Hierarchical Sampling Strategy

• Hierarchical sampling strategies are methods that construct multilevel representations of random fields using fine-to-coarse decompositions and interpolation operators.
• They employ mixed SPDE solvers and finite element methods to yield scalable, sparse linear systems ideal for large-scale uncertainty quantification and MLMC.
• This approach lowers computational costs by sidestepping dense eigenvalue problems while ensuring statistically consistent random field realizations across different resolutions.

A hierarchical sampling strategy refers to a suite of methodologies in computational mathematics and statistical modeling that leverage multi-level, structured decompositions of the sample space or the target random field to enable computationally scalable inference or uncertainty quantification. These strategies are designed to exploit problem hierarchies—spatial, structural, or probabilistic—in order to efficiently generate samples at various levels of granularity, often with the goal of facilitating large-scale simulations, uncertainty propagation, or multilevel Monte Carlo analysis.

1. Multilevel Decomposition of Random Fields

The hierarchical sampling technique begins by constructing a multilevel representation of the stochastic field. This is achieved by starting from a fine discretization (e.g., a mesh for a spatial domain) and then systematically “agglomerating” elements to build a hierarchy of coarser grids. Element-based algebraic multigrid (AMGe) techniques are employed to define a sequence of discrete spaces for both $H(\mathrm{div})$ and $L^2$ functions.

For a given spatial level $\ell$, the fine-scale random field sample $\theta_\ell(\omega)$ is decomposed as:

$$\theta_\ell(\omega) = P_\theta\, \theta_{\ell+1}(\omega) + \delta \theta_\ell(\omega),$$

where $P_\theta$ is an interpolation (or prolongation) operator from the coarse level $\ell+1$ to level $\ell$, and $\delta\theta_\ell(\omega)$ is the fine-scale correction. This recursive construction forms a hierarchy in which realizations at differing spatial resolutions are consistent, which is essential for multilevel Monte Carlo (MLMC) methods and related multiscale uncertainty quantification procedures.

2. Hierarchical Sampling Procedure via Mixed SPDE Solvers

Hierarchical sampling is implemented by solving a stochastic partial differential equation (SPDE) discretized with a mixed finite element method. This approach yields a diagonal mass matrix in the piecewise-constant space, a property that ensures scalability for large degrees of freedom.

A nested sequence of finite element spaces is built using agglomerated meshes, and corresponding interpolation operators $P_\theta$ (scalar field) and $P_u$ (vector field in $H(\mathrm{div})$) are defined between levels. The two-level block system governing the hierarchical decomposition is given by: $$\begin{bmatrix} A_\ell & A_\ell P \ P^\mathrm{T}A_\ell & 0 \end{bmatrix} \begin{bmatrix} \delta U_\ell \ U_{\ell+1} \end{bmatrix} = \begin{bmatrix} F_\ell \ 0 \end{bmatrix},$$ where $U_\ell = [u_\ell; \theta_\ell(\omega)]$ and $F_\ell = [0; -g W_\ell^{1/2} \xi_\ell(\omega)]$. The Schur complement of this block system directly yields the (proper) coarse operator, guaranteeing that coarse-level fields correspond to valid random field samples marginally consistent with the fine-level distribution.

3. Scalability and Computational Efficiency

Avoiding the computation of a dense eigenvalue decomposition, as required in the classical Karhunen–Loève (KL) expansion, is a central motivation. Instead, the hierarchical approach solves sparse linear systems associated with the discretized SPDE. With the mixed finite element formulation yielding a diagonal mass matrix and AMGe-built coarse spaces delivering sparse transfer operators, the computational cost per random field sample is nearly linear in the total number of degrees of freedom.

Scalable iterative solvers, such as preconditioned conjugate gradient algorithms with auxiliary space algebraic multigrid (e.g., hypre’s HypreADS), ensure mesh-independent solver convergence rates. Empirical weak scaling studies demonstrate that the hierarchical sampler maintains near-linear computational scalability up to massively parallel architectures.

4. Numerical Consistency and MLMC Integration

Hierarchical sampling produces statistically robust and consistent random field realizations across multiple grid levels. Two- and three-dimensional numerical experiments confirm that coarse and fine grid samples are appropriately correlated, though fine-level details are naturally “blurred” at coarser resolutions.

This consistency is critical for integrating the hierarchical sampler into MLMC simulations for problems such as subsurface porous media flow. Within MLMC, variance reduction benefits from the ability to draw consistent samples for the same underlying randomness on different spatial grids, allowing optimal allocation of computational effort (with more samples at cheaper, coarser levels).

5. Application to Forward Uncertainty Propagation

The hierarchical SPDE-based sampling is particularly impactful for problems involving forward propagation of uncertainty in spatially correlated random fields, such as the modeling of hydraulic conductivity (or permeability) as log-normal random fields in groundwater or subsurface flow. By ensuring that the sampling method provides consistent field realizations across grid levels, MLMC estimators of quantities of interest (e.g., effective permeability) can be computed with controlled variance and at a substantially reduced computational cost relative to traditional single-level techniques.

This approach is especially valuable when simulating systems at large scale, where fine-level discretizations commonly involve millions of degrees of freedom and classical methods (KL-expansion-based sampling) are computationally infeasible.

6. Methodological Comparison and Advantages

Relative to established alternatives:

Method	Complexity	Grid/Resolution Consistency	Scalability
Karhunen–Loève Expansion	Dense eigenproblem	Difficult	Infeasible for large N
SPDE Sampling (Single Grid)	Sparse linear	N/A	Scalable
Hierarchical SPDE Sampling	Sparse linear	Consistent by design	Scalable, MLMC-ready

Hierarchical SPDE sampling sidesteps the mesh dependence common in standard SPDE-based (single-grid) methods and directly addresses the resolution consistency problem, thus uniquely supporting multi-resolution Monte Carlo schemes.

7. Broader Significance and Future Implications

By decomposing random fields into coarse and fine components via multilevel finite element spaces and carefully constructed interpolation operators, hierarchical sampling strategies enable computationally tractable uncertainty quantification in high-dimensional random field models. The methodology is not only theoretically appealing—guaranteeing statistical consistency and near-linear scalability—but also immediately transferrable to a spectrum of applications demanding uncertainty analysis (e.g., environmental modeling, computational physics).

A plausible implication is that continued developments along these lines will further expand the scale, domain complexity, and realism addressable by probabilistic simulation in engineering and scientific applications, and serve as a methodological backbone for future advances in multilevel and multifidelity computational approaches (Osborn et al., 2017).