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Quasi-Monte Carlo based meshfree methods for parabolic equations on parameterized Riemannian manifolds

Zhijian He, Zhiyuan Hui, Changtao Sheng, Bihao Su, Tao Zhou

arXiv:2610.23341v1math.NAcs.NA

TL;DR

Pointwise parabolic PDE approximation on parameterized manifolds faces geometric and high-dimensional discretization challenges, while existing analysis lacks a unified p-Ret RQMC error theory. The paper combines a parameter-coordinate p-Ret discretization with MC and RQMC sampling, and establishes improved variance rates for closed, Dirichlet, and Array-RQMC settings under stated assumptions.

  • Problem

    No unified RQMC error analysis covers p-Ret Feynman–Kac discretizations, Dirichlet first-exit discontinuities, and Hilbert-ordered Array-RQMC on manifolds.

    Source: §1 · p. 2

  • Method

    The paper uses a parameter-coordinate p-Ret discretization to define a shared discrete Feynman–Kac target sampled by direct and Array-RQMC methods.

    Source: §1 · p. 3, §6 · p. 38

  • Results

    On closed manifolds, direct RQMC improves on the Monte Carlo variance rate, while Dirichlet and Array-RQMC analyses also establish variance rates better than O(n^-1) under suitable assumptions.

    Source: §1 · p. 3, §6 · p. 38

  • Takeaways & Limitations

    Array-RQMC uses stepwise randomized point sets whose dimension does not grow with the number of time steps, unlike full path-space direct RQMC.

    Source: §6 · p. 39

    Further improvements may require sharper control of numerical first-exit error, verifiable Array-RQMC stability conditions, and extensions beyond globally parameterized manifolds.

    Source: §6 · p. 39

Abstract

from arXiv · show

We develop and analyze a randomized path sampling framework for the pointwise approximation of linear parabolic equations on parameterized Riemannian manifolds, treating both closed manifolds and manifolds with Dirichlet boundary. Based on the Feynman-Kac representation, we construct a parameter-coordinate retraction-based discretization, referred to as the p-Ret scheme, for the manifold diffusion generated by the uniformly elliptic operator, and use it to formulate the Monte Carlo (MC) and randomized quasi-Monte Carlo (RQMC) methods. We derive the local weak consistency of the p-Ret transition and first-order weak convergence of the temporal discretization on closed manifolds. For fixed $h$, we show that RQMC sampling yields asymptotically faster variance decay than standard MC. On closed manifolds, the variance is $o(n^{-1})$ and, under additional regularity, satisfies the sharper bound $O(n^{-2+\varepsilon})$ for any $\varepsilon>0$. For Dirichlet problems, despite the discontinuities induced by numerical first exit, nested-uniformly scrambled Sobol' sampling attains $O(n^{-1-1/s+\varepsilon})$ variance for any $\varepsilon>0$ under suitable regularity assumptions, where $s$ denotes the nominal path-space dimension. More notably, we further develop an Array-RQMC strategy based on stepwise low-dimensional randomized point sets and a Hilbert space-filling curve that maps high-dimensional particle states to a one-dimensional ordering, and establish a variance rate of $o(n^{-1})$ under suitable assumptions. Numerical results demonstrate that both RQMC methods reduce variance and root mean square error (RMSE) relative to standard MC and remain effective for manifolds with intrinsic dimension up to $50$.

1. Introduction

The paper develops randomized path sampling for pointwise parabolic PDE solutions on parameterized manifolds, where mesh generation and growing global systems challenge conventional methods. It combines p-Ret discretization with MC and RQMC, analyzes their errors, and treats both closed and Dirichlet settings.

