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| 1 | Efficient Collocational Approach for Parametric Uncertainty Analysis显示文摘A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented.The uncertain parameters are modeled as random variables,and the governing equations are treated as stochastic.The solutions,or quantities of interests,are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters.A high-order stochastic collocation method is employed to solve the solution statistics,and more importantly,to reconstruct the polynomial expansion.While retaining the high accuracy by polynomial expansion,the resulting“pseudo-spectral”type algorithm is straightforward to implement as it requires only repetitive deterministic simulations.An estimate on error bounded is presented,along with numerical examples for problems with relatively complicated forms of governing equations. | Dongbin Xiu | 2007 | Communications in Computational Physics2007，2，2： | 6 |
| 2 | Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos显示文摘 | Dongbin Xiu George Em Karniadakis | 2002 | Computer Methods in Applied Mechanics and Engineering2002，，43： | 2 |
| 3 | Fast Numerical Methods for Stochastic Computations:A Review显示文摘This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations.The focus is on efficient high-order methods suitable for practical applications,with a particular emphasis on those based on generalized polynomial chaos(gPC)methodology.The framework of gPC is reviewed,along with its Galerkin and collocation approaches for solving stochastic equations.Properties of these methods are summarized by using results from literature.This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multi-dimensional random spaces. | Dongbin Xiu | 2009 | Communications in Computational Physics2009，5，2： | 2 |
| 4 | Modeling uncertainty in flow simulations via generalized polynomial chaos显示文摘 | Dongbin Xiu George Em Karniadakis | 2003 | Journal of Computational Physics2003，，1： | 1 |
| 5 | A new stochastic approach to transient heat conduction modeling with uncertainty显示文摘 | Xiu Dongbin Karniadakis G E | 2003 | International Journal of Heat and Mass Transfer2003，46，24： | 1 |
| 6 | Modeling uncertainty in flow simulations via generalized polynomial chaos显示文摘 | Dongbin Xiu Karniadakis G E | 2003 | Journal of Computational Physics2003，187，： | 1 |
| 7 | Galerkin Method for Wave Equations with Uncertain Coefficients显示文摘Polynomial chaos methods(and generalized polynomial chaos methods)have been extensively applied to analyze PDEs that contain uncertainties.However this approach is rarely applied to hyperbolic systems.In this paper we analyze the properties of the resulting deterministic system of equations obtained by stochastic Galerkin projection.We consider a simple model of a scalar wave equation with random wave speed.We show that when uncertainty causes the change of characteristic directions,the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues.A consistent method of imposing the boundary conditions is proposed and its convergence is established.Numerical examples are presented to support the analysis. | David Gottlieb Dongbin Xiu | 2008 | Communications in Computational Physics2008，3，2： | 1 |
| 8 | Modelling Uncertainty in Steady State Diffusion Problems Via Generalized Polynomial Chaos 显示文摘 | Dongbin Xiu | 2002 | Comput Methods Appl Mech Engrg2002，191，： | 1 |
| 9 | High-order collocation methods for differential equations with random inputs显示文摘 | Xiu Dongbin Jan S Hesthaven | 2005 | Siam J Sci Comput2005，27，3： | 1 |
| 10 | Stochastic solution for the two-dimensional advection-diffusion显示文摘 | WAN Xiaoliang XIU Dongbin Karniadakis George | 2003 | SIAM J Sci Comput2003，26，2： | 1 |
| 11 | The Wiener-Askey polynomial chaos for stochastic differential equations显示文摘 | XIU Dongbin KARNIADAKIS G | 2002 | SIAM Journal on Scientific Computing2002，24，2： | 1 |
| 12 | A New Stochastic Approach to Transient Heat Conduction Modeling with Uncertainty显示文摘 | Dongbin Xiu Karniadakis G E | 2003 | International Journal of Heat and Mass Transfer2003，46，24： | 1 |
| 13 | A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems显示文摘We present an efficient numerical strategy for the Bayesian solution of inverse problems.Stochastic collocation methods,based on generalized polynomial chaos(gPC),are used to construct a polynomial approximation of the forward solution over the support of the prior distribution.This approximation then defines a surrogate posterior probability density that can be evaluated repeatedly at minimal computational cost.The ability to simulate a large number of samples from the posterior distribution results in very accurate estimates of the inverse solution and its associated uncertainty.Combined with high accuracy of the gPC-based forward solver,the new algorithm can provide great efficiency in practical applications.A rigorous error analysis of the algorithm is conducted,where we establish convergence of the approximate posterior to the true posterior and obtain an estimate of the convergence rate.It is proved that fast(exponential)convergence of the gPC forward solution yields similarly fast(exponential)convergence of the posterior.The numerical strategy and the predicted convergence rates are then demonstrated on nonlinear inverse problems of varying smoothness and dimension. | Youssef Marzouk Dongbin Xiu | 2009 | Communications in Computational Physics2009，6，9： | 1 |
| 14 | Modelling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos显示文摘 | Xiu Dongbin | 2002 | Comput Methods Appl Mech Engrg2002，191，： | 1 |
| 15 | On numerical properties of the ensemble Kalman filter for data assimilation显示文摘 | Jia Li Dongbin Xiu | 2008 | Computer Methods in Applied Mechanics and Engineering2008，，： | 1 |
| 16 | The Wiener-Askey polynomial chaos for stochastic differential equations显示文摘 | Xiu Dongbin Karniadkis G E | 2002 | SIAM Journal on Scientific Computing2002，24，2： | 1 |
| 17 | An Efficient Spectral Method for Acoustic Scattering from Rough Surfaces显示文摘An efficient and accurate spectral method is presented for scattering problems with rough surfaces.A probabilistic framework is adopted by modeling the surface roughness as random process.An improved boundary perturbation technique is employed to transform the original Helmholtz equation in a random domain into a stochastic Helmholtz equation in a fixed domain.The generalized polynomial chaos(gPC)is then used to discretize the random space;and a Fourier-Legendre method to discretize the physical space.These result in a highly efficient and accurate spectral algorithm for acoustic scattering from rough surfaces.Numerical examples are presented to illustrate the accuracy and efficiency of the present algorithm. | Dongbin Xiu Jie Shen | 2007 | Communications in Computational Physics2007，2，1： | 1 |
| 18 | High-order collocation methods for differential equation with random inputs 显示文摘 | Xiu Dongbin Hesthaven J S | 2005 | SIAM J on Scientific Computing2005，27，3： | 1 |
| 19 | Reducing parameter space for neural network training显示文摘For neural networks(NNs)with rectified linear unit(ReLU)or binary activation functions,we show that their training can be accomplished in a reduced parameter space.Specifically,the weights in each neuron can be trained on the unit sphere,as opposed to the entire space,and the threshold can be trained in a bounded interval,as opposed to the real line.We show that the NNs in the reduced parameter space are mathematically equivalent to the standard NNs with parameters in the whole space.The reduced parameter space shall facilitate the optimization procedure for the network training,as the search space becomes(much)smaller.We demonstrate the improved training performance using numerical examples. | Tong Qin Ling Zhou Dongbin Xiu | 2020 | Theoretical & Applied Mechanics Letters2020，10，3： | 1 |
| 20 | Recognition and quantification of sleep apnea by analysis of heart rate variability parameters 显示文摘 | Xiu Dongbin SPENCER J | 2007 | Journal of Computational Physics2007，，5： | 1 |
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