Estimating functional uncertainty using polynomial chaos and adjoint equations

Size: px
Start display at page:

Download "Estimating functional uncertainty using polynomial chaos and adjoint equations"

Transcription

1 0. Estimating functional uncertainty using polynomial chaos and adjoint equations February 24, Florida State University, Tallahassee, Florida, Usa 2 Moscow Institute of Physics and Technology, Moscow, Russia 1 inavon@fsu.edu, 2 inavon@fsu.edu Estimating functional uncertainty using polynomial chaos and adjoint equations 1/33

2 Lecture Plan 1 The polynomial chaos expansion for uncertainty quantification 2 Intrusive polynomial chaos 3 Non-intrusive polynomial chaos 4 Adjoint acceleration of Monte-Carlo method 5 Adjoint with polynomial chaos 6 Test problem for the thermal conduction model 7 Numerical results 8 Future research directions

3 1. The polynomial chaos expansion for uncertainty quantification Polynomial Chaos Local sensitivity analysis and moment methods are suitable in the limit of small uncertainty (Cacuci,2009). Polynomial chaos, also referred to as Wiener chaos expansion is a non-sampling method used to determine evolution of uncertainty in a dynamical framework where there is probabilistic uncertaintanty in the system parameters. { } Let ξ i (w) be a set of independent standard Gaussian i=1 random variables on Ω, the sample space and P a probability measure. Estimating functional uncertainty using polynomial chaos and adjoint equations 3/33

4 1. The polynomial chaos expansion for uncertainty quantification Polynomial Chaos Than we can represent any random variable X : Ω R with finite variance, i.e. X L 2 (Ω) as i 1 X (w) = a 0 Γ 0 + a i1 Γ 1 (ξ i1 ) + a i1 i 2 Γ 2 (ξ i1, ξ i2 )+ i 1 i=1 i 2 i 1 =1 i 2 =1 i 3 =1 i 1 =1 i 2 =1 a i1 i 2 i 3 Γ 3 (ξ i1, ξ i2, ξ i3 ) +..., where Γ b is the Wiener polynomial chaos expansion (Wiener 1998) and a () R. In practice one truncates the PCE in both order p and dimension n. The orthogonality of the Γ p w.r.t. inner product < u(w), v(w) >= Ω u(w)v(w)dp requires the usage of the multivariate Hermite polynomials. Estimating functional uncertainty using polynomial chaos and adjoint equations 4/33

5 1. The polynomial chaos expansion for uncertainty quantification Polynomial Chaos Cameron and Martin (1947) proved convergence of this Wiener - Hermite PCE for a general square integrable process χ(x, t, w). Estimating functional uncertainty using polynomial chaos and adjoint equations 5/33

6 Monte-Carlo acceleration by adjoint We calculate tail probability of an objective (cost) function. J(ξ) objective function depends on a vector of uncertain variables ξ. PDF of ξ is known. J c is a known constante. The goal is the calculation of P(J > J c ). Brute force Monte-Carlo is most commonly use but is computationally expensive and inefficient. If probability is small only small fraction of samples would fall into tail region (i.e. insufficient sampling). Estimating functional uncertainty using polynomial chaos and adjoint equations 6/33

7 Monte-Carlo acceleration by adjoint P(J > J c ) P N = 1 N N i=1 I (J(ξ i) > J c ). The indicator function is defined as follow { 1 if J(ξ i ) > J c is true, I (J(ξ i ) > J c ) = 0 otherwise. Variance of the estimator P N is Var[P N ] = 1 N Var[I (J(ξ i) > J c )] = P(J>Jc) P2 (J>J c) N. Estimating functional uncertainty using polynomial chaos and adjoint equations 7/33

8 Monte-Carlo acceleration by adjoint Relative error of MC method can be characterized by ratio of standard deviation and the mean of estimator P N i.e. Var[P n ] P(J > J c ) = 1 P(J > J c ) N P(J > J c ) If tail probability is 0.1, MC samples should be used. Estimating functional uncertainty using polynomial chaos and adjoint equations 8/33

9 Adjoint sensitivity gradient for accelerating Monte-Carlo Variance reduction techniques for calculating tail probability P(J > J c ) based on a single adjoint calculation. It approximates the objective function as a linear function of the random variables describing the sources of uncertainties. Adjoint evaluated at ξ 0 and sensitivity gradient J(ξ 0 ) is obtained from the adjoint solution J(ξ) J L (ξ) = J(ξ 0 ) + J(ξ 0 ) (ξ ξ 0 ). Estimating functional uncertainty using polynomial chaos and adjoint equations 9/33

10 Adjoint sensitivity gradient for accelerating Monte-Carlo Approximate the target tail probability P(J > J C ) with the estimator P(J > J C ) P CV N = P(J L > J C )+ 1 N N (I (J(ξ i ) > J C ) I (J L (ξ i ) J C )). i=1 Estimating functional uncertainty using polynomial chaos and adjoint equations 10/33

11 Adjoint sensitivity gradient for accelerating Monte-Carlo Monte-Carlo method can give entire probability density function of any system variable - but is too expensive since a large number of samples are required for reassemble accuracy. For small errors the variance of a cost functional J may be calculated from the input data f i variances using: σ 2 l = N i=1 ( J f i σ fi ) 2. The adjoint method enables time for J f i calculation to be approximately double the time of J calculation. The CPU burden does not depend on parameters containing the errors, thus providing high computational efficiency. Estimating functional uncertainty using polynomial chaos and adjoint equations 11/33

12 If input data f i error probability distribution is know, we may calculate the J error distribution and moment using adjoint - gradients serving as a rapid meta model. J(f ) = J( f ) + i J f i fi. If the adjoint problem is solved and the gradient is stored, we may deal with a very big ensemble for distribution and moments calculation without any computational burden. See Alekseev and Navon (2003) The second order adjoint - The adjoint MC may not be appropriate in vicinity of the optimal solution where the gradient is zero. Second order estimations may be used for functional shift J = J( f ) N i=1 2 J fi, 2 σf 2 i,. Estimating functional uncertainty using polynomial chaos and adjoint equations 12/33

13 Adjoint sensitivity gradient for accelerating Monte-Carlo The variance may be approximated with σ 2 y = 1 2 ( 2 J( f ) ) 2. σ fi, σ fj, f i, f j, i These estimations are governed by the Hessian j H i,j = 2 J f i, f j,, which may be calculated via the second order adjoint problem. Estimating functional uncertainty using polynomial chaos and adjoint equations 13/33

14 Uncertainty Quantification (UQ) Uncertainty in model parameters UQ has 2 intimately coupled components. First pertains to forward propagation of uncertainty from model parameters to model outputs. Second component involves estimation of parametric uncertainties themselves based on available data Estimating functional uncertainty using polynomial chaos and adjoint equations 14/33

15 UQ methods Local sensitivity analysis and moment methods (Cacuci 2003) suitable in the limit of small uncertainty. For large degrees of uncertainty - methods using probability theory including polynomial chaos (PC) amongst other methods. Uncertainty can be epistemic when it results from lack of knowledge about quantity whose true value exhibits no actual variability. Uncertainty resulting from variability is termed aleatoric. Both epistemic and aleatoric uncertainties can be handled using probability theory in Bayesian framework. Estimating functional uncertainty using polynomial chaos and adjoint equations 15/33

16 Polynomial Chaos 8. Future research directions The underlying concept of polynomial chaos is the uncertainty in a random parameter can be represented as a series of orthogonal polynomials (Wiener 1938) This representation of the stochastic process will converge to the true uncertainty of the parameter (Cameron and Martin 1947). The Polynomial Chais (PC) involves a probability space (Ω, Σ, P) where Ω is a sample space, Σ is a σ - algebra on Ω and P is a probability measure on (Ω, σ). Let {ξ i (w)} i=1 be a set of independent standard Gaussian random variables (RVs) on Ω. Estimating functional uncertainty using polynomial chaos and adjoint equations 16/33

17 Polynomial Chaos Then we can represent any RV X : Ω R with finite variance, i.e., x L 2 (Ω) as X (w) = a 0 Γ 0 + a i1 Γ 1 (ξ i1 ) + i 1 i=1 i 2 i 1 =1 i 2 =1 i 3 =1 i 1 i 1 =1 i 2 =1 a i1 i 2 i 3 Γ 3 (ξ i1, ξ i2, ξ i3 ) +..., where Γ p is the Wiener PC of order p. This is a polynomial chaos expansion (PCE) X (w) = a k Ψ k (ξ 1, ξ 2,...). i=1 a i1 i 2 Γ 2 (ξ i1, ξ i2 )+ Estimating functional uncertainty using polynomial chaos and adjoint equations 17/33

18 Polynomial Chaos 8. Future research directions The orthogonality of the Γ p (or Ψ k ) w.r.t. inner product < u(w), v(w) >= Ω u(w)v(w)dp requires they be multivariate Hermite polynomials. Cameron and Martin (1947) proved the convergence of the Wiener - Hermite (WH) PCE for a general square integrable stochastic process X (X, t, w). In practical computational context one truncates the PCE in both order p and dimension n. item Next we introduce two different approaches for solving differential equations involving uncertain parameters using PCE. Estimating functional uncertainty using polynomial chaos and adjoint equations 18/33

19 Polynomial Chaos Intrusive method - we substitute PCE of the solution into the differential equation. Then by a Galerkin projection in the probability space we obtain a set of coupled equations governing the coefficients of the PCE of the solution. In a second approach, the non - intrusive method (based on approximating the PCE - we do not substitute the PCE into the differential equation - but use samples of the solutions of the differential equations to construct the coefficients of the PCE. Unlike the intrusive approach - this approach does not require solving a different (and usually more complicated) set of coefficient equations. This approach can use existing software and codes of solving deterministic differential equations. Estimating functional uncertainty using polynomial chaos and adjoint equations 19/33

20 Large-errors - Adjoint Polynomial Chaos First stage: Variables (providing major input in error of cost function) are determined using gradient - obtained via adjoint equations. This allows reducing dimension of random variables space Second stage: Coefficient of PCE or Hermite polynomials are determined using a least - squares nonintrusive variant of (PC). In a final stage - moments and PDF are obtained using the PC expansion (for large - errors) and adjoint based gradient for small errors. Input data parameters containing errors are expressed as f i =< f i > +ξ i σ i, where ξ i are normally distributed random variables with unit variance and σ i is the standard deviation of f i. Estimating functional uncertainty using polynomial chaos and adjoint equations 20/33

21 Large-errors - Adjoint Polynomial Chaos The linear estimate of the uncertainty of the functional J is represented as N J = i Jσ i ξ i, i=1 where the gradient i J is obtained using the adjoint method. New random values (RV) ξ i which satisfy the condition i Jσ i > σ, where σ is a certain critical magnitude of the error - are taken to be the leading variables providing main input to the total error of the functional. Estimating functional uncertainty using polynomial chaos and adjoint equations 21/33

22 Large-errors - Adjoint Polynomial Chaos To estimate nonlinear impact of large errors the cost functional as function of RVs is expand over Hermite polynomials as J = b 0 + N b i1 Hi 1 1 (ξ i1 ) + i 1 =1 N i 1 i 2 i 1 =1 i 2 =1 i 3 =1 N i 1 i 1 =1 i 2 =1 b i1 i 2 i 3 H 3 i 1 i 2 i 3 (ξ i1, ξ i2, ξ i3 ), b i1 i 2 H 2 i 1 i 2 (ξ i1, ξ i2 )+ where H p i 1...i n (ξ 1,..., ξ n ) = e ξ ξ/2 ( 1) p p ( ξ 1 ) i 1...( ξ n) in e ξ ξ/2, with ξ ξ/2 = (ξ ξ2 N )/2. Estimating functional uncertainty using polynomial chaos and adjoint equations 22/33

23 Large-errors - Adjoint Polynomial Chaos We used Hermite polynomials of the third and fourth order only. The Cameron - Martin theorem allows calculation of the mean and variance M E[y(ξ)] = b 0, σy 2 = bi 2. i=1 The random variables space is divided into 2 subspace ξ R Nm ξ R N Nm large errors small errors So the expansion is truncated assuming the following form N N i 1 J = b 0 + b i1 Hi 1 1 (ξ i1 ) + b i1 i 2 H 2 (ξ i 1 i i1, ξ i2 )+ 2 i 1 =1 N i 1 i 2 i 1 =1 i 2 =1 Estimating functional uncertainty using polynomial chaos 3 and adjoint equations 1 23/33

24 Large-errors - Adjoint Polynomial Chaos So the expansion is truncated assuming the following form J = b 0 + N i 1 N b i1 Hi 1 1 (ξ i1 ) + i 1 =1 i 2 i 1 =1 i 2 =1 i 3 =1 N i 1 i 1 =1 i 2 =1 b i1 i 2 i 3 H 3 i 1 i 2 i 3 (ξ i1, ξ i2, ξ i3 ) + b i1 i 2 H 2 i 1 i 2 (ξ i1, ξ i2 )+ N i 1 =N m b i1 H 1 i 1 (ξ i1 ). The first 3 terms account for large errors while the last term represents small errors and may be stated as N i=n m i Jσ i ξ i providing possibility of fast calculations using adjoint based gradient. Estimating functional uncertainty using polynomial chaos and adjoint equations 24/33

25 Large-errors - Adjoint Polynomial Chaos The adjoint equations enable us to calculate the gradient (N variables) of the cost functional by running a code whose computational cost is close to that of the forward code. This can accelerate computation of PC coefficients for large N. The adjoint PC approach provides approