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## Adaptive-sparse polynomial dimensional decomposition for highdimensional stochastic computing

Venue: | Comput. Methods Appl. Math |

Citations: | 2 - 1 self |

### Citations

3935 |
Dynamic Programming
- Bellman
- 1957
(Show Context)
Citation Context ...hich have found many successful applications. However, for truly highdimensional systems, they require astronomically large numbers of terms or coefficients, succumbing to the curse of dimensionality =-=[1]-=-. Therefore, alternative computational methods IGrant sponsor: U.S. National Science Foundation; Grant Nos. CMMI0653279, CMMI-1130147. ∗Corresponding author. Email addresses: vaibhav-yadav@uiowa.edu (... |

1084 |
Random Number Generation and Quasi-Monte Carlo Methods
- Niederreiter
- 1992
(Show Context)
Citation Context .... 4.2. Quasi Monte Carlo Simulation The basic idea of the quasi MCS is to replace the random or pseudo-random samples in crude MCS by well-chosen deterministic samples that are highly equidistributed =-=[21]-=-. The qausi Monte Carlo samples are often selected from a low-discrepancy sequence [9, 16, 21, 34] or by a lattice rule [33] to minimize the integration errors. The estimation of the PDD expansion coe... |

791 |
Stochastic Finite Elements: A Spectral Approach
- Ghanem, Spanos
- 1991
(Show Context)
Citation Context ...ctive is to compare the performance of the proposed adaptive-sparse PDD methods with that of the existing truncated PDD method. Readers interested in contrasting the truncated PDD method with the PCE =-=[14]-=- and other classical methods are referred to the authors’ prior work [24–26, 32]. Classical Legendre polynomials were used to define the orthonormal polynomials in Example 1, and all expansion coeffic... |

398 | The Wiener-Askey polynomial chaos for stochastic differential equations
- Xiu, Karniadakis
(Show Context)
Citation Context ... similar numerical calculations, is all too often expensive to evaluate. The most promising stochastic methods available today are perhaps the collocation [6, 10] and polynomial chaos expansion (PCE) =-=[14, 39]-=- methods, including sparse-grid techniques [18], which have found many successful applications. However, for truly highdimensional systems, they require astronomically large numbers of terms or coeffi... |

285 |
Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Volume 2: Programs, Birkhäuser
- Cullum, Willoughby
- 1985
(Show Context)
Citation Context ...ing of 800 eight-noded, second-order shell elements, shown in Figure 7(b), was constructed for FEA, to determine the natural frequencies of the FGM plate. No damping was included. A Lanczos algorithm =-=[3]-=- was employed for calculating the eigenvalues. The probability distributions of the first six natural frequencies of the functionally graded material plate were eval2Functionally graded materials are ... |

251 |
On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals
- Halton
- 1960
(Show Context)
Citation Context ...andom or pseudo-random samples in crude MCS by well-chosen deterministic samples that are highly equidistributed [21]. The qausi Monte Carlo samples are often selected from a low-discrepancy sequence =-=[9, 16, 21, 34]-=- or by a lattice rule [33] to minimize the integration errors. The estimation of the PDD expansion coefficients, which are high-dimensional integrals, comprises three simple steps: (1) generate a low-... |

233 |
The distribution of points in a cube and approximate evaluation of integrals
- Sobol’
- 1967
(Show Context)
Citation Context ...andom or pseudo-random samples in crude MCS by well-chosen deterministic samples that are highly equidistributed [21]. The qausi Monte Carlo samples are often selected from a low-discrepancy sequence =-=[9, 16, 21, 34]-=- or by a lattice rule [33] to minimize the integration errors. The estimation of the PDD expansion coefficients, which are high-dimensional integrals, comprises three simple steps: (1) generate a low-... |

199 |
An Introduction to the theory of random signals and noise
- Davenport, Root
- 1958
(Show Context)
Citation Context ...SiC(ξ) = F−1SiC [ Φ(α(ξ)) ] , where Φ is the distribution function of a standard Gaussian random variable and FSiC is the marginal distribution function of φ̃SiC(ξ). The Karhunen-Loève approximation =-=[5]-=- was employed to discretize α(ξ) and hence φSiC(ξ) into 28 standard Gaussian random variables. In addition, the constituent material properties, ESiC, EAl, νSiC, νAl, ρSiC, and ρAl, were modeled as in... |

