OpenLB-UQ: Non-Intrusive UQ for LBM CFD

• OpenLB-UQ is a non-intrusive uncertainty quantification framework that runs deterministic LBM simulations at multiple stochastic samples to reconstruct statistical outputs.
• The framework leverages Monte Carlo Sampling, Quasi Monte Carlo, and stochastic collocation with generalized polynomial chaos to efficiently capture uncertainty in incompressible fluid flows.
• It demonstrates practical applications in urban wind simulations by combining measured inflow uncertainties with HPC-enabled sample and domain-level parallelization.

OpenLB-UQ is a modular, non-intrusive uncertainty quantification framework built on top of the OpenLB lattice Boltzmann library for incompressible fluid flow simulations. In the general framework, uncertainty is propagated through repeated deterministic OpenLB solves using Monte Carlo Sampling (MCS), Quasi Monte Carlo (QMC), and stochastic collocation generalized polynomial chaos (SC-gPC), with hybrid sample-level and domain-level parallelization on high-performance computers (Zhong et al., 19 Aug 2025). In a specific urban-wind instantiation, the framework combines a recursive regularized lattice Boltzmann method (LBM) with stochastic collocation based on generalized polynomial chaos (gPC), a measurement-driven inflow uncertainty model, and postprocessing for means, standard deviations, vertical profiles, and confidence intervals in a real city-scale configuration in Reutlingen (Zhong et al., 25 Aug 2025).

1. Definition and conceptual position

OpenLB-UQ extends OpenLB from a deterministic computational fluid dynamics environment into an uncertainty-aware workflow. Its central design choice is non-intrusiveness: instead of modifying the deterministic solver to solve stochastic equations directly, it runs the deterministic solver at a set of stochastic samples or collocation nodes and reconstructs statistics from the resulting ensemble (Zhong et al., 19 Aug 2025). This places it in the class of forward UQ frameworks that preserve the original solver kernel while adding a sampling, collocation, quadrature, and postprocessing layer.

In the incompressible-flow formulation, the deterministic Navier--Stokes equations are promoted to a stochastic system by replacing input quantities with random variables $\bm{Z}=(Z_1,\dots,Z_{d_Z}) \in \mathbb{R}^{d_Z}$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$:

$$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$

The framework supports one-dimensional uncertainty, multi-dimensional uncertainty, and uncertainties in initial or boundary conditions or in viscosity (Zhong et al., 19 Aug 2025).

This positioning differs from Optimal Uncertainty Quantification (OUQ), where uncertainty bounds are formulated as optimization problems over admissible models and measures rather than propagated through a specified stochastic input law. In OUQ, the object is to compute sharp upper and lower bounds such as

$$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$

where $\mathcal{A}$ is an admissible information set (Owhadi et al., 2010). OpenLB-UQ, by contrast, assumes an explicit stochastic parametrization of uncertain inputs and computes statistical outputs through sampling or spectral projection. This suggests that OpenLB-UQ is primarily a propagation framework, whereas OUQ is a bound-based framework under incomplete information.

2. Numerical foundation and software architecture

The deterministic core of OpenLB-UQ is the lattice Boltzmann method implemented in OpenLB. In the general incompressible-flow paper, the discrete evolution equation is

$$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$

with BGK collision operator

$$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$

and equilibrium distribution

$$f_i^{\mathrm{eq}} = w_i\rho\left( 1+\frac{\bm{u}\cdot\bm{c}_i}{c_s^2} +\frac{(\bm{u}\cdot\bm{c}_i)^2}{2c_s^4} -\frac{\bm{u}\cdot\bm{u}}{2c_s^2} \right).$$

Macroscopic fields are recovered by

$$\rho=\sum_{i=0}^{q-1}f_i,\qquad \rho\bm{u}=\sum_{i=0}^{q-1}\bm{c}_i f_i.$$

The validation cases use a $D2Q9$ lattice (Zhong et al., 19 Aug 2025).

In the urban-wind application, the deterministic solver is a recursive regularized LBM with a D3Q19 discrete velocity set, a third-order expanded equilibrium, and a Smagorinsky--Lilly subgrid-scale model. The effective relaxation time is

$(\Omega,\mathcal{F},\mathbb{P})$0

with SGS viscosity

$(\Omega,\mathcal{F},\mathbb{P})$1

where $(\Omega,\mathcal{F},\mathbb{P})$2 and $(\Omega,\mathcal{F},\mathbb{P})$3 (Zhong et al., 25 Aug 2025).

