Year 1:

1. Higher-Order Moment Analysis of Spline Chaos Expansion:

Year 2:

1. A Generalized Polynomial Chaos Expansion (GPCE) for Global Sensitivity Analysis with Dependent Random Variables

2. Time-dependent Uncertainty Quantification Analysis of Complex Dynamical Systems:

Year 3: 

1. A Generalized Polynomial Dimensional Decomposition for Uncertainty Quantification with Dependent Random Input
2. Uncertainty Quantification of Dynamical Systems with Dependent Random Variables

Year 1: 

Higher-Order Moment Analysis of Spline Chaos Expansion:  A UQ analysis entailing high-order moments calculated from SCE approximations of a real-valued, general output function of input random variable was conducted. The approximation quality of SCE was assessed in terms of the modulus of smoothness of the function. When the largest element of the mesh from SCE approaches zero, the modulus of smoothness vanishes, resulting in the higher-order moment convergence of SCE to the correct limit. Therefore, the moment of SCE of an arbitrary order converges to the exact moment for any degree of splines as the element size decreases. Moreover, the weaker modes of convergence, such as those in probability and in distribution, transpire naturally.

Numerical computations of moments by SCE and polynomial chaos expansion (PCE), conducted for a collection of simple yet relevant examples, indicate the following:

1. When the output function is smooth or nonsmooth and the random domain is bounded or unbounded, SCE and PCE both provide convergent estimates of the variance. However, their relative convergent rates may differ, depending on the smoothness of the function.
2. When the function is globally smooth, PCE is likely to provide more efficient estimates of variance than SCE for the same computational effort. However, for a nonsmooth function, the trend reverses, and the convergence properties of PCE in estimating variance may degrade appreciably.
3. Higher-order moments, such as skewness and kurtosis, calculated using SCE converge for all examples considered in this study. In contrast, moments of PCE of orders larger than two may or may not converge, depending on the regularity of the output function or the probability measure of input random variables.

Year 2: 

A Generalized Polynomial Chaos Expansion (GPCE) for Global Sensitivity Analysis with Dependent Random Variables:  A GPCE method was developed for variance-based global sensitivity analysis of uncertain systems with statistically dependent input random variables following a general, non-product-type probability distribution. The method is built upon three key components:(1) a general three-step process for constructing multivariate orthonormal polynomials consistent with the probability measure of input variables; (2) a Fourier series expansion of a square-integrable output function with respect to these multivariate polynomials; and (3) novel analytical formulae for global sensitivity indices in terms of the expansion coefficients and expectations of the products between relevant orthonormal polynomials.  For statistically independent input variables, the proposed GPCE simplifies to the classical PCE, recovering established expressions of sensitivity analysis. By tackling dependent random variables head-on, GPCE circumvents the need for a measure transformation between dependent and independent variables, thereby retaining the original output function and hence calculating the sensitivity indices accurately and efficiently. Once the GPCE approximation is established — typically as part of uncertainty quantification analysis — the sensitivity indices can be generated with minimal extra effort.  Therefore, both uncertainty quantification and sensitivity analyses can be performed simultaneously. 

Results from four illustrative examples involving elementary mathematical functions show that GPCE yields exact or rapidly converging sensitivity indices for both Gaussian and non-Gaussian input distributions. Furthermore, application to an industrial-scale problem demonstrates GPCE’s practical utility, integrating seamlessly with legacy simulation code to identify influential and non-influential variables during random eigenvalue analysis of a complex mechanical system.

Time-dependent Uncertainty Quantification Analysis of Complex Dynamical Systems:  A novel computational methodology, supported by robust numerical algorithms, was developed for time-dependent uncertainty quantification (UQ) analysis on complex dynamical systems. The proposed approach is based on (1) a new stochastic adaptation of the nonlinear autoregressive with exogenous input (NARX) model, utilizing dimension-wise tensor product expansion to effectively capture the behavior of dynamical systems, (2) a polynomial dimensional decomposition (PDD) technique to propagate uncertainty in input random variables to the NARX coefficients, and (3) a unique integration between NARX and PDD, resulting in the PDD-NARX approximation for time-dependent UQ analysis. The PDD-NARX method distinguishes itself from conventional deterministic system identification tools by considering uncertainties originating from both the system’s dynamic properties (e.g., mass, stiffness, and damping) and external forces (e.g., amplitude and frequency content of excitation time series). Unlike traditional methods, which rely on an intuitive selection of NARX basis functions, this approach employs dimensional decomposition and importance factors to systematically construct the NARX model function. Furthermore, PDD, due to its hierarchical, dimension-wise expansion, is better equipped to manage high-dimensional UQ problems than many existing methods, including the widely recognized polynomial chaos expansion. 