190 |
Orthogonal Polynomials: Computation and Approximation
- Gautschi
- 2004
(Show Context)
Citation Context ...ted weights depend on the probability distribution of Xi. They are readily available, for example, the Gauss-Hermite or GaussLegendre quadrature rule, when Xi follows Gaussian or uniform distribution =-=[11]-=-. For an arbitrary probability distribution of Xi, the Stieltjes procedure [11] can be employed to generate the measure-consistent Gauss quadrature formulae [11]. An nvpoint Gauss quadrature rule exac... |

176 |
The jackknife estimate of variance,
- Efron, Stein
- 1981
(Show Context)
Citation Context ...) represent a Hilbert space of square-integrable functions y with respect to the induced generic measure fX(x)dx supported on RN . The ANOVA dimensional decomposition, expressed by the recursive form =-=[8, 30, 35]-=- y(X) = ∑ u⊆{1,··· ,N} yu(Xu), (1) y∅ = ∫ RN y(x) fX(x)dx, (2) yu(Xu) = ∫ RN−|u| y(Xu, x−u) fX−u (x−u)dx−u − ∑ v⊂u yv(Xv), (3) is a finite, hierarchical expansion in terms of its input variables with ... |

153 |
Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates
- Sobol’
- 2001
(Show Context)
Citation Context ...e dimensional structure lurking behind a complex response. Indeed, these sensitivity indices have been used to rank variables, fix unessential variables, and reduce dimensions of large-scale problems =-=[29, 36]-=-. The authors propose to exploit these indices, developed in conjunction with PDD, for adaptive-sparse PDD approximations as follows. The global sensitivity index of y(X) for a subset Xu, ∅ , u ⊆ {1, ... |

137 |
Discrépances de suites associées à un système de numération (en dimension s
- Faure
- 1982
(Show Context)
Citation Context ...andom or pseudo-random samples in crude MCS by well-chosen deterministic samples that are highly equidistributed [21]. The qausi Monte Carlo samples are often selected from a low-discrepancy sequence =-=[9, 16, 21, 34]-=- or by a lattice rule [33] to minimize the integration errors. The estimation of the PDD expansion coefficients, which are high-dimensional integrals, comprises three simple steps: (1) generate a low-... |

126 |
Solution of stochastic partial differential equations using Galerkin finite element techniques
- Deb, Babus̆ka, et al.
(Show Context)
Citation Context ...ia expensive finite-element analysis (FEA) or similar numerical calculations, is all too often expensive to evaluate. The most promising stochastic methods available today are perhaps the collocation =-=[6, 10]-=- and polynomial chaos expansion (PCE) [14, 39] methods, including sparse-grid techniques [18], which have found many successful applications. However, for truly highdimensional systems, they require a... |

82 | An adaptive multi-element generalized polynomial chaos method for stochastic differential equations,
- Wan, Karniadakis
- 2005
(Show Context)
Citation Context ...MR) [20] and the anchored decomposition [43], employed in conjunction with the sparse-grid collocation methods, for solving stochastic problems in fluid dynamics. Several adaptive variants of the PCE =-=[2, 19, 37]-=- method have also appeared. It is important to clarify that the cut-HDMR and anchored decompositions are the same as the referential dimensional decomposition (RDD) [28, 30]. Therefore, both adaptive ... |

64 |
General foundations of high dimensional model representations,
- Rabitz, Alis
- 1999
(Show Context)
Citation Context ...es the probability measure in Equations (1)-(3), leading to the recursive form y(X) = ∑ u⊆{1,··· ,N} wu(Xu; c), (4) w∅ = y(c), (5) wu(Xu; c) = y(Xu, c−u) − ∑ v⊂u wv(Xv; c), (6) also known as cut-HDMR =-=[23]-=-, anchored decomposition [17], and anchored-ANOVA decomposition [15], with the latter two referring to the reference point as the anchor. Xu and Rahman introduced Equations (4)-(6) with the aid of Tay... |

47 |
Simple cubature formulas with high polynomial exactness, ConDo wn loa de d 1 1/2 6/1 2 t o 1 .14 8.2 .35 . R ed ist rib uti on su bje ct to SI AM lic en se or co py rig ht; se e h ttp ://w ww .si am .or g/j ou rna ls/ ojs a.p hp
- Novak, Ritter
- 1999
(Show Context)
Citation Context ...ical techniques: (1) the full-grid integration technique; (2) the sparse-grid integration technique using the extended FSI rule; and (3) the sparsegrid integration technique using Smolyak’s algorithm =-=[22]-=-. The Smolyak’s algorithm is included because it is commonly used as a preferred sparse-grid numerical technique for approximating high-dimensional integrals. Define an integer l ∈ N such that all thr... |

