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Global Sensitivity Analysis of Spatial Sets: Finite-Element-Based Estimation and the Role of Observation Windows

Farbod Chamanian, Chiara Piazzola, Elisabeth Ullmann

arXiv:2609.12178v1math.NAcs.NAmath.STstat.TH

TL;DR

The paper asks how random inputs influence threshold-defined spatial sets in PDE models and whether their importance changes by region. It develops kernel-based and generalized Sobol’ sensitivity methods with FEM estimators; in the combustion example, rankings are similar across methods but can flip with the observation window.

  • Problem

    The paper examines how random inputs affect whether temperature stays below a critical value in selected regions of a hydrogen-combustion domain, and whether input importance changes across the domain.

    Source: §1 · p. 2

  • Method

    It analyzes threshold-exceedance sets directly with HSIC-ANOVA and their indicator fields with generalized Sobol’ indices, developing FEM-based estimators for both.

    Source: §1 · pp. 2–3, §7 · p. 21

  • Results

    In the combustion example, HSIC-ANOVA and spatially integrated Sobol’ indices give qualitatively similar rankings, while rankings can flip across observation windows.

    Source: §7 · p. 21

  • Takeaways & Limitations

    The observation window matters when ranking inputs for this combustion model, and different spatial locations may require different dimension-reduced order models.

    Source: §7 · p. 21

    The choice of input and output kernels in HSIC-ANOVA requires further mathematical analysis.

    Source: §7 · p. 21

Abstract

from arXiv · show

In this work we consider numerical models with random input parameters, where the model output is a spatially distributed random field. We carry out a statistical sensitivity analysis of spatial sets which arise for instance when the model output exceeds a critical threshold. We consider two approaches: (i) a kernel-based sensitivity analysis working with the Hilbert--Schmidt Independence Criterion, and (ii) a function-valued sensitivity analysis working with generalized Sobol' indices, where the model output is the indicator function of the spatial set of interest. We develop efficient estimators for the sensitivity indices, especially for finite-element-based numerical models, where we approximate volume integrals in the sensitivity measures by finite element quadrature, thereby avoiding Monte Carlo sampling over the spatial domain. This improves the state-of-the-art for the kernel-based indices and makes these methods practically accessible for expensive numerical models. For the generalized Sobol' indices we utilize the same finite element quadrature for efficient computation. We present numerical experiments for a hydrogen combustion process, modeled by a coupled system of convection-diffusion-reaction equations in a two-dimensional spatial domain, and ask whether the temperature remains below a critical value in selected regions of the combustion domain. Our results show that in this example the kernel-based and the generalized Sobol' indices give qualitatively similar input importance rankings. Moreover, the ranking depends on the chosen observation window and can flip between different windows. The algorithms for implementing the proposed methods are provided in the supplementary materials and in an open source code repository.

1 Introduction

The paper studies global sensitivity of threshold-defined spatial sets in PDE models and develops two derivative-free approaches with finite-element estimators. In the combustion example, input rankings can change with the observation window.

1 Introduction

  • For a hydrogen-combustion PDE model, the study asks how random inputs affect whether temperature stays below a critical value in selected regions and whether input rankings vary spatially.The example has five random inputs and four output random fields.Source: §1 · p. 2
  • The two derivative-free approaches analyze either the threshold-exceedance random set directly or its indicator function using generalized Sobol’ indices.The set-valued output lacks an easily defined derivative, while the function-valued input-output map is not differentiable.Source: §1 · p. 2
  • Sensitivity analysis for functional and set-valued PDE outputs is less developed and more computationally complex than analysis for scalar outputs.Prior work includes spatial Sobol’ maps, functional-output methods, and kernel-based sensitivity analysis for sets [20].Source: §1 · p. 2
  • Generalized Sobol’ indices for the indicator field and HSIC-ANOVA indices provide complementary approaches, with the former based on covariance-operator traces and the latter handling set-valued outputs directly.Typical pick-freeze Sobol’ estimation cost grows with the number of inputs, whereas HSIC-ANOVA estimation cost does not depend on it.Source: §1 · pp. 2–3
  • FEM-based estimators approximate spatial integrals for both sensitivity approaches, and the combustion example’s rankings can flip across observation windows.The paper also provides algorithms in supplementary materials and an open-source code base.Source: §1 · p. 3

2 Problem description

The problem formulation maps independent random inputs through a process model to a random field and then to a feasible spatial set. The set can also be represented by its indicator field for function-valued sensitivity analysis.

2 Problem description

  • Independent components of the random input vector drive the process model, whose output is a random field over the spatial domain.The input distribution is a product of the component distributions.Source: §2 · p. 4
  • An observation window selects the spatial region where the output must satisfy a prescribed constraint, defining a feasible set for each input realization.Unless otherwise stated, the full spatial domain is used as the observation window.Source: §2 · p. 4
  • The feasible set is modeled as a random set induced by the input distribution through a measurable set-valued map.Its probability law is the push-forward of the input law.Source: §2 · p. 4
  • The set-valued approach analyzes the set directly, while the function-valued approach applies generalized Sobol’ indices to its indicator field.The two indices use different maps and probability measures; the conditional measures for generalized Sobol’ indices may increase estimation cost.Source: §2 · pp. 4–5
  • The set-valued framework assumes realizations have finite, uniformly bounded perimeter, supporting compactness of the set space used in the analysis.The compactness argument uses bounded-variation indicator functions and their convergence.Source: §2 · p. 5

3 Kernel-based sensitivity indices

The kernel-based approach measures dependence between model inputs and set-valued outputs through HSIC and an ANOVA decomposition. Its normalized first-order indices support ranking individual inputs.

3.1 Mathematical preliminaries

  • Kernel mean embeddings represent probability distributions in an RKHS, and HSIC measures dependence by comparing the joint distribution with the product of marginals.With characteristic kernels, zero HSIC implies independence.Source: §3.1 · pp. 6–7
  • The kernel assumptions include boundedness and continuity, and characteristic kernels are not guaranteed by simply multiplying characteristic kernels.The latter qualification is noted for kernels on associated product spaces.Source: §3.1 · p. 6
  • The kernel framework permits outputs on measurable spaces without vector-space structure, allowing the output kernel to operate directly on spatial sets.The paper’s set kernel follows the construction for Lebesgue-measurable sets in [20].Source: §3.1 · p. 6, §3.2 · p. 7

3.2 Kernel-based Sensitivity Index

  • An ANOVA-type input kernel decomposes HSIC into contributions from subsets of inputs when the inputs are mutually independent.This structure enables normalized sensitivity indices for ranking input contributions.Source: §3.2 · pp. 7–8
  • The paper studies first-order HSIC-ANOVA indices, whose normalized values indicate how much of total HSIC is explained by an input alone.The input kernel used for estimation is chosen for low computational complexity and no hyperparameters.Source: §3.2 · p. 8

4 Spatially-integrated Sobol’ sensitivity indices

The function-valued approach applies generalized Sobol’ indices to the indicator random field of a spatial set. Trace formulas yield spatially integrated expressions that can be estimated using pick-freeze sampling.

4.1 Mathematical Preliminaries

  • Generalized Sobol’ indices are formulated for function-valued random variables using covariance operators and their trace.For an indicator field, the trace equals the expected squared spatial L2 deviation and can be expressed through integrated pointwise variance.Source: §4 · p. 8, §4.1 · p. 9
  • The indicator field’s covariance function is bounded and square-integrable on the bounded domain, but it need not be continuous.The measurability of the function-valued map must be established case by case, requiring assumptions on the constraint, observation window, and process model.Source: §4.1 · p. 9

4.2 Spatially-integrated Sobol’ Index

  • The spatially integrated Sobol’ indices express generalized Sobol’ indices in a practically computable form based on spatially integrated variance terms.They are not novel indices and are not equal to volume integrals of pointwise Sobol’ indices.Source: §4.2 · pp. 9, 11
  • A pick-freeze representation estimates the spatially integrated indices by integrating covariance terms involving the indicator field and an input-resampled field.The paper uses this expression in its numerical experiments and presents its FEM-based calculation in Section 5.Source: §4.2 · p. 11

5 Estimation of the sensitivity indices

The section develops finite-element-based estimators for HSIC-ANOVA and spatially integrated Sobol’ indices, using the finite-element mesh to approximate spatial integrals without additional Monte Carlo sampling over the domain.

5.1 Estimation of the HSIC-ANOVA sensitivity indices

  • HSIC-ANOVA uses a tensor-product Order-1 Sobolev ANOVA input kernel, chosen for low computational cost and no hyperparameters.Its evaluation is a quadratic polynomial, unlike alternatives requiring hyperparameter choices or more expensive calculations.Source: §5.1 · p. 12

5.1.1 The estimator

  • The HSIC estimator uses i.i.d. input-output pairs, zeroes the kernel-matrix diagonals, and is unbiased.The set-kernel scaling parameter σ is chosen according to the stated procedure.Source: §5.1.1 · p. 13

5.1.2 Estimator cost

  • Computing all first-order HSIC-ANOVA indices requires n model evaluations, and estimator cost is independent of the number of inputs.The double sum in the estimator does not increase the required model evaluations beyond n.Source: §5.1.2 · p. 13

