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

**Award No.: **CMMI-1130147

**Duration: **September 1, 2011 – August 31, 2015

**Principal Investigator: **Professor Sharif Rahman

**Graduate Student: **V. Yadav, X. Ren

**SUMMARY**

The major goal of this research project is to create theoretical foundations and numerical algorithms of innovative computational methods for solving a general random eigenvalue problem in modeling and simulation of high-dimensional stochastic dynamic systems. The proposed effort involves: (1) development of a new polynomial dimensional decomposition (PDD) method for predicting the statistical moments and probability distributions of eigensolutions of stochastic dynamic systems, including innovative dimension-reduction and sampling techniques for estimating the expansion coefficients (Task 1); (2) quantification and comparison of approximation errors from the referential and ANOVA dimensional decomposition (RDD and ADD) methods and of approximation errors from the PDD and polynomial chaos expansion (PCE) methods (Task 2); and (3) development of new multiplicative PDD methods, hybridization, and respective error analyses (Task 3). The new methods and computational tools to be developed in this project will aid in accurate and efficient probabilistic characterization of dynamic system responses, such as natural frequencies and modes shapes. The statistics and rare-event probabilities predicted by these methods may potentially lead to new or improved designs in the presence of uncertainties due to insufficient information, limited understanding of underlying phenomena, and inherent randomness. Therefore, the research results will be of significant benefit to several commercial and industrial applications, such as civil, automotive, and aerospace infrastructure. Potential engineering applications include analysis and design of civil structures; noise-vibration-harshness, and crashworthiness of ground vehicle systems; and fatigue durability of aerospace structures. For the microelectronics industry, applications in reliability of microelectronics and interconnects, reliability of micro-electro-mechanical systems for sensors and actuators are relevant. It is anticipated that the implementation of the new stochastic methods developed herein will lead to next-generation scientific simulation codes, facilitating nuanced mathematical analysis of increasingly complex dynamical systems.

**Task 1: Development of a new polynomial dimensional decomposition (PDD) method for predicting the statistical moments and probability distributions of eigensolutions of stochastic dynamic systems, including innovative dimension-reduction and sampling techniques for estimating the expansion coefficients**

**A PDD Method for Random Eigenvalue Analysis:**A new, enhanced version of polynomial dimensional decomposition (PDD) method was developed for solving high-dimensional random eigenvalue problem subject to independent random input following an arbitrary probability distribution. The method is based on Fourier-polynomial expansions of analysis-of-variance (ANOVA) component functions by orthonormal polynomial bases, the Stieltjes procedure for generating the recursion coefficients of orthogonal polynomials, and the Gauss quadrature rule for a specified probability measure for determining the expansion coefficients. Compared with the previous versions of PDD, the new method no longer depends on classical orthogonal polynomials or standard Gauss quadrature rules. Instead, non-classical orthogonal polynomials for defining basis functions and non-standard Gauss quadrature rules for calculating the expansion coefficients are generated for arbitrary probability measures. Since PDD stems from the ANOVA decomposition, the PDD inherits all the desirable properties ANOVA, including the orthogonal structure of the decomposition. Due to the dimensional hierarchy of PDD, subsequent stochastic analysis of large-scale complex systems can be conducted efficiently. Therefore, a long-standing stochastic problem associated with the curse of dimensionality will be alleviated to some magnitude with positive ramifications in engineering and applied sciences.**Efficient Calculation of PDD Expansion Coefficients:**Two innovative numerical schemes or techniques, one exploiting dimension-reduction integration, and the other employing sampling techniques, were developed for estimating the expansion coefficients accurately and efficiently. The dimension-reduction technique entails a low-dimensional approximation of an eigensolution, resulting in a finite sum of at most R-dimensional integrations, where R < N, where N is the number of random variables. It generates the expansion coefficients approaching their exact values as R approaches N and is more efficient than performing one N-dimensional integration, particularly when R << N. Hence, the computational effort in calculating the coefficients is significantly lowered using the dimension-reduction integration. When R = 1, 2, or 3, the computational efforts involve one-, at most two-, and at most three-dimensional integrations, respectively. The integration points and associated weights depend on the probability distribution of the random input and are readily available using the Stieltjes procedure. In contrast, the sampling techniques, including crude Monte Carlo simulation (MCS) or quasi MCS, were developed for estimating the expansion coefficients in two simple steps: (1) generate a point set consistent with the probability measure of the random input; and (2) approximate the high-dimensional integrals (expansion coefficients) as the averages of eigensolutions, evaluated at all members of the point set. In crude MCS, the point set contains a sequence of pseudo-random numbers, following the probability distributions of random input, whereas in quasi MCS, the point set comprises a low-discrepancy sequence. The advantage of one MCS over the other depends on the smoothness properties of the integrand and the dimension of the integral. Due to nonintrusive evaluation of the expansion coefficients, both techniques can be easily adapted to solving large-scale random eigenvalue problems requiring external, deterministic simulation codes.**Statistical Moments and Tail Distributions of Random Eigensolutions:**By exploiting the orthogonal structure of PDD and the properties of orthogonal polynomials, explicit formulae were derived for calculating the mean and variance of a random eigensolution in terms of the expansion coefficients. Regardless of truncation parameters, the PDD yields the exact mean, but progressively accurate variances that are mean-square convergent. However, no closed-form formulae or analytical expressions are possible for calculating the rare-event probabilities of a general eigenvalue problem. Since a PDD approximation furnishes a surrogate expressed as a sum of simple analytical functions, any probabilistic characteristic of an eigensolution, including its rare event probabilities, can be easily estimated by performing MCS of the surrogate. Numerical results, including frequency distributions of a functionally graded plate, indicate that the PDD method developed provides accurate, convergent, and computationally efficient estimates of the tail probabilistic characteristics of eigenvalues. Finally, the random eigensolutions of a sport utility vehicle comprising 40 random variables were evaluated, demonstrating the ability of the PDD method to solve industrial-scale problems.