Numerical results demonstrate that low-order PDD-NARX approximations provide accurate and computationally efficient estimates of the probabilistic characteristics of simple dynamical systems. Moreover, the probabilistic vehicle dynamic analysis of a pick-up truck traversing road bumps underscores the effectiveness of the PDD-NARX method in addressing industrial-scale complex problems.

Year 3:

A Generalized Polynomial Dimensional Decomposition for Uncertainty Quantification with Dependent Random Input: A novel and practical adaptation of generalized polynomial dimensional decomposition (GPDD) was developed for uncertainty quantification (UQ) in the presence of dependent input random variables with arbitrary, non-product-type probability distributions. Instead of relying on a Rodrigues-type formula, which exists only for select probability measures, a new four-step computational algorithm is introduced to generate approximate, measure-consistent multivariate orthogonal polynomials in subsets of input variables. Unlike generalized polynomial chaos expansion (GPCE), which requires the full joint input distribution to construct its orthogonal polynomial basis, GPDD operates using only low-variate marginal distributions, allowing efficient dimensionwise construction even when the joint distribution is unknown. For high-dimensional stochastic problems characterized by strongly nonlinear output but weak input interactions, GPDD is expected to deliver substantial computational advantages over GPCE due to its hierarchical, dimensionwise structure.

Numerical experiments involving both Gaussian and non-Gaussian probability measures on rectangular and non-rectangular domains show that the proposed algorithm produces highly accurate orthogonal polynomials. Results from representative mathematical and structural examples demonstrate that GPDD provides accurate and computationally efficient estimates of statistical moments and reliability, while an engineering application involving stochastic stress analysis of a vehicle suspension control arm with 34 random variables further highlights GPDD’s practical effectiveness for high-dimensional UQ problems.

Uncertainty Quantification of Dynamical Systems with Dependent Random Variables: A new computational framework for time-dependent uncertainty quantification (UQ) of dynamical systems subject to random input following statistically dependent arbitrary probability distributions. The framework is founded on (1) a stochastic adaptation of the Nonlinear Auto-Regressive with eXogenous input (NARX) model, coupled with a three-step basis-selection algorithm, for describing the underlying system dynamics; (2) a generalized polynomial chaos expansion (GPCE) employing measure-consistent multivariate orthonormal polynomials in random input for propagating uncertainty to the NARX coefficients. The resulting integrated GPCE–NARX representation enables accurate and computationally efficient UQ of time-dependent stochastic systems. By isolating temporal dynamics through NARX and managing time-invariant stochastic variability via GPCE, the proposed framework effectively mitigates the long-time integration challenges of existing UQ methods.

The accuracy and computational efficiency of the GPCE–NARX framework have been demonstrated through three numerical examples. For a single-degree-of-freedom benchmark problem, error analysis of the statistical moments indicates that the GPCE–NARX surrogate provides significantly greater approximation accuracy than time-frozen GPCE-based approaches. Furthermore, applications to a nonlinear quarter-car model and an eight-story building subjected to base excitation demonstrate that GPCE–NARX accurately captures the temporal evolution of response statistics, including means and standard deviations, and probability density functions at all time instances. In all cases, the GPCE–NARX results exhibit excellent agreement with crude Monte Carlo solutions at a fraction of the associated computational cost, while the time-frozen GPCE shows noticeable discrepancies at longer time horizons.

Supervision of Graduate Students: Two PhD students, Mr. Md. Talukdar and M. Ebadollahi, are appointed as half-time graduate research assistants. They are both currently supported by this NSF-funded project. The PI serves as the advisor of these two students.

Insertion of Research Results into Undergraduate and Graduate Courses: In 2022, the PI developed a new intermediate-level course, Uncertainty Quantification and Design Optimization, to introduce undergraduate and graduate students to contemporary methods of probabilistic analysis and design optimization under uncertainty. For its third offering in Fall 2026 (August–December), the PI revised the lecture notes to incorporate novel methods of uncertainty quantification and robust/reliability-based design optimization developed through this and previous NSF projects. In Spring 2026 (January–May), the PI also overhauled another intermediate-level course, Computer-Aided Engineering, by incorporating deterministic shape and topology optimization methods for industrial-scale complex systems.

International Collaboration: The PI initiated an international collaboration with Professor J. G. Kim’s group at Kyung Hee University, South Korea. In Fall 2025 (November-December), Mr. S. H. Han, a Ph.D. student from Kyung Hee University, visited The University of Iowa using his own funding. During the visit, The PI advised him on neural network and uncertainty quantification of complex dynamical systems.

Invited Seminars and Lectures: The PI’s research group received invitations to present three lectures on uncertainty quantification and design optimization at the following prominent conferences:

1. Engineering Mechanics Institute Conference (EMI2026), Boulder, CO, from June 2-5, 2026.
2. 9th Computational Stochastic Mechanics Conference (CSM9) in Corfu, Greece, from June 21-25, 2026.