The software architecture is explicitly layered. The UQ layer is responsible for distribution definition, sampling or collocation selection, quadrature rules, statistical postprocessing, and orchestration. The core classes include Distribution, PolynomialBasis, SamplingStrategy, QuadratureRule, and an UncertaintyQuantification manager. Supported methods include MonteCarloSampling, QuasiMonteCarloSampling, LatinHypercubeSampling, and StochasticCollocation; quadrature implementations include Gauss quadrature, Genz--Keister, Clenshaw--Curtis, tensor-product grids, and sparse Smolyak grids (Zhong et al., 19 Aug 2025). The OpenLB layer retains the regular deterministic simulation workflow, including SuperLattice, BlockLattice, geometry, and boundary-condition handling.

This decomposition reflects a general pattern in modern UQ software. UM-Bridge, for example, provides a language-agnostic client-server abstraction in which a model is exposed as a forward operator $(\Omega,\mathcal{F},\mathbb{P})$4 over HTTP and JSON (Seelinger et al., 2024). OpenLB-UQ is not built as an external UM-Bridge service in the cited papers, but the architectural resemblance is clear: both separate deterministic model execution from UQ orchestration.

3. Uncertainty quantification methodologies

OpenLB-UQ supports three non-intrusive forward UQ methods: MCS, QMC, and SC-gPC (Zhong et al., 19 Aug 2025). For a scalar quantity of interest $(\Omega,\mathcal{F},\mathbb{P})$5, MCS and QMC estimate the mean and variance by

$(\Omega,\mathcal{F},\mathbb{P})$6

$(\Omega,\mathcal{F},\mathbb{P})$7

QMC replaces random samples with low-discrepancy sequences such as Sobol or Halton and is reported to have convergence behavior roughly

$(\Omega,\mathcal{F},\mathbb{P})$8

The spectral branch uses generalized polynomial chaos. A quantity of interest is approximated as

$(\Omega,\mathcal{F},\mathbb{P})$9

with projection coefficients

$$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$0

For orthonormal bases, the mean is the zeroth coefficient and the variance is

$$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$1

In the non-intrusive stochastic collocation variant, the coefficients are approximated via quadrature:

$$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$2

Legendre polynomials are used for uniform variables and Hermite polynomials for Gaussian variables (Zhong et al., 19 Aug 2025).

The urban-wind paper uses the same SC-gPC principle but emphasizes its collocation interpretation. For an uncertain input vector $$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$3, the model output is approximated by

$$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$4

with coefficients computed by quadrature over collocation nodes and weights. The mean and variance of a quantity of interest $$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$5 are then computed directly from the quadrature ensemble:

$$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$6

$$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$7

Because the method is non-intrusive, the deterministic LBM solver is untouched (Zhong et al., 25 Aug 2025).

A central limitation is also stated explicitly: SC-gPC is subject to the curse of dimensionality and is practically limited beyond about $$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$8 unless sparse grids or dimension reduction are used (Zhong et al., 19 Aug 2025). This is a structural constraint of the method rather than a peculiarity of OpenLB-UQ.

4. Benchmarks, convergence, and performance characteristics

The initial OpenLB-UQ paper validates the framework on two canonical incompressible-flow benchmarks and one higher-dimensional stochastic test. The cylinder benchmark uses a 2D channel of size $$\partial_t \bm{u}(\bm{Z}) + (\bm{u}(\bm{Z})\cdot\nabla)\bm{u}(\bm{Z}) - \nu(\bm{Z})\nabla^2 \bm{u}(\bm{Z}) = -\frac{1}{\rho}\nabla p(\bm{Z}), \qquad \nabla\cdot\bm{u}(\bm{Z})=0.$$9 with uncertain inlet velocity

$$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$0

and inlet profile

$$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$1

The principal quantity of interest is the drag coefficient

$$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$2

For this problem, SC-gPC showed exponential or spectral convergence in polynomial order, and 5th-order SC-gPC was sufficient for near-machine-precision statistical convergence. The reported drag statistics are:

• MC: $$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$3, $$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$4
• QMC: $$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$5, $$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$6
• SC-gPC: $$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$7, $$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$8
• Deterministic reference: $$\mathcal{U}(\mathcal{A})=\sup_{(f,\mu)\in \mathcal{A}} \mu\big(f \le 0\big),$$9 (Zhong et al., 19 Aug 2025)

The Taylor--Green vortex with uncertain viscosity provides an analytically tractable validation case. The Reynolds number is modeled as

$\mathcal{A}$0

with viscosity $\mathcal{A}$1. The primary global quantity of interest is the normalized kinetic energy

$\mathcal{A}$2

The relative error in mean kinetic energy is

$\mathcal{A}$3

In this case, SC-gPC error decayed exponentially with polynomial order until deterministic spatial discretization error dominated, whereas MCS showed the expected $\mathcal{A}$4 rate for mean and standard deviation (Zhong et al., 19 Aug 2025).