1. Introduction

  • Mesh generation on curved and high-dimensional manifolds is challenging, while global spatial discretizations produce rapidly growing systems as intrinsic dimension increases.These challenges motivate a meshfree probabilistic approach that approximates the solution pointwise through stochastic trajectories.Source: §1 · pp. 1–2
  • Manifold RQMC faces high path-space dimension and, with Dirichlet stopping, discontinuity interfaces; Array-RQMC uses sequential low-dimensional point sets and Hilbert ordering.The paper identifies a gap in unified p-Ret RQMC error analysis spanning these cases.Source: §1 · pp. 2–3
  • The framework combines p-Ret manifold diffusion with a common discrete Feynman–Kac functional for MC, direct RQMC, and Array-RQMC, covering closed and Dirichlet manifolds.The shared time-discrete target separates sampling strategy from temporal discretization.Source: §1 · p. 3
  • p-Ret uses r−1 independent uniform variables per step, reducing nominal direct-RQMC dimension over K steps from rK to (r−1)K.An r-dimensional Euler–Maruyama discretization uses r random inputs per step.Source: §1 · p. 3
  • On closed manifolds, direct RQMC has variance o(n^-1), with a sharper O(n^-2+ε) bound under additional regularity.For Dirichlet problems, first-exit discontinuities are handled by layerwise decomposition and Owen’s boundary growth condition.Source: §1 · pp. 3–4
  • Hilbert-ordered Array-RQMC achieves variance o(n^-1) for fixed K under suitable stability and regularity assumptions.It uses stepwise low-dimensional randomized point sets.Source: §1 · p. 4

2. The p-Ret discretization with MC and direct RQMC sampling

This section sets up a parameter-coordinate p-Ret path discretization and its discrete Feynman–Kac estimators for closed manifolds and Dirichlet problems. MC and direct RQMC use the same discrete target, with different random-input constructions.

2.1. Geometric setting and Feynman–Kac representations.

  • The geometric setting assumes a smooth compact connected manifold with a global regular parameterization, and defines closed and Dirichlet parabolic problems using a uniformly elliptic diffusion operator.The Feynman–Kac representation expresses the closed-problem solution through manifold diffusion paths.Source: §2 · p. 4, §2.1 · pp. 4–5

2.2. The p-Ret discretization in parameter coordinates.

  • The p-Ret update uses a parameter-coordinate second-order retraction with a Christoffel-symbol quadratic correction to approximate the manifold diffusion.The driving-noise dimension r need not equal the manifold dimension m.Source: §2.2 · pp. 5–6
  • The one-step p-Ret transition is locally weakly consistent with generator L, with the cited expansion controlling its first and second moments.The covariance term satisfies E[∆i∆j] = h aij + O(h2), while the third- and fourth-order moments are O(h2).Source: §2.2 · pp. 6–7

2. The p-Ret discretization with MC and direct RQMC sampling

  • The p-Ret discretization supplies the shared manifold-diffusion dynamics underlying the section’s discrete Feynman–Kac functionals and sampling schemes.The section introduces the construction and estimators for linear parabolic equations on parameterized manifolds.Source: §2 · p. 4, §2.3 · p. 7

2.3.1. Discrete Feynman–Kac functionals.

  • The discrete Feynman–Kac functionals use p-Ret paths for closed problems and stopped paths with numerically localized first exit for Dirichlet problems.The boundary functional includes the source contribution over the partial exit step.Source: §2.3.1 · pp. 7–8

2.3.2. Monte Carlo and direct RQMC sampling.

  • Each p-Ret step maps r−1 uniform inputs through a recursive inverse-beta transformation to a uniform direction on the sphere; direct RQMC uses a scrambled Sobol’ point set in dimension s=(r−1)K.A full point generates one path, and unused input blocks are discarded after numerical boundary exit; the nominal dimension grows as h decreases.Source: §2.3.2 · p. 8

2.3.3. Estimators and implementation.

  • MC and direct RQMC estimators use the same discrete functional, differing only in how their uniform inputs are generated.For direct RQMC, each particle retains the blocks of its original (r−1)K-dimensional point during simulation.Source: §2.3.3 · p. 9

3. Error estimates for temporal discretization and path-space sampling

This section analyzes temporal discretization and path-space sampling errors for pointwise approximation of parabolic problems on closed manifolds and manifolds with Dirichlet boundary. It establishes first-order temporal weak accuracy on closed manifolds, develops MC and direct RQMC sampling bounds, and accounts for boundary localization and the regularity analysis needed for RQMC in Dirichlet problems.

3.1. Error decomposition and preliminary estimates.

  • Closed-manifold temporal weak error is first order; for Dirichlet problems, numerical boundary error E∂ adds to the interior error, which is first order if E∂=O(h).The boundary contribution includes errors in exit-time and exit-location approximation and boundary-data evaluation.Source: §3.1 · pp. 10–12
  • MC sampling has variance O(n^-1) and sampling RMSE O(n^-1/2), providing the reference rates for the RQMC analysis.These sampling estimates are relative to the time-discrete solution and apply to both closed and Dirichlet discrete functionals.Source: §3.1 · p. 12

3.2. Randomized quasi–Monte Carlo error estimates.

  • For fixed h, nested-uniformly scrambled Sobol’ sampling gives an asymptotic variance improvement over MC, although the basic bound specifies no fixed algebraic rate.The result assumes the first n=2^ℓ points in dimension s.Source: §3.2 · p. 13
  • The Gaussian change of variables addresses potentially singular derivatives near the unit-cube boundary; the sphere transformation then has bounded first derivatives and mixed derivatives.The bounded-derivative argument uses the Gaussian and trigonometric factors in the recursive sphere transformation.Source: §3.2 · pp. 13–15
  • Under Assumption 3.3, the path regularity analysis establishes the stated polynomial-growth regularity for the Gaussian-transformed integrand.The assumption fixes h and requires identification-compatible Cs extensions for the specified coefficients and data.Source: §3.2 · p. 15
  • For the closed-manifold estimator, mixed derivatives of the path integrand are continuous and polynomially growing, placing it in Gp and supporting the variance estimate.The derivative regularity follows from the assumptions and the path recursion; the estimator is also unbiased for the closed-manifold solution.Source: §3.2 · pp. 17–18
  • For closed manifolds, nested-uniformly scrambled Sobol’ sampling gives an explicit variance bound for the direct RQMC estimator for sufficiently large n under Assumptions 3.2–3.3.The estimate applies when the point set has n = 2^ℓ points and the closed-manifold integrand satisfies the stated Gp regularity.Source: §3.2 · p. 17
  • For Dirichlet problems, first numerical exit makes the fixed-K RQMC integrand generally discontinuous, so the stopped functional is decomposed by first-exit layer.Each layer corresponds to exit during a particular p-Ret step, with a final survival contribution for paths reaching T without exit.Source: §3.2 · pp. 18–19
  • Under Assumption 3.4, the first-exit and survival-set boundaries have finite upper codimension-one Minkowski content.The condition imposes regularity and finite graph-cover structure on the switching interfaces, controlling the geometry of discontinuities.Source: §3.2 · p. 20
  • Under Assumption 3.5, smooth extensions of the Dirichlet layer weights belong to L2 and satisfy Owen’s boundary growth condition.The assumption includes polynomial growth of derivatives of an extension of the coordinate boundary data and of the Gaussian-transformed boundary fractions.Source: §3.2 · p. 22
  • For every exit layer, the argument uses an upper tube estimate implied by finite upper (sk−1)-dimensional Minkowski content; since sk ≤ s, the dimension-s bound gives the slowest rate.Source: §3.2 · p. 24
  • For a manifold with Dirichlet boundary, under Assumptions 3.2, 3.4, and 3.5 and temporal weak error (3.7), the scrambled Sobol’ point-set result holds for any ε > 0.The constants C and C_RQMC,∂,ε are positive; C is independent of h, and C_RQMC,∂,ε is independent of n.Source: §3.2 · p. 24
  • For a closed manifold, under the assumptions of Lemmas 3.1 and 3.6 and the stated κh condition, the bound holds for any ε > 0, with constants independent of h and n, respectively.Source: §3.2 · p. 24

4. Array-RQMC sampling and error estimates

Array-RQMC replaces a single high-dimensional path-space randomization with sequential low-dimensional randomized point sets and Hilbert rank matching. Under stated regularity and stability assumptions, the analysis establishes unbiased time-discrete estimators and combines sampling-error bounds with temporal discretization bounds for closed and Dirichlet problems.