Journal Articles

1. Rahman, S., “Probability-Informed Initial Bases for Constructing Measure-Consistent Multivariate Orthonormal Polynomials with Dependent Inputs,” submitted to Probabilistic Engineering Mechanics, 2026.
2. Ebadollahi, M. and Rahman, S., “A GPCE–NARX Framework for Uncertainty Quantification in Dynamical Systems with Dependent Random Variables,” submitted to Mechanical Systems and Signal Processing, 2026.
3. Han, S.-H., Ebadollahi, M., Han, S., Kim, J.-G., and Rahman, S., “Time-Dependent Uncertainty Quantification of Dynamical Systems using a Spectral Neural Network Framework,” submitted to Probabilistic Engineering Mechanics, 2026.
4. Talukdar, M. and Rahman, S., “Toward Uncertainty Quantification with Arbitrarily Dependent Probability Distributions of Random Input,” Computer Methods in Applied Mechanics and Engineering, Vol. 454, Article 118845, pp. 1-31, 2026.
5. Rahman, S., “A Generalized Polynomial Chaos Expansion for Global Sensitivity Analysis with Dependent Random Variables”, submitted to SIAM Journal on Uncertainty Quantification, 2025.
6. Ebadollahi, M. and Rahman, S., “Time-dependent Uncertainty Quantification Analysis of Complex Dynamical Systems,” Probabilistic Engineering Mechanics, Vol. 80, Article 103776, pp. 1-19, 2025.
7. Rahman, S., “Higher-Order Moments of Spline Chaos Expansion,” Probabilistic Engineering Mechanics, Vol. 77, Article No. 103666, 2024.
8. Lee, D. and Rahman, S., “High-Dimensional Stochastic Design Optimization under Dependent Random Variables by a Dimensionally Decomposed Generalized Polynomial Chaos Expansion,” International Journal for Uncertainty Quantification, Volume 13, No. 4, pp. 23-59, 2023.

Papers in Conferences, Symposiums, and Congresses

1. Rahman, S., “Higher-Order Moments of Spline Chaos Expansion,” Proceedings of the 9th Computational Stochastic Mechanics Conference, Corfu, Greece, June 2026.
2. Talukdar, M. R. and Rahman, S., Robust Design Optimization under Dependent Random Variables by a Generalized Polynomial Dimensional Decomposition,” Proceedings of the 2026 Engineering Mechanics Institute Conference, Boulder, Colorado, June 2026.
3. Ebadollahi, M. and Rahman,S., “A GPCE-NARX Framework for Uncertainty Quantification Analysis of Dynamical Systems in the Presence of Dependent Random Variables,” Proceedings of the 2026 Engineering Mechanics Institute Conference, Boulder, Colorado, June 2026.
4. Talukdar, M. R. and Rahman, S., “Practical Uncertainty Quantification Analysis in the Presence of Statistically Dependent Random Variables,” Proceedings of the 14th International Conference on Structural Safety and Reliability (ICOSSAR25), Los Angeles, CA, June 1-6, 2025.
5. Ebadollahi, M. and Rahman, S., “Time-dependent Uncertainty Quantification Analysis of Complex Dynamical Systems,” Proceedings of the 14th International Conference on Structural Safety and Reliability, Los Angeles, CA, June 1-6, 2025.
6. Ebadollahi, M. and Rahman, S., “A PDD-NARX Approximation for Uncertainty Quantification in Complex Dynamical Systems,” Proceedings of the 2024 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Washington, DC, August 25-28, 2024.
7. Rahman, S., “A Spline Chaos Expansion for Uncertainty Quantification in Linear Dynamical Systems,” Proceedings of 2024 Engineering Mechanics Institute Conference and Probabilistic Mechanics & Reliability Conference (EMI/PMC 2024), Chicago, IL, May 28-31, 2024.
8. Lee, D. and Rahman, S., “A Generalized Polynomial Chaos Expansion for High-Dimensional Design Optimization under Dependent Random Variables,” Proceedings of 2024 Engineering Mechanics Institute Conference and Probabilistic Mechanics & Reliability Conference (EMI/PMC 2024), Chicago, IL, May 28-31, 2024.
9. Lee, D. and Rahman, S., “A Generalized Polynomial Chaos Expansion for High-Dimensional Design Optimization under Dependent Random Variables,” Proceedings of 2024 Engineering Mechanics Institute Conference and Probabilistic Mechanics & Reliability Conference (EMI/PMC 2024), Chicago, IL, May 28-31, 2024.

Others

Several computer codes have been developed to perform the various numerical simulations required for this project. These codes were created exclusively for research purposes. For further information, please contact the PI (S. Rahman) at sharif-rahman@uiowa.edu.