46 |
Sparse grid collocation schemes for stochastic natural convection problems,
- Ganapathysubramanian, Zabaras
- 2007
(Show Context)
Citation Context ...ia expensive finite-element analysis (FEA) or similar numerical calculations, is all too often expensive to evaluate. The most promising stochastic methods available today are perhaps the collocation =-=[6, 10]-=- and polynomial chaos expansion (PCE) [14, 39] methods, including sparse-grid techniques [18], which have found many successful applications. However, for truly highdimensional systems, they require a... |

37 |
Theorems and examples on high dimensional model representation,
- Sobol
- 2003
(Show Context)
Citation Context ...) represent a Hilbert space of square-integrable functions y with respect to the induced generic measure fX(x)dx supported on RN . The ANOVA dimensional decomposition, expressed by the recursive form =-=[8, 30, 35]-=- y(X) = ∑ u⊆{1,··· ,N} yu(Xu), (1) y∅ = ∫ RN y(x) fX(x)dx, (2) yu(Xu) = ∫ RN−|u| y(Xu, x−u) fX−u (x−u)dx−u − ∑ v⊂u yv(Xv), (3) is a finite, hierarchical expansion in terms of its input variables with ... |

35 | Fully symmetric interpolatory rules for multiple integrals over infinite regions with gaussian weight,
- Genz, Keister
- 1996
(Show Context)
Citation Context ...rule, that is capable of exploiting dimension-reduction integration is proposed. Fully symmetric interpolatory rule. The fully symmetric interpolatory (FSI) rules developed by Genz and his associates =-=[12, 13]-=-, is a sparse-grid integration technique for performing high-dimensional numerical integration. Applying this rule to the |v|-dimensional integrations in Equations (38) and (39), the PDD expansion coe... |

32 | Quadrature error bounds with applications to lattice rules,
- Hickernell
- 1996
(Show Context)
Citation Context ... Equations (1)-(3), leading to the recursive form y(X) = ∑ u⊆{1,··· ,N} wu(Xu; c), (4) w∅ = y(c), (5) wu(Xu; c) = y(Xu, c−u) − ∑ v⊂u wv(Xv; c), (6) also known as cut-HDMR [23], anchored decomposition =-=[17]-=-, and anchored-ANOVA decomposition [15], with the latter two referring to the reference point as the anchor. Xu and Rahman introduced Equations (4)-(6) with the aid of Taylor series expansion, calling... |

30 | A generalized dimension-reduction method for multi-dimensional integration in stochastic mechanics,
- Xu, Rahman
- 2004
(Show Context)
Citation Context ...position [15], with the latter two referring to the reference point as the anchor. Xu and Rahman introduced Equations (4)-(6) with the aid of Taylor series expansion, calling them dimension-reduction =-=[40]-=- and decomposition [41] methods for statistical moment and reliability analyses, respectively, of mechanical systems. Compared with ADD, RDD lacks orthogonal features, but its component functions are ... |

27 |
Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach,
- Blatman, Sudret
- 2008
(Show Context)
Citation Context ...MR) [20] and the anchored decomposition [43], employed in conjunction with the sparse-grid collocation methods, for solving stochastic problems in fluid dynamics. Several adaptive variants of the PCE =-=[2, 19, 37]-=- method have also appeared. It is important to clarify that the cut-HDMR and anchored decompositions are the same as the referential dimensional decomposition (RDD) [28, 30]. Therefore, both adaptive ... |

23 | Dimension-wise integration of high-dimensional functions with applications to finance
- Griebel, Holtz
(Show Context)
Citation Context ...rsive form y(X) = ∑ u⊆{1,··· ,N} wu(Xu; c), (4) w∅ = y(c), (5) wu(Xu; c) = y(Xu, c−u) − ∑ v⊂u wv(Xv; c), (6) also known as cut-HDMR [23], anchored decomposition [17], and anchored-ANOVA decomposition =-=[15]-=-, with the latter two referring to the reference point as the anchor. Xu and Rahman introduced Equations (4)-(6) with the aid of Taylor series expansion, calling them dimension-reduction [40] and deco... |

20 |
Decomposition methods for structural reliability analysis,
- Xu, Rahman
- 2005
(Show Context)
Citation Context ...ive variants of the PCE [2, 19, 37] method have also appeared. It is important to clarify that the cut-HDMR and anchored decompositions are the same as the referential dimensional decomposition (RDD) =-=[28, 30]-=-. Therefore, both adaptive methods essentially employ RDD for multivariate function approximations, where the mean values of random input are treated as the reference or anchor point − a premise origi... |