The four-dimensional Taylor--Green test introduces stochastic initial perturbations

$\mathcal{A}$5

and measures convergence using the 1-Wasserstein distance

$\mathcal{A}$6

The paper reports that the numerical statistical solutions converge with increasing resolution, the Wasserstein distance decreases toward the reference statistical solution, and the observed convergence rate decreases over time toward approximately $\mathcal{A}$7, consistent with Monte Carlo-type sampling behavior (Zhong et al., 19 Aug 2025).

Performance studies on the HoreKa HPC system show that sample-level parallelism is highly effective because the samples are independent, while domain-level MPI parallelism helps for large grids. For 100 Monte Carlo samples in the cylinder case, the reported speedups are about $\mathcal{A}$8 at $\mathcal{A}$9, about $$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$0--$$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$1 at $$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$2, about $$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$3 at $$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$4, and about $$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$5 or more at $$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$6. The paper states that the total UQ wall-clock time can be reduced by a factor greater than 30 in favorable cases (Zhong et al., 19 Aug 2025).

These results place OpenLB-UQ among UQ frameworks that leverage embarrassingly parallel sampling effectively. A plausible implication is that, for forward-propagation problems with moderate stochastic dimension and expensive deterministic solves, OpenLB-UQ’s primary computational advantage comes from sample concurrency rather than from intrusive stochastic reformulation.

5. Urban wind flow simulations and uncertain data assimilation

The urban-wind paper specializes OpenLB-UQ to a real urban geometry in Reutlingen, Germany, with isolated real building geometry, two buildings in a circular domain, radius $$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$7 m, height $$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$8 m, about $$f_i(\bm{x}+\bm{c}_i\Delta t, t+\Delta t)-f_i(\bm{x},t)=\Omega_i(\bm{f}(\bm{x},t)),$$9 cells, and a simulation horizon of 48 hours of measured inflow data (Zhong et al., 25 Aug 2025). Physical and numerical parameters include $$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$0 m, $$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$1 s, $$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$2, and $$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$3.

The uncertain boundary data are derived from hourly measurements between 2024-11-07 20:00 and 2024-11-09 19:00. The inflow uncertainty model consists of a multiplicative relative perturbation for wind speed and an additive perturbation for direction:

$$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$4

$$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$5

These define a stochastic logarithmic inflow profile

$$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$6

with roughness length $$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$7 m, reference height $$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$8, and $$\Omega_i(\bm{f})=-\frac{1}{\tau}\left(f_i-f_i^{\mathrm{eq}}\right),$$9 (Zhong et al., 25 Aug 2025).

The stochastic collocation setup uses a dense tensor-product Gauss--Hermite rule with polynomial order $$f_i^{\mathrm{eq}} = w_i\rho\left( 1+\frac{\bm{u}\cdot\bm{c}_i}{c_s^2} +\frac{(\bm{u}\cdot\bm{c}_i)^2}{2c_s^4} -\frac{\bm{u}\cdot\bm{u}}{2c_s^2} \right).$$0, 11 quadrature points per stochastic dimension, and total $$f_i^{\mathrm{eq}} = w_i\rho\left( 1+\frac{\bm{u}\cdot\bm{c}_i}{c_s^2} +\frac{(\bm{u}\cdot\bm{c}_i)^2}{2c_s^4} -\frac{\bm{u}\cdot\bm{u}}{2c_s^2} \right).$$1 collocation points. The workflow is: read measured hourly wind speed and direction; define uncertain variables $$f_i^{\mathrm{eq}} = w_i\rho\left( 1+\frac{\bm{u}\cdot\bm{c}_i}{c_s^2} +\frac{(\bm{u}\cdot\bm{c}_i)^2}{2c_s^4} -\frac{\bm{u}\cdot\bm{u}}{2c_s^2} \right).$$2; construct a quadrature rule; evaluate uncertain inflow boundary conditions at each collocation node; run one deterministic LBM simulation per node; and postprocess the ensemble to compute statistics (Zhong et al., 25 Aug 2025).

