UQ-Platform: Modular UQ Framework

• UQ-Platform is a modular uncertainty quantification framework that integrates tailored intrusive, semi-intrusive, and non-intrusive methods for multi-physics simulations.
• It employs generalized polynomial chaos (gPC) expansions to maintain global stochastic consistency while allowing independent local module updates via projection and recovery operators.
• The framework demonstrates high numerical accuracy (error <10⁻⁴) and achieves 2.3–3.5× computational speedup on benchmarks like thermally-driven cavity flow problems.

The UQ-Platform, as introduced in "A Flexible Uncertainty Quantification Framework for General Multi-Physics Systems" (Mittal et al., 2014), defines a module-based hybrid framework for uncertainty quantification (UQ) in general nonlinear multi-physics simulations. Its central innovation is in enabling stochastic modularization: each physical module of a coupled multi-physics problem is assigned a tailored probabilistic UQ methodology—intrusive, semi-intrusive, or non-intrusive—while maintaining global stochastic consistency via generalized polynomial chaos (gPC) expansions. The framework’s capabilities are empirically demonstrated on a thermally-driven cavity flow problem, reporting strong numerical accuracy and substantial computational speedups. The approach is distinguished by its flexibility, scalability, and systematic separation between modular UQ strategies and global stochastic propagation.

1. Modular Decomposition and Framework Architecture

The UQ-Platform partitions a coupled multi-physics system—mathematically expressed as a discretized system of algebraic or differential equations—into individual physics modules. Each module maintains its own local solution variables and independent stochastic input space, with the following workflow:

• Each module iteratively solves for local variables, treating the latest (possibly stochastic) outputs from interfaced modules as inputs.
• At the inter-module level, iterative Gauss–Seidel or staggered schemes update each module’s variables at every stochastic point.

For a two-module system with solution vectors $u_1$ and $u_2$ and stochastic parameters $\xi_1, \xi_2$, the module updates take the form: $$\begin{align*} u_1^{(\ell+1)}(\xi_1, \xi_2) &= m_1(u_1^{(\ell)}(\xi_1, \xi_2), u_2^{(\ell)}(\xi_1, \xi_2), \xi_1) \ u_2^{(\ell+1)}(\xi_1, \xi_2) &= m_2(u_1^{(\ell+1)}(\xi_1, \xi_2), u_2^{(\ell)}(\xi_1, \xi_2), \xi_2) \end{align*}$$ where $m_1$ and $m_2$ denote stochastic module operators.

Each module’s stochastic representation is embedded in a local, usually lower-dimensional, gPC space. The mapping between global (full system) and local (module-specific) stochastic spaces is handled by:

• Restriction/projection maps: Translate global gPC coefficients to module-specific gPC spaces.
• Prolongation/recovery operators: Combine updated modular results into the global solution.

This modular approach enables independent development, maintenance, and upgradability of modules, supporting "plug-and-play" physics and solver methodologies.

2. Integrated Probabilistic UQ Methods

The platform supports three principal categories of UQ methodologies within each module:

• Non-intrusive methods: Black-box approaches relying on repeated model evaluations. The module’s deterministic solver is run at pre-defined sampling points (e.g., regression points for polynomial chaos or pseudospectral quadrature nodes). Coefficients of the gPC expansion are estimated via least-squares regression:

$$\min_{\hat{U}} \| U - \hat{U} \Psi \|_F, \qquad \hat{U} = U \Psi^T (\Psi \Psi^T)^{-1}$$

where $U$ is the sampled solution matrix and $\Psi$ the gPC basis functions evaluated at sample points.

• Semi-intrusive methods: Intermediate approaches that slightly modify deterministic modules to collect additional information, such as parameter sensitivities or derivatives. Augmenting regression problems with gradient data delivers statistical accuracy comparable to non-intrusive methods but at significantly reduced sample counts (typically reduced by a factor $\sim$1/$(s+1)$ in $s$ stochastic dimensions).
• Intrusive methods: Techniques (e.g., stochastic Galerkin) that reformulate the deterministic module equations directly onto the polynomial chaos basis. The intrusive approach projects all uncertain quantities into the chaos space, transforming the governing equations into a coupled deterministic system for the chaos coefficients:

$u^p(\xi) = \sum_{|j|=0}^p \hat{u}^j \psi^j(\xi)$

Each coefficient vector $\hat{u}^j$ is solved from the projected system.

Each module makes an independent selection of the most appropriate UQ strategy, allowing modules with incompatible solver or physics characteristics to coexist in a unified stochastic simulation.