15 |
An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations,
- Ma, Zabaras
- 2010
(Show Context)
Citation Context ...4 ar X iv :1 40 2. 33 30 v1s[ ma th. NA ]s13sFe b 2 01 4 well. Addressing some of the aforementioned concerns have led to adaptive versions of the cut-high-dimensional model representation (cut-HDMR) =-=[20]-=- and the anchored decomposition [43], employed in conjunction with the sparse-grid collocation methods, for solving stochastic problems in fluid dynamics. Several adaptive variants of the PCE [2, 19, ... |

12 |
Improving the rejection sampling method in quasi-Monte Carlo methods,
- Wang
- 2000
(Show Context)
Citation Context ...129 163 201 4 17 39 81 151 257 407 609 871 1201 5 37 93 201 401 749 1317 2193 3481 5301 mentioned here that many authors, including Halton [16], Faure [9], Niederreiter [21], and Sobol [34], and Wang =-=[38]-=-, have extensively studied how to generate the best low-discrepancy point sets and to facilitate variance reduction. For a bounded variation of the integrand, the quasi MCS has a theoretical error bou... |

11 | Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration,
- Li, Ghanem
- 1998
(Show Context)
Citation Context ...MR) [20] and the anchored decomposition [43], employed in conjunction with the sparse-grid collocation methods, for solving stochastic problems in fluid dynamics. Several adaptive variants of the PCE =-=[2, 19, 37]-=- method have also appeared. It is important to clarify that the cut-HDMR and anchored decompositions are the same as the referential dimensional decomposition (RDD) [28, 30]. Therefore, both adaptive ... |

11 |
Adaptive anova decomposition of stochastic incompressible and compressible flows,
- Yang, Choi, et al.
- 2012
(Show Context)
Citation Context ... NA ]s13sFe b 2 01 4 well. Addressing some of the aforementioned concerns have led to adaptive versions of the cut-high-dimensional model representation (cut-HDMR) [20] and the anchored decomposition =-=[43]-=-, employed in conjunction with the sparse-grid collocation methods, for solving stochastic problems in fluid dynamics. Several adaptive variants of the PCE [2, 19, 37] method have also appeared. It is... |

10 |
A polynomial dimensional decomposition for stochastic computing,
- Rahman
- 2008
(Show Context)
Citation Context ...1, · · · , j|u|), constitutes an orthonormal basis in L2(×p=|u|p=1 Ωip , ×p=|u|p=1 Fip , ×p=|u|p=1 Pip ). The orthogonal polynomial expansion of a non-constant |u|- variate component function becomes =-=[25, 26]-=- yu(Xu) = ∑ j|u|∈N|u|0 j1,··· , j|u|,0 Cuj|u|ψuj|u| (Xu), ∅ , u ⊆ {1, · · · ,N}, (7) with Cuj|u| := ∫ RN y(x)ψuj|u| (xu) fX(x)dx, ∅ , u ⊆ {1, · · · ,N}, j|u| ∈ N|u|0 , (8) representing the correspondi... |

10 |
Extended polynomial dimensional decomposition for arbitrary probability distributions,
- Rahman
- 2009
(Show Context)
Citation Context ...1, · · · , j|u|), constitutes an orthonormal basis in L2(×p=|u|p=1 Ωip , ×p=|u|p=1 Fip , ×p=|u|p=1 Pip ). The orthogonal polynomial expansion of a non-constant |u|- variate component function becomes =-=[25, 26]-=- yu(Xu) = ∑ j|u|∈N|u|0 j1,··· , j|u|,0 Cuj|u|ψuj|u| (Xu), ∅ , u ⊆ {1, · · · ,N}, (7) with Cuj|u| := ∫ RN y(x)ψuj|u| (xu) fX(x)dx, ∅ , u ⊆ {1, · · · ,N}, j|u| ∈ N|u|0 , (8) representing the correspondi... |

7 | Global sensitivity analysis by polynomial dimensional decomposition,
- Rahman
- 2011
(Show Context)
Citation Context ...e dimensional structure lurking behind a complex response. Indeed, these sensitivity indices have been used to rank variables, fix unessential variables, and reduce dimensions of large-scale problems =-=[29, 36]-=-. The authors propose to exploit these indices, developed in conjunction with PDD, for adaptive-sparse PDD approximations as follows. The global sensitivity index of y(X) for a subset Xu, ∅ , u ⊆ {1, ... |