The reported quantities of interest include mean velocity magnitude fields, standard deviation fields, vertical profiles at three virtual probes, 95% confidence intervals, sample realizations, and $$f_i^{\mathrm{eq}} = w_i\rho\left( 1+\frac{\bm{u}\cdot\bm{c}_i}{c_s^2} +\frac{(\bm{u}\cdot\bm{c}_i)^2}{2c_s^4} -\frac{\bm{u}\cdot\bm{u}}{2c_s^2} \right).$$3-criterion isocontours. The probes are placed at P1 at the outer corner of the left building, P2 in the channel between buildings, and P3 at the outer corner of the right building (Zhong et al., 25 Aug 2025).

The main physical finding is that uncertainty is spatially heterogeneous. Standard deviation concentrates in wakes behind buildings, shear layers along building edges, and channels between buildings where acceleration and mixing are strong. The probe results show that P1 and P3 lie in shear-layer regions, whereas P2 lies in the channel between buildings, and the confidence intervals differ markedly between these locations (Zhong et al., 25 Aug 2025). The mean field captures the dominant urban flow pattern, while the standard deviation field indicates where predictions are least certain.

The computational claims are equally explicit: the framework is non-intrusive, highly parallel, efficient for smooth quantities of interest, and practical for large-scale studies. The paper reports that 121 samples were completed in about 2.1 days on a supercomputer, with postprocessing taking only about 1 hour (Zhong et al., 25 Aug 2025). This suggests that the urban-wind version of OpenLB-UQ is aimed at operationally relevant ensemble sizes rather than brute-force Monte Carlo.

6. Relation to broader UQ infrastructure and research directions

OpenLB-UQ belongs to a broader ecosystem of modular UQ infrastructures for large simulation codes. UM-Bridge addresses the coupling problem by exposing models as network services over HTTP and JSON, with native integrations for Python, C++, Matlab, R, and Julia, and with support for UQ packages including MUQ, PyApprox, QMCPy, and UQPy (Seelinger et al., 2024). For OpenLB-UQ-style workflows, this demonstrates one route to decoupling model development from UQ algorithm development.

At HPC scale, scheduling becomes a distinct concern. A performance study of UQ task scheduling reports that combining UM-Bridge with HyperQueue reduces scheduling overhead by up to three orders of magnitude and yields a maximum reduction of 38% in CPU time for long-running simulations compared to a naive SLURM approach (Loi et al., 28 Mar 2025). A later UM-Bridge load-balancing paper reports average node idle time close to a millisecond in a multilevel Bayesian inversion setting with heterogeneous tsunami simulations (Loi et al., 24 Jun 2026). These results are not OpenLB-UQ results, but they are directly relevant to any OpenLB-UQ deployment in which many model evaluations must be dispatched dynamically on HPC systems.

Within the OpenLB-UQ papers themselves, the stated extensibility directions are clear. The general framework is designed with abstract interfaces so that more distributions, more quadrature rules, and additional UQ strategies can be added without changing existing solver code, and the paper explicitly mentions multilevel Monte Carlo as a possible future extension (Zhong et al., 19 Aug 2025). The urban-wind paper presents the framework as a practical tool for real-time urban wind analysis, with uncertainty-aware predictions for pedestrian safety, wind comfort, pollutant dispersion, and structural loading (Zhong et al., 25 Aug 2025).

A recurring misconception is that uncertainty quantification in CFD is exhausted by reporting a deterministic mean field or by adding error bars after the fact. OpenLB-UQ instead makes uncertainty propagation part of the simulation workflow itself: uncertain parameters are sampled or collocated before solver execution, and the output consists of statistical fields, confidence intervals, and ensemble-derived diagnostics (Zhong et al., 19 Aug 2025). Another potential confusion concerns terminology: the “UQ” in OpenLB-UQ denotes uncertainty quantification and is unrelated to the “UQ rings” of noncommutative ring theory (Danchev et al., 13 Jun 2026).

Taken together, the current literature presents OpenLB-UQ as a built-in, open-source, non-intrusive UQ framework for incompressible LBM-based CFD, with validated convergence on canonical problems, hybrid HPC parallelization, and a demonstrated urban-wind application driven by measured inflow uncertainty (Zhong et al., 19 Aug 2025). Its practical significance lies in turning OpenLB into a platform for statistically reliable uncertainty propagation rather than only deterministic flow simulation, while its present limitations remain those stated in the papers: SC-gPC suffers from the curse of dimensionality, the implemented methods target moderate-dimensional uncertainty, and richer multilevel or dynamically scheduled UQ workflows remain an open extension path (Zhong et al., 19 Aug 2025).