3. Generalized Polynomial Chaos (gPC) as a Coupling Language

gPC expansions provide the mathematical substrate for representing and propagating uncertainty across the platform. For any square-integrable solution component $u_i(\xi_1, ..., \xi_m)$, the gPC expansion of order $p$ is: $$u_i^{p}(\xi_1, ..., \xi_m) \approx \sum_{|j|=0}^p \hat{u}_i^{(j)} \psi^{(j)}(\xi)$$ where $\psi^{(j)}(\xi)$ is the multivariate orthogonal polynomial basis, and $P+1 = \binom{p+s}{p}$ is the total number of basis functions for stochastic dimension $s$.

Key operational aspects:

• Moment extraction: The mean is $E[u_i] \approx \hat{u}_i^{(0)}$, and the covariance is $$Cov[u_i] \approx \sum_{j=1}^P \hat{u}_i^{(j)}(\hat{u}_i^{(j)})^T$$.
• Transformation operators: Projection (restriction) and recovery (prolongation) maps enable conversion between global and module-local gPC coefficient matrices:

$$\tilde{U}_{(i, module)}(\xi_{external}) = \hat{U}_i P \Pi(\xi_{external})$$

Here, $\Pi(\xi_{external})$ is a sparse matrix evaluating basis function products, and $P$ is a permutation operator.

The gPC formalism allows efficient, non-intrusive computation of statistics and observables derived from the stochastic solution.

4. Thermally-Driven Cavity Flow Application and Numerical Performance

The robustness and efficiency of the UQ-Platform are established on a benchmark thermally-driven cavity flow problem:

• Physics split: The incompressible Boussinesq system is partitioned into a momentum-pressure module (with state variables $u_1, u_2, p$) and a temperature (energy) module ($T$), each with its discretization and local stochastic dimensions.
• Uncertain parameters: The Rayleigh number (Ra) and hot-wall temperature amplitude ($h$), expanded via Karhunen–Loève and then gPC.
• Hybrid propagation: Two contrasting cases are considered: both modules are propagated by an intrusive stochastic Galerkin scheme; alternatively, momentum is solved intrusively, energy non-intrusively.

Quantitative results:

• Statistical accuracy: Probability distributions and low-order moments (velocity, temperature fields) are captured to error $<10^{-4}$.
• Computational gains: The modular UQ formulation achieves 2.3–3.5× speedup (compared to a monolithic, fully-coupled Galerkin system), primarily via dimensionality reduction in each module’s chaos system and resultant decrease in system size and cost.

5. Scalability, Flexibility, and Parallelism

The modular architecture confers several distinctive benefits:

• Dimensionality decomposition: If the overall system is $s = s_1 + s_2$ dimensions, each module only carries the gPC expansion in its local $s_i$, reducing curse-of-dimensionality effects.
• Interchangeability: Modules can be independently upgraded or swapped with minimal impact on global coupling, facilitating distributed development and code reusability.
• Mixed UQ strategy support: Heterogeneous methods (intrusive in one module, non-intrusive in another, for example) are seamlessly coupled, retaining global stochastic consistency via gPC.
• Intrinsic parallelism: The module-level stochastic propagations are lower-dimensional and independent, allowing aggressive parallelization both within and across modules.
• Adaptability: The approach accommodates linear and nonlinear physics, and—by future extension to multi-resolution expansions—situations with non-smooth (e.g., discontinuous) solution features.

6. Directions for Framework Advancement

Future work identified in the original contribution includes:

• Alternative uncertainty representations: Incorporating multi-element gPC, Haar wavelets, and other multiresolution approaches to address non-smoothness, long-time integration, and local adaptivity.
• Parallelization strategies: Optimizing inter-module communication and memory management, especially for exascale, high-fidelity problems.
• Error management: Precise interplay and control of numerical discretization errors versus stochastic propagation errors within each module.
• Community infrastructure: As a promising archetype for next-generation UQ environments, further research is indicated to enhance computational robustness, extend to additional physics, and support broad adoption in complex multiphysics applications.

Summary Table: Core Design Features

Feature	Description	Implementation Mechanism
Modular UQ	Each physics module uses suited UQ method; global iterative update	Restriction/prolongation via gPC
UQ method support	Intrusive, semi-intrusive, non-intrusive	Independent selection by module
Stochastic basis	Global and local generalized polynomial chaos (gPC) expansions	Orthonormal polynomials, projection
Coupling	Lower → higher-dimensional mapping and solution assembly	Projection/permutation matrix algebra
Parallelism	Stochastic propagation by module, sample-level parallelism	Module/task/parallel implementation

The UQ-Platform as outlined in (Mittal et al., 2014) establishes a versatile foundation for scalable, high-accuracy uncertainty propagation in nonlinear multi-physics systems. Its modular, gPC-based coupling strategy enables both sophisticated UQ analysis and practical computational efficiencies, with demonstrated performance on physically relevant benchmarks and a clear roadmap for extension to complex, high-dimensional, and multi-resolution regimes.