy is explicitly verified.

    Authors: We agree that explicit verification of solution accuracy is necessary to support the performance claims, given the potential sensitivity of dense suspension dynamics to small force perturbations. The original manuscript validates the methods through application to representative dense Stokesian Janus-particle systems, demonstrating convergence in 3–4 MVPs and the reported speedups via runtime comparisons. However, it does not include the specific quantitative error metrics noted. In the revised manuscript, we will add LCP residual norms for both Mono-PQN and Bi-PQN solvers, contact-force error statistics relative to a full-fidelity baseline, and side-by-side particle-trajectory comparisons for selected cases to confirm that the approximations preserve the essential dynamics. revision: yes

  2. Referee: [Method description] The bifidelity surrogate construction is described as replacing selected expensive PDE-based MVPs with low-fidelity approximations, yet no analysis or numerical test is supplied showing that the resulting LCP solution remains sufficiently close (in the appropriate norm) to the monofidelity solution for the subsequent rigid-body dynamics to be statistically indistinguishable.

    Authors: The bifidelity surrogate is constructed to selectively replace expensive MVPs while preserving the structure and convergence properties of the proximal quasi-Newton iteration for the LCP. The manuscript reports robust, size-independent convergence and overall performance gains on the target applications, but does not provide a dedicated error analysis or direct comparison of mono- versus bi-fidelity LCP solutions. We will add to the revised manuscript both a brief theoretical discussion of the approximation error under the assumptions of the low-fidelity model and numerical tests that quantify the difference between the resulting LCP solutions in appropriate norms, together with ensemble comparisons of the rigid-body trajectories to establish statistical indistinguishability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method extends standard proximal quasi-Newton techniques with empirical validation

full rationale

The paper describes the development of Mono-PQN and Bi-PQN as custom extensions of proximal quasi-Newton methods for efficient LCP solution in rigid body collision resolution. The core claims rest on algorithmic construction (reducing MVP count to 3-4) and numerical benchmarking against baselines on Stokesian particle systems, with reported speedups derived from wall-clock timings rather than any self-referential fitting or prediction. No equations or steps reduce by construction to inputs, no load-bearing self-citations are invoked for uniqueness or ansatzes, and the bifidelity approximation is presented as a practical surrogate without tautological equivalence to the full-fidelity result. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities. The approach builds on existing proximal quasi-Newton optimization but details of any assumptions or extensions are not stated.

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Reference graph

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