4.1. Stepwise Array-RQMC construction and Hilbert rank matching.

  • At each layer, Array-RQMC uses an independently randomized low-dimensional point set and reorders particles by scalar Hilbert keys before matching points by rank.Inactive Dirichlet records are placed after active records, which are the only ones propagated; the array and point set retain size n.Source: §4 · p. 25, §4.1 · pp. 25–26

4.2. Error estimates.

  • The layerwise error analysis controls rank-coupled integrands through Hilbert-order stability and continuity assumptions, then applies scrambled-net variance bounds.For Dirichlet problems, first-exit discontinuities require a stopped-integrand stability assumption; transversality and finite-complexity control of exit interfaces can help verify it.Source: §4.2 · pp. 27, 29, 30
  • For time-discrete closed-manifold and Dirichlet estimators, the proof gives unbiasedness and o(n^-1) variance; the Dirichlet result assumes Assumptions 3.2 and 4.2, and temporal weak error (3.7).For the closed case, the o(n^-1) term may depend on fixed h; C is positive and independent of h.Source: §4.2 · p. 31

5. Numerical experiments

The experiments test temporal convergence, sampling variance, and spatial RMSE for MC, direct RQMC, and Array-RQMC on closed and bounded manifolds. Across the examples, p-Ret shows approximately first-order temporal convergence, while RQMC methods generally improve sampling performance over MC, with benefits that weaken as intrinsic dimension grows.

5. Numerical experiments.

  • The experiments assess temporal error, sampling variance at fixed initial points, and RMSE over prescribed test sets.Unless otherwise stated, T = 0.20; sampling tests use K = 64 time steps and 50 randomized replications, while spatial errors use 100 fixed test points.Source: §5 · pp. 31–32
  • For products of Lawson Klein bottles, p-Ret temporal errors show clear first-order decay at intrinsic dimensions m = 2, 8, 20, and 50, with fitted orders close to O(h).Source: §5.3 · p. 36

5.1. A general diffusion operator on a closed star-shaped torus.

  • On the star-shaped torus, both RQMC estimators improve variance decay and spatial RMSE over MC, with Array-RQMC performing best.The fitted variance rates are O(n−1.07), O(n−1.27), and O(n−1.77) for MC, direct RQMC, and Array-RQMC; the corresponding RMSE rates are O(n−0.5), O(n−0.59), and O(n−0.79). Temporal error shows first-order decay in h.Source: §5.1 · pp. 32–33, Figure 3 · p. 33, Figure 2 · p. 33

5.2. A general diffusion operator on a wavy cylinder with boundary.

  • On the wavy cylinder with boundary, Array-RQMC has the smallest tested-range RMSE and the fastest fitted variance decay among the three estimators.Its fitted RMSE rate is approximately O(n−0.63), and its variance rate is approximately O(n−1.59), compared with O(n−1.01) for MC and O(n−1.13) for direct RQMC; temporal decay is approximately first order, with additional numerical boundary-exit error.Source: §5.2 · pp. 34–35, Figure 4 · p. 34, Figure 5 · p. 35

5.3. A Laplace–Beltrami problem on high-dimensional products of Lawson Klein bottles.

  • On Lawson-product manifolds, Array-RQMC generally gives the strongest variance and RMSE reduction, though both RQMC methods weaken as intrinsic dimension increases.For m = 2, Array-RQMC variance has fitted rate O(n−1.71); temporal convergence remains close to O(h) through intrinsic dimension 50, while fixed temporal error can contribute to RMSE flattening at large n.Source: §5.3 · pp. 36–38, Figure 7 · p. 37

6. Conclusion

The framework separates manifold-diffusion approximation from sampling, supporting a meshfree, parallel approach. Its analysis distinguishes closed and Dirichlet manifolds, while experiments show RQMC benefits that diminish as intrinsic dimension grows.

6. Conclusion

  • Local p-Ret dynamics propagate the manifold diffusion, while MC, direct RQMC, and Array-RQMC sample the same discrete Feynman–Kac functional.Source: §6 · p. 38
  • On closed manifolds, direct RQMC improves on the Monte Carlo variance rate and approaches quadratic decay under stronger regularity; Dirichlet first exits create path-space discontinuities.Source: §6 · p. 38
  • Array-RQMC replaces full path-space randomization with sequential low-dimensional randomizations and retains variance o(n^-1) for fixed K under stated stability assumptions.Source: §6 · p. 38
  • Both RQMC approaches reduce variance and RMSE in low and moderate intrinsic dimensions, but their advantage decreases as dimension grows.Source: §6 · p. 38
  • Further work may sharpen numerical first-exit error control, establish verifiable Array-RQMC stability conditions, and extend the construction beyond globally parameterized manifolds.Source: §6 · p. 39
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