For further information on Task 1, see a conference presentation.

**Task 2: Quantification and comparison of approximation errors from the referential and ANOVA dimensional decomposition (RDD and ADD) methods and of approximation errors from the PDD and polynomial chaos expansion (PCE) methods**

**Approximation Errors from Truncated RDD and ADD:**Two variants of dimensional decomposition, namely, RDD and ADD, of a multivariate function, both representing finite sums of lower-dimensional component functions, were studied. The approximations resulting from the truncated RDD and ADD are explicated, including clarifications of parallel developments and synonyms used by various researchers. For the RDD approximation, a direct form, previously developed by the PI’s group, was found to provide a vital link to subsequent error analysis. New theorems were proven about the expected errors from the bivariate and general RDD approximations, so far available only for the univariate RDD approximation, when the reference point is selected randomly. They furnish new formulae for the lower and upper bounds of the expected error committed by an arbitrarily truncated RDD, providing a means to grade RDD against ADD approximations. The formulae indicate that the expected error from the*S*-variate RDD approximation of a function of*N*variables, where 0*≤ S < N < ∞*, is at least 2^{(S+1) }times larger than the error from the*S*-variate ADD approximation. Consequently, ADD approximations are exceedingly more precise than RDD approximations at higher-variate truncations. The analysis also finds the RDD approximation to be sub-optimal for an arbitrarily selected reference point, whereas the ADD approximation always results in minimum error. Therefore, RDD approximations should be used with caveat.**Approximation Errors from PDD and PCE Methods:**The PDD and PCE methods were evaluated for solving random eigenvalue problems commonly encountered in stochastic dynamic systems. A new theorem, proven herein, demonstrates that the infinite series from PCE can be reshuffled to derive the infinite series from PDD and*vice versa*. However, compared with PCE, which contains terms arranged with respect to the order of polynomials, PDD is structured with respect to the degree of interaction between a finite number of random variables. Therefore, significant differences exist regarding the accuracy, efficiency, and convergence properties of their truncated series. An alternative form of the PCE approximation expressed in terms of the PDD expansion coefficients was developed. As a result, the probabilistic eigensolutions from both the PDD and PCE methods can be obtained from the same PDD coefficients, leading to closed-form expressions of the second-moment properties of eigenvalues and respective errors. For a class of REPs, where the cooperative effects of input variables on an eigenvalue get progressively weaker or vanish altogether, the error perpetrated by the PCE approximation is larger than that committed by the PDD approximation, when the expansions orders are equal. Given the same expansion orders, the PDD approximation including main and cooperative effects of all input variables cannot be worse than the PCE approximation, although the inclusion of all cooperative effects undermines the salient features of PDD. Numerical analyses of mathematical functions systems reveal markedly higher convergence rates of the PDD approximation than the PCE approximation.**Adaptive-Sparse PDD Methods for Random Eigenvalue Analysis:**Two novel adaptive-sparse PDD methods were developed for solving general high-dimensional uncertainty quantification problems in computational science and engineering. The methods entail global sensitivity analysis for defining the relevant pruning criteria, a unified computational algorithm for retaining component functions with largest orders of their orthogonal polynomial expansions, two distinct ranking schemes for grading component functions, and a full- or sparse-grid dimension-reduction integration and quasi Monte Carlo simulation for estimating the expansion coefficients. The fully adaptive-sparse PDD method is comprehensive and rigorous, leading to the second-moment statistics of a stochastic response that converges to the exact solution when the tolerances vanish. A partially adaptive-sparse PDD method, obtained through regulated adaptivity and sparsity, is economical and is, therefore, expected to solve practical problems with numerous variables. Compared with past developments, the adaptive-sparse PDD methods do not require its truncation parameter(s) to be assigned a priori or arbitrarily. The numerical results reveal that an adaptive-sparse PDD method achieves a desired level of accuracy with considerably fewer coefficients compared with existing PDD approximations. For a required accuracy in calculating the tail probabilistic characteristics of natural frequencies of a functionally graded plate, the new bivariate adaptive-sparse PDD method is more efficient than the existing bivariately truncated PDD method by almost an order of magnitude. Finally, stochastic dynamic analysis of a disk brake system was performed, demonstrating the ability of the new method to tackle practical engineering problems.