5.1.3 FE approximation of the volume integral in the set kernel

  • For the set kernel, the method replaces nested Monte Carlo integration of symmetric differences with finite-element indicator approximations and mass-matrix inner products.The indicators are interpolated at mesh vertices; the resulting integral is evaluated using their coefficient vectors and the FE mass matrix.Source: §5.1.3 · p. 14

5.2 Estimation of the spatially-integrated sensitivity indices

  • The spatially integrated Sobol’ estimator combines covariance estimation with spatial integration, using a pick-freeze scheme and FE-mesh approximations for the volume integrals.The covariance terms use n Monte Carlo samples.Source: §5.2 · p. 14, §5.2.1 · pp. 14–15

5.2.2 Estimator cost

  • Computing all first-order spatially integrated Sobol’ indices requires n · (2 · nU + 1) model evaluations, with cost growing linearly in the number of inputs.One input-output pair per Monte Carlo sample estimates the indicator field; remaining pairs support the input-specific index calculations.Source: §5.2.2 · p. 15

5.2.3 FE approximation of the volume integral

  • The FEM-SpIn estimator inserts finite-element approximations of the random fields into the covariance estimator, with the mass matrix representing spatial integration.The field approximations use interpolation conditions.Source: §5.2.3 · p. 15
  • The FEM-SpIn estimator is biased and may incur rounding errors when its subtracted quantities are similar in size.For piecewise-linear continuous elements, the authors do not expect rounding-error cancellations because the coefficient vectors are binary and the quantities differ by a factor of order O(1/n).Source: §5.2.3 · p. 15

6 Numerical experiments

The experiments apply sensitivity analysis to a finite-element model of hydrogen combustion and assess temperature-threshold feasible sets across spatial observation windows. Both index types give similar input rankings within each window, while rankings for several inputs change between windows.

6.1 Process model

  • The hydrogen-combustion experiment models fuel, oxidizer, product, and temperature fields with coupled nonlinear convection-diffusion-reaction equations and five random inputs.The modeled inputs are A, E, Ti, To, and ϕ.Source: §6.1 · pp. 15–16

6.2 Discretization and numerical solution

  • The CDR model is solved with finite elements for space and Crank–Nicolson time integration, using a triangular mesh with 1,868 cells and 995 vertices.The simulation runs to 0.05 seconds in 500 time steps, with four continuous piecewise-linear fields and Newton iterations for nonlinear systems.Source: §6.2 · p. 17

6.3 Feasible set and observation windows

  • The feasible set contains observation-window points where temperature at tobs does not exceed Tcrit; restricting Xobs also restricts the spatial integration used by the indices.The demonstrated settings use tobs = 0.05 s and Tcrit = 700 K, and the observation window can be selected for the physical area of interest.Source: §6.3 · p. 17, §6.4.2 · p. 19

6.4.1 Temperature field and indicator function

  • The feasible set varies substantially with input parameters: a relatively cold flame leaves most of the domain feasible, while a hot flame leaves a much smaller region.The figures show temperature fields alongside finite-element approximations of their feasible-set indicator functions.Source: §6.4.1 · p. 18, Figure 2 · p. 18, Figure 3 · p. 18

6.4.2 Sensitivity indices for different observation windows

  • Across observation windows, both index types rank the inputs similarly, but the relative importance of inputs other than ϕ changes with the window.ϕ remains the most important input; near the left boundary, To and Ti rise in importance while E and A fall.Source: §6.4.2 · pp. 19–20, Figure 5 · p. 20
  • Using 10,000 data points reduces HSIC-ANOVA box-plot variability while preserving relative importance profiles and similar average index values.For the full domain, E and To are second most important and Ti is least important; near the left boundary, Ti and To follow ϕ.Source: §6.4.2 · p. 20, Figure 6 · p. 21

7 Conclusion

The study develops estimators for two global sensitivity measures of spatial sets and applies them to hydrogen combustion. The two measures yield qualitatively similar rankings, but input importance can change with the observation window; further work remains on kernel choices and other settings.

7 Conclusion

  • The paper develops efficient estimators for HSIC-ANOVA and spatially integrated Sobol’ indices for FEM-based models and provides open-source Python code.The datasets and code associated with the simulations are available in public repositories.Source: §7 · p. 21
  • In the hydrogen combustion example, the two indices give qualitatively similar rankings, while input importance can change or flip with the observation window.The fuel-to-oxidizer ratio ranks highest across windows; different spatial locations may require different reduced-order models.Source: §7 · p. 21
  • Further work includes analyzing HSIC-ANOVA input and output kernel choices and studying more complex, time- or solution-dependent spatial-set constraints.The paper also identifies comparisons with kernel-based indices by Barr and Rabitz and correlated-input analysis as topics for further investigation.Source: §7 · p. 21

Declaration of AI use

The authors used Gemini for source identification and proof-checking, and ChatGPT as an auxiliary software-development aid. They state that the published code contains no AI-generated code and that they take responsibility for the results and implementation.

Declaration of AI use

  • Gemini 3.5 Flash was used to identify sources on Lebesgue-measurable sets and BV functions and to proof-check.The authors assume responsibility for all theoretical and numerical results.Source: Declaration of AI use · p. 22
  • ChatGPT helped explain error messages and identify relevant software packages and package-method functionality.Illustrative snippets were used only to understand package interfaces; authors independently implemented the required functionality.Source: Declaration of AI use · p. 22
  • The authors state that the published codebase contains no AI-generated code and that they reviewed, tested, and verified its implementation.They assume responsibility for the computational implementation and reported results.Source: Declaration of AI use · p. 22

A Supplementary Material

The supplementary materials provide algorithms and routines for estimating both sensitivity-index types and generating data. They also record the combustion test-problem configurations and show the algorithms’ dependencies.

A Supplementary Material

  • The supplement contains routines for estimating spatially integrated Sobol’ indices, HSIC-ANOVA indices, and generating data.These materials are organized in Sections A.1, A.2, and A.3, respectively.Source: A Supplementary Material · p. 22
  • The specific configurations for the CDR test problem are recorded in Section A.4.Source: A Supplementary Material · p. 22
  • Figure 7 maps dependencies for data generation and estimation of the spatially integrated Sobol’ and HSIC-ANOVA indices.The Sobol’ algorithm is in the left column and HSIC-ANOVA in the right.Source: A Supplementary Material · p. 22

A.1 Spatially-integrated Sobol’ indices

The SpIn Sobol’ estimator loads FEM solutions, forms indicator functions, and computes integrated spatial covariances using the mesh mass matrix. Its inputs include sample and model settings, a test domain, and index-order selections.

A.1 Spatially-integrated Sobol’ indices

  • The estimation procedure is detailed in Algorithms 1 and 2.Source: §A.1 · p. 23
  • Algorithm 1 takes sample count, input dimension, data and mesh directories, a constraint, a test domain, index orders, and a total-Sobol’ flag.Source: §A.1 · p. 23
  • The algorithm selects mesh cells whose vertices all lie in the test domain, then loads FEM solutions for the unconditional and input-conditioned cases.Source: §A.1 · p. 23
  • It computes indicator functions from the loaded solutions and passes them with the mass matrix to the integrated-spatial-covariance routine.Source: §A.1 · p. 23
  • For each input, the algorithm also loads the corresponding alternative solutions, computes their indicators, and forms covariance lists using the same routine.Source: §A.1 · p. 23

A 35: else

Algorithm 2, integrated-spatial-cov, takes two FEM-function lists and the mass matrix as inputs.

A 35: else

  • Algorithm 2, integrated-spatial-cov, takes two FEM-function lists and the mass matrix as inputs.Source: A 35: else · p. 24

A.2 HSIC-ANOVA indices

The FEM-optimized HSIC-ANOVA estimator builds spatial and input kernel quantities from finite-element solutions and input data.

A.2 HSIC-ANOVA indices

  • Algorithm 3 estimates the HSIC sensitivity index with FEM optimization, with Algorithms 4–7 detailing its component calculations.Source: §A.2 · p. 24
  • The estimator selects singleton or full input-index sets and restricts mesh cells to those whose vertices lie in the test domain.Source: §A.2 · p. 24
  • For each FEM solution, it constructs an indicator function from the constraint, then transforms sampled inputs to Uniform([0, 1]).Source: §A.2 · pp. 24–25
  • It uses FEM coefficient vectors and the mass matrix to compute spatially integrated symmetric differences for the gamma matrix.Source: §A.2 · p. 25
  • The input-kernel calculation receives the selected input index and transformed input data; its kernel is defined for values in [0, 1].Source: §A.2 · p. 26

A.3 Data generation

The appendix describes data-generation routines for the two index families and their configurations for the hydrogen-combustion experiments.

A.3 Data generation

  • The SpIn Sobol’ routine uses a pick-freeze data-generation scheme, while the HSIC-ANOVA routine does not require that scheme.Source: §A.3 · p. 26

A.4 Specific configurations for the hydrogen combustion problem

  • The CDR pipeline solves the PDE for sampled inputs and stores temperature-field FEM solutions; SpIn data use pick-freeze structure, unlike HSIC data.Source: §A.4 · p. 27

Spatially-integrated Sobol’ indices

A.4 Specific configurations for the hydrogen combustion problem

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