**Sponsor:** U.S. National Science Foundation

**Project No.:** CMMI-1462385

**Duration:** August 15, 2015 – July 31, 2020

**Principal Investigator:** Professor Sharif Rahman

**Graduate Student:** Mr. Dongjin Lee

**SUMMARY**

The objectives of this proposal are to build a solid mathematical foundation, devise efficient numerical algorithms, and develop practical computational tools for stochastic design optimization of large-scale complex systems subject to random input following arbitrary dependent probability distributions. The proposed effort will involve: (1) a new theoretical development of the generalized analysis-of-variance (ANOVA) dimensional decomposition (ADD) for dependent random variables, leading to the generalized polynomial dimensional decomposition (PDD) of a high-dimensional stochastic response; (2) new formulae and scalable algorithms associated with the generalized PDD method for calculating the statistical moments and reliability, followed by design sensitivity analysis; and (3) new reliability-based and robust optimization algorithms for shape and topology designs from a single or at most a few stochastic simulations. Due to innovative calculation of the expansion coefficients, the generalized decomposition method will be efficiently implemented regardless of the size of the stochastic design problem. The innovative formulation of the statistical moment and reliability analyses and design sensitivities, which requires a single or at most a few stochastic simulations for all possible designs, will markedly accelerate the optimization process, potentially producing breakthrough solutions to stochastic design problems.

**Task 1:**

**Generalized ADD:**A generalized ADD for dependent random variables, representing a finite sum of lower-dimensional component functions of a multivariate function, was studied. The classical annihilating conditions, when appropriately weakened, reveal two important properties of the generalized ADD: the component functions have zero means and are hierarchically orthogonal. A simple, alternative approach was developed for deriving the coupled system of equations satisfied by the component functions. The coupled equations, which subsume as a special case the classical ADD, reproduces the component functions for independent probability measures. New generalized formulae were developed for the second-moment characteristics of a general stochastic function, including three distinct global sensitivity indices, relevant to dependent probability distributions. Gaining insights from the generalized ADD, two generalized effective dimensions, one in the superposition sense and the other in the truncation sense, were established. Numerical results from a simple yet insightful mathematical example indicate that the statistical dependence among random variables induces higher-order terms in the generalized ADD that may not be present in the original function or in the classical ADD. In addition, the component functions depend significantly on the correlation coefficients of random variables. Consequently, the global sensitivity indices may also vary widely, producing distinct rankings of random variables.

**Task 2:**

**Wiener-Hermite Polynomial Expansion:**A new generalized polynomial chaos expansion (PCE) of a square-integrable random variable, comprising multivariate Hermite polynomials in dependent Gaussian random variables, was developed. Derived analytically, the second-moment properties of multivariate Hermite polynomials reveal a weakly orthogonal system with respect to an inner product comprising a general Gaussian probability measure. When the Gaussian variables are statistically independent, the multivariate Hermite polynomials elevate to a strongly orthogonal system, leading to the classical PCE. Nonetheless, when the Gaussian variables are statistically dependent, the exponential integrability of norm still allows the Hermite polynomials to constitute a complete set and hence a basis in a Hilbert space. The completeness is vitally important for the convergence of the generalized PCE to the correct limit. The optimality of the generalized PCE and the approximation quality due to truncation were established. New analytical formulae were developed to calculate the mean and variance of a generalized PCE approximation of a general output variable in terms of the expansion coefficients and statistical properties of Hermite polynomials. However, unlike in the classical PCE, calculating the coefficients of the generalized PCE requires solving a coupled system of linear equations. Moreover, the variance formula of the generalized PCE contains additional terms -- a consequence of statistical dependence among Gaussian input variables -- that are not present in that of the classical PCE. The additional terms vanish as they should when the Gaussian variables are statistically independent, regressing the generalized PCE to the classical PCE. A possible extension of the generalized PCE for non-Gaussian variables was studied. Numerical examples developed from an elementary function, a stochastic ODE, and a random eigenvalue analysis illustrate the construction and use of a generalized PCE approximation in estimating the statistical properties of output variables.**A Generalized Polynomial Chaos Expansion:**A new generalized polynomial chaos expansion (PCE), referred to as GPCE, of a square-integrable random variable, comprising measure-consistent multivariate orthonormal polynomials in dependent random variables with non-product-type probability measures, was developed. There are two main novelties: First, a degree-wise splitting of the polynomial space of all input random variables into orthogonal subspaces, each spanned by measure-consistent multivariate orthogonal polynomials, was constructed, resulting in the GPCE developed without the need for a tensor-product structure. Under prescribed assumptions, the set of measure-consistent orthogonal polynomials was proved to form a basis of each subspace, leading to an orthogonal sum of such sets of basis functions to span the space of all polynomials. Second, a whitening transformation is proposed to decorrelate orthogonal polynomials into orthonormal polynomials for an arbitrary probability measure. The transformation is valid whether or not the orthogonal polynomials of the same degree are mutually orthogonal. The orthogonal sum of measure-consistent polynomials, whether orthogonal or orthonormal, is dense in a Hilbert space of square-integrable functions, leading to mean-square convergence of the generalized PCE to the correct limit, including when there are infinitely many random variables. The optimality of the generalized PCE and the approximation quality due to truncation were demonstrated or discussed. For independent probability measures, the proposed generalized PCE reduces to the existing classical or generalized PCE. Analytical formulae are proposed to calculate the mean and variance of a truncated generalized PCE of a general output variable in terms of the expansion coefficients. An example stemming from a stochastic boundary-value problem illustrates the construction and use of a generalized PCE approximation in estimating the statistical properties of an output variable for 12 distinct non-product-type probability measures of input variables.

**Task 3:**

**A Generalized Polynomial Dimensional Decomposition:**A new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, was developed for uncertainty quantification analysis of complex systems subject to dependent input random variables. The method involves a Fourier-like series expansion of a high-dimensional random output function in terms of a hierarchically ordered, measure-consistent multivariate orthogonal polynomials in dependent input variables. The proposed GPDD, which can be viewed as a generalized version of the existing PDD, exists for any square-integrable function and converges in mean-square to the correct limit, provided that a few assumptions on the input probability measures are met. The GPDD method does not require independence of random variables; yet it generates a convergent sequence of low-variate, low-order approximations for efficiently estimating the probabilistic characteristics of a general stochastic response of interest. New formulae are developed to calculate the mean and variance of a GPDD approximation of a general output variable in terms of the expansion coefficients and second-moment properties of multivariate orthogonal polynomials. However, unlike in the existing PDD, calculating the coefficients of GPDD requires solving a coupled system of linear equations. Moreover, the variance formula of GPDD contains additional terms -- a consequence of statistical dependence among input random variables -- that are not present in that of PDD. The additional terms vanish as they should when the input variables are statistically independent, regressing the proposed PDD to the existing PDD. Two uncertainty quantification problems -- the one involving a stochastic differential equation from three-dimensional diffusion analysis and the other entailing an 11-dimensional random eigenvalue analysis from structural dynamics -- were solved, illustrating the construction and use of GPDD approximations in estimating various statistical properties of output variables.**Dimensionwise Multivariate Orthogonal Polynomials in General Probability Spaces:**A new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, of a square-integrable output random variable, encompassing hierarchically structured, multivariate orthogonal polynomials in dependent input random variables, was developed. A dimensional decomposition of polynomial spaces into respective subspaces, each spanned by a set of measure-consistent orthogonal polynomials, was established. The outcome is a polynomial surrogate of the generalized analysis-of-variance dimensional decomposition (ADD), ultimately leading to GPDD of a general L2-function without enforcing any tensor-product structure. Under requisite assumptions, the set of measure-consistent orthogonal polynomials constitutes a complete basis of each subspace, producing a union-sum collection of such sets of basis functions to span the space of all polynomials. Moreover, the aforementioned collection is L2-dense in a Hilbert space of square-integrable functions, resulting in mean-square convergence of GPDD to the correct limit, including for the instance of infinitely many random variables. New results determining the statistical properties of random orthogonal polynomials were derived. The optimality and approximation quality of a truncated GPDD were verified or explained. For independent random variables, the proposed PDD reduces to the existing PDD, justifying the appellation GPDD introduced in this work. By exploiting the hierarchical structure of a function, if it exists, the GPDD approximation is expected to efficiently solve high-dimensional stochastic problems in the presence of dependent random variables with non-product-type probability measures.**Robust Design Optimization under Dependent Random Variables by a Generalized Polynomial Chaos Expansion:**New computational methods were developed for robust design optimization (RDO) of complex engineering systems subject to input random variables with arbitrary, dependent probability distributions. The methods are built on a generalized polynomial chaos expansion (GPCE) for determining the second-moment statistics of a general output function of dependent input random variables, an innovative coupling between GPCE and score functions for calculating the second-moment sensitivities with respect to the design variables, and a standard gradient-based optimization algorithm, establishing direct GPCE, single-step GPCE, and multi-point single-step GPCE design processes. New analytical formulae are unveiled for design sensitivity analysis that is synchronously performed with statistical moment analysis. Numerical results confirm that the proposed methods yield not only accurate, but also computationally efficient optimal solutions of several mathematical and simple RDO problems. Finally, the success of conducting stochastic shape optimization of a steering knuckle demonstrates the power of the multi-point single-step GPCE method in solving industrial-scale engineering problems.**Reliability-based Design Optimization under Dependent Random Variables by a Generalized Polynomial Chaos Expansion:**A novel computational method was developed for reliability-based design optimization (RBDO) of complex engineering systems subject to input random variables with arbitrary, dependent probability distributions. The method involves a generalized polynomial chaos expansion (GPCE) of a stochastic response for reliability analysis, novel integrations of GPCE and score functions for calculating the sensitivities of reliability with respect to the design variables, and standard gradient-based optimization algorithms, encompassing a multi-point, single-step design process. For the RBDO method developed, the failure probability and its design sensitivities are determined concurrently from a single stochastic simulation or analysis. Precisely for this reason, the multi-point, single-step framework of the proposed method affords the ability of solving industrial-scale problems with large design spaces. Numerical results stemming from mathematical functions or elementary engineering problems indicate that the new method provides more accurate or computationally efficient design solutions than existing methods. Furthermore, shape design optimization of a complex engineering system was performed, demonstrating the usefulness of the new method to tackle practical RBDO problems.

**Supervision of a Graduate Student: **A graduate student, Mr. Dongjin Lee, was involved in this project. The student is making satisfactory progress toward completion of his Ph.D. study. The PI is the advisor of the graduate student.

**Insertion of Research Results into Undergraduate and Graduate Courses: **General concepts of engineering design optimization under uncertainty were discussed in an intermediate-level course, *Computer-Aided Engineering*, which was last offered in the Spring 2020 semester. In addition, the PI will introduce robust and reliability-based design optimization methods into the course syllabus of a graduate-level course, *Probabilistic Mechanics and Reliability*, which will be offered in the Fall 2021 semester.

**Invited Seminars and Lectures: **Since the beginning of this project, the PI or his associates presented ten invited lectures or posters on GPCE, GPDD, design sensitivity analysis, and GPCE/GPDD--based design optimization methods at ASCE, ASME, and ICOSSAR conferences and NSF workshops.

**Journal Articles**

- Lee, D. and Rahman, S., “Robust Design Optimization under Dependent Random Variables by a Generalized Polynomial Chaos Expansion,” submitted to Structural and Multidisciplinary Optimization, 2020.
- Lee, D. and Rahman, S., “Practical Uncertainty Quantification Analysis Involving Statistically Dependent Random Variables,” Applied Mathematical Modelling, Vol. 84, 2020, pp. 324-356.
- Rahman, S., “Dimension-wise Multivariate Orthogonal Polynomials in General Probability Spaces,” Applied Mathematics and Computation, Vol. 362, Article 124538, 2019, pp. 1-19.
- Rahman, S., “Uncertainty Quantification under Dependent Random Variables by a Generalized Polynomial Dimensional Decomposition,” Computer Methods in Applied Mechanics and Engineering, Vol. 344, 2019, pp. 910-937.
- Ren, X. and Rahman, S., “Stochastic Design Optimization Accounting for Structural and Distributional Design Variables,” Engineering Computations, Vol. 35, No. 8, 2018, pp. 2654-2695.
- Rahman, S., “Mathematical Properties of Polynomial Dimensional Decomposition,” SIAM/ASA Journal on Uncertainty Quantification, Vol. 6, 2018, pp. 816-844.
- Rahman, S., “A Polynomial Chaos Expansion in Dependent Random Variables,” Journal of Mathematical Analysis and Applications, Vol. 464, pp. 749-775, 2018.
- Rahman, S., “Wiener-Hermite Polynomial Chaos Expansion for Dependent Variables,” Journal of Mathematical Analysis and Applications, Vol. 454, 2017, pp. 303-334.
- Ren, X., Yadav, V., and Rahman, S., “Reliability-based Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition,” Structural and Multidisciplinary Optimization, Vol. 53, 2016, pp. 425-452.
- Rahman, S., “The
*f*-Sensitivity Index,” SIAM/ASA Journal on Uncertainty Quantification, Vol. 4, 2016, pp. 130-162. - Rahman, S., “A Surrogate Method for Density-based Global Sensitivity Analysis,” Reliability Engineering & Systems Safety, Vol. 155, 2016, pp. 224-235.
- Rahman, S. Yadav, V., and Ren, X., “High-Dimensional Stochastic Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition,” Lecture Notes in Computational Science and Engineering, Springer, Vol. 109, pp. 247-264, 2016.

**Papers in Conferences, Symposiums, and Congresses**

- Lee, D. and Rahman, S., “Robust Design Optimization with a Generalized Polynomial Chaos Expansion under Dependent Input Variables,” Proceedings of 2020 Engineering Mechanics Institute and Probabilistic Methods Conference, New York, NY, May 26-29, 2020.
- Lee, D. and Rahman, S., “A Generalized Polynomial Chaos Expansion for Stochastic Design Optimization under Dependent Random Variables,” Proceedings of 15th U.S. National Congress on Computational Mechanics, Austin, Texas, July 28-August 1, 2019.
- Rahman, S., “A Polynomial Dimensional Decomposition in Dependent Random Variables,” Proceedings of 8th International Conference on Computational Stochastic Mechanics, Paros, Greece, June 10-13, 2018.
- Lee, D. and Rahman, S., “High-Dimensional Stochastic Sensitivity Analysis and Design Optimization for Dependent Random Variables,” Proceedings of 13
^{th}World Congress on Computational Mechanics, New York, NY, July 22-27, 2018. - Rahman, S., “Stochastic Optimization for Design under Uncertainty with Dependent Probability Measures,” Proceedings of 2018 NSF Design Circle Workshop, Corvallis, OR, March 22-24, 2018.
- Rahman, S., “A Generalized Wiener-Hermite Polynomial Expansion for Multivariate Gaussian Probability Measures,” Proceedings of 12th ICOSSAR, Vienna, Austria, August 6-12, 2017.
- Ren, X. and Rahman, S., “An Adaptive-Sparse Polynomial Dimensional Decomposition for Robust Design Optimization,” Proceedings of 12th ICOSSAR, Vienna, Austria, August 6-12, 2017.
- Ren, X. and Rahman, S., “Robust Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition,” Proceedings of 2016 ASME International Design Engineering Technical Conferences, Charlotte, NC, August 21-24, 2016.
- Rahman, S., “The
*f*-Sensitivity Index,” Proceedings of 2016 ASCE EMI and Probabilistic Mechanics and Reliability Conferences, Nashville, TN, May 22-25, 2016. - Ren, X. and Rahman, S., “Stochastic Design Optimization Accounting for Structural and Distributional Design Variables,” Proceedings of 2016 ASCE EMI and Probabilistic Mechanics and Reliability Conferences, Nashville, TN, May 22-25, 2016.

**Poster Presentations**

- Rahman, S., “Stochastic Optimization for Design under Uncertainty with Dependent Probability Measures,” Posters of 2015 NSF ESD-SYS Grantees Conference, Clemson, SC, November 14-17, 2015.

A number of computer codes are being developed for performing various numerical simulations required in this project. For further information, please contact the PI (S. Rahman) at sharif-rahman@uiowa.edu.