For further information on Task 2, see a conference presentation.

**Task 3: Development of new multiplicative PDD methods, hybridization, and respective error analyses**

**Multiplicative Polynomial Dimensional Decomposition Method:** A new class of computational methods, referred to as the multiplicative polynomial dimensional decomposition (PDD) methods, was developed for solving high-dimensional stochastic problems commonly encountered in engineering and applied sciences. When a stochastic response is dominantly of multiplicative nature, the standard PDD approximation, predicated on additive function decomposition, may not provide sufficiently accurate probabilistic solutions of a complex system. To circumvent this problem, two multiplicative versions of PDD, referred to as factorized PDD and logarithmic PDD, were developed. Both versions involve a hierarchical, multiplicative decomposition of a multivariate function, a broad range of orthonormal polynomial bases for Fourier-polynomial expansions of component functions, and a dimension-reduction or sampling technique for estimating the expansion coefficients. A number of problems involving mathematical functions or uncertain dynamic systems were solved to corroborate how and when a multiplicative PDD is more efficient or accurate than the additive PDD. The results show that, indeed, both the factorized and logarithmic PDD approximations can effectively exploit the hidden multiplicative structure of a stochastic response when it exists. Since a multiplicative PDD recycles the same component functions of the additive PDD, no additional cost is incurred. Finally, the random eigensolutions of a sport utility vehicle comprising 40 random variables were evaluated, demonstrating the ability of the new methods to solve industrial-scale problems.

For further information on Task 3, see a conference presentation.

**Supervision of Graduate Students: **Two Ph.D. students, V. Yadav and X. Ren, are involved in this project. Mr. Yadav has graduated in 2013. Mr. Ren is making satisfactory progress towards completion of his Ph.D. program. The PI is the sole advisor of both graduate students.

**Insertion of Research Results into Undergraduate and Graduate Courses: **The newly developed decomposition methods from Tasks 1 were already integrated into the syllabus of a graduate course, 58:257 *Probabilistic Mechanics and Reliability*, which was last offered in the Fall 2012 semester. The PI will integrate new stochastic methods developed in Tasks 2 and 3 for the Fall 2014 offering of the above-mentioned course. The PI strongly believes that the insertion of new research results will motivate learning by students in the areas of computational dynamics, stochastic methods, and high-performance computing.

**Development of Symposia: **Since the beginning of the project (2011), the PI developed several technical sessions on stochastic methods and computational mechanics in conjunction with various national and international conferences. He also chaired a technical session on probabilistic methods at *Engineering Mechanics Institute Annual Conferences* in 2012 (Boston, MA).

**Invited Seminars and Lectures: **During the project period, the PI and his students have given ten invited lectures or seminars on PDD and related methods at national and international conferences and workshops. The dissemination of research results from this project has already started to impact both education and training of researchers and practicing engineers in the industry.

**Journal Articles**

- Rahman, S., “Global Sensitivity Analysis from Information Divergence Measures,” submitted to SIAM/ASA Journal on Uncertainty Quantification, 2014.
- Ren, X., Rahman, S., and Yadav, V., “Reliability-based Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition,” submitted to Structural and Multidisciplinary Optimization, 2014.
- Yadav, V. and Rahman, S., “A Hybrid Polynomial Dimensional Decomposition for High-Dimensional Stochastic Analysis,” accepted in Probabilistic Engineering Mechanics, 2014.
- Rahman, S., “A Generalized ANOVA Dimensional Decomposition for Dependent Probability Measures,” accepted in SIAM/ASA Journal on Uncertainty Quantification, 2014.
- Rahman, S., “Approximation Errors in Truncated Dimensional Decompositions,” Mathematics of Computation, Vol. 83, No. 290, 2014, pp. 2799-2819.
- Rahman, S., “Uncertainty Quantification by Alternative Decompositions of Multivariate Functions,” SIAM Journal on Scientific Computing, Vol. 35, 2013, pp. A3024-A3051.
- Yadav, V. and Rahman, S. and “Adaptive-Sparse Polynomial Dimensional Decomposition Methods for High-Dimensional Stochastic Computing,”Computer Methods in Applied Mechanics and Engineering, Vol 274, 2014 pp. 56-83.
- Yadav, V. and Rahman, S. and “Uncertainty Quantification of High-Dimensional Complex Systems by Multiplicative Polynomial Dimensional Decompositions,” International Journal for Numerical Methods in Engineering, Vol. 94, 2013, pp. 221-247.
- Rahman, S. and Yadav, V., “Orthogonal Polynomial Expansions for Solving Random Eigenvalue Problems,” International Journal for Uncertainty Quantification, Vol. 1, No. 2, 2011, pp. 163-187.

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

- Rahman, S., “High-Dimensional Stochastic Computing by Adaptive-Sparse Polynomial Dimensional Decomposition,” Proceedings of 3rd Workshop on Sparse Grids and Applications, Stuttgart, Germany, September 2014.
- Ren, X., Yadav, V., and Rahman, S., “Reliability-based Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition,” Proceedings of EMI2014 Engineering Mechanics Institute Conference, Hamilton, ON, August 2014.
- Rahman, S. and Ren, X., “Stochastic Design Optimization of Complex Systems,” 4th International Symposium on Inverse Problems, Design and Optimization, Albi, France, June 2013.
- Yadav, V. and Rahman, S., “Adaptive Polynomial Dimensional Decompositions for Uncertainty Quantification in High Dimensions,” Proceedings of 11th International Conference on Structural Safety & Reliability (ICOSSAR), New York, NY, June 2013. (ICOSSAR Paper Slides)
- Rahman, S., “Stochastic Isogeometric Analysis for Uncertainty Quantification,” Proceedings of 12th U.S. National Congress on Computational Mechanics (USNCCM), Raleigh, NC, July 2013.
- Yadav, V. and Rahman, S., “Multiplicative Polynomial Dimensional Decompositions for Uncertainty Quantification of High-Dimensional Complex Systems,” Proceedings of 14th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Indianapolis, IN, September 2012. (AIAA/ISSMO Paper Slides)
- Rahman, S., “Approximation Errors in Truncated Dimensional Decompositions for Uncertainty Quantification,” Proceedings of 14th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Indianapolis, IN, September 2012.
- Yadav, V. and Rahman, S., “An Adaptive-Sparse Polynomial Dimensional Decomposition Method for High-Dimensional Stochastic Computing,” Proceedings of 11th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, South Bend, IN, June 2012. (EMI/PMC Paper Slides)
- Rahman, S., “A Generalized ANOVA Dimensional Decomposition for Arbitrary Probability Measures,” Proceedings of 11th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, South Bend, IN, June 2012.

**Others**

The computer programs and related files created during this research are organized according to the following journal papers published. Click on the title of each paper to open a list of computer programs created for solving numerical and engineering problems published in the paper.

- Yadav, V. and Rahman, S. "A hybrid polynomial dimensional decomposition for uncertainty quantification of high-dimensional complex system". Accepted in Probabilistic Engineering Mechanics, 2014.
- Yadav, V. and Rahan, S. "Adaptive-sparse polynomial dimensional decomposition methods for high-dimensional stochastic computing." Computer Methods in Applied Mechanics and Engineering. Vol. 274, 2014, pp. 56-83.
- Yadav, V. and Rahman, S. “Uncertainty Quantification of High-Dimensional Complex Systems by Multiplicative Polynomial Dimensional Decompositions,” International Journal for Numerical Methods in Engineering, Vol. 94, 2013, pp. 221-247.
- Rahman, S. and Yadav, V., “Orthogonal Polynomial Expansions for Solving Random Eigenvalue Problems,” International Journal for Uncertainty Quantification, Vol. 1, No. 2, 2011, pp. 163-187.