6 | Sparse grid quadrature in high dimensions with applications in finance and insurance (Ph.D. Dissertation),
- Holtz
- 2008
(Show Context)
Citation Context ...xpensive to evaluate. The most promising stochastic methods available today are perhaps the collocation [6, 10] and polynomial chaos expansion (PCE) [14, 39] methods, including sparse-grid techniques =-=[18]-=-, which have found many successful applications. However, for truly highdimensional systems, they require astronomically large numbers of terms or coefficients, succumbing to the curse of dimensionali... |

5 | Stochastic dynamic systems with complex-valued eigensolutions,
- Rahman
- 2007
(Show Context)
Citation Context ...he dynamic instabilities caused in the disk brake system. It is worth mentioning that a similar brake-squeal analysis with only five input random variables was performed using a univariate RDD method =-=[24]-=-. However, verification or improvement of the univariate solution was not possible due to inherent limitations of the method used. The adaptive-sparse PDD approximations developed in this work have ov... |

5 |
Lattice Methods for Multiple Integration, Oxford science publications,
- Sloan, Joe
- 1994
(Show Context)
Citation Context ...de MCS by well-chosen deterministic samples that are highly equidistributed [21]. The qausi Monte Carlo samples are often selected from a low-discrepancy sequence [9, 16, 21, 34] or by a lattice rule =-=[33]-=- to minimize the integration errors. The estimation of the PDD expansion coefficients, which are high-dimensional integrals, comprises three simple steps: (1) generate a low-discrepancy point set PL :... |

4 |
Statistical moments of polynomial dimensional decomposition,
- Rahman
- 2010
(Show Context)
Citation Context ...ections. 2.4.2. Statistical Moments of PDD Applying the expectation operator on ỹS ,m(X) and (ỹS ,m(X) − y∅)2 and noting the zero-mean and orthogonal properties of PDD component functions, the mean =-=[27]-=- E [ ỹS ,m(X) ] = y∅ (16) of the S -variate, mth-order PDD approximation matches the exact mean E [ y(X) ] , regardless of S or m, and the approximate variance [27] σ2S ,m := E [( ỹS ,m(X) − E [ỹS ... |

4 | Approximation errors in truncated dimensional decompositions,
- Rahman
- 2014
(Show Context)
Citation Context ...ive variants of the PCE [2, 19, 37] method have also appeared. It is important to clarify that the cut-HDMR and anchored decompositions are the same as the referential dimensional decomposition (RDD) =-=[28, 30]-=-. Therefore, both adaptive methods essentially employ RDD for multivariate function approximations, where the mean values of random input are treated as the reference or anchor point − a premise origi... |

2 | Orthogonal polynomial expansions for solving random eigenvalue problems,
- Rahman, Yadav
- 2011
(Show Context)
Citation Context ...ues of m in Equation (20). In contrast, many more terms and expansion coefficients are required to be included in the PCE approximation to capture such high nonlinearity. In reference to a past study =-=[32]-=-, consider two mean-squared errors, eS ,m := E[y(X) − ỹS ,m(X)]2 and ep := E[y(X) − y̌p(X)]2, owing to the S -variate, mth-order PDD approximation ỹS ,m(X) and pth-order PCE approximation y̌p(X), re... |

2 |
Uncertainty quantification of high-dimensional complex systems by multiplicative polynomial dimensional decompositions,
- Yadav, Rahman
- 2013
(Show Context)
Citation Context ... shown in Figure 7(a), made of a functionally graded material (FGM)2, where silicon carbide (SiC) particles varying along the horizontal coordinate ξ are randomly dispersed in an aluminum (Al) matrix =-=[42]-=-. The result is a random inhomogeneous plate, where the effective elastic modulus E(ξ), effective Poisson’s ratio ν(ξ), and effective mass density ρ(ξ) are random fields. They depend on two principal ... |

1 | Novel computational methods for high-dimensional stochastic sensitivity analysis,
- Rahman, Ren
- 2014
(Show Context)
Citation Context ...ximation for obtaining an exponential family of approximate distributions. Readers interested in this alternative approach are referred to the authors’ ongoing work on stochastic sensitivity analysis =-=[31]-=-. It is important to emphasize that the two truncation criteria proposed are strictly based on variance as a measure of output uncertainty. They are highly relevant when the second-moment properties o... |

1 | in: Wikipedia, - brake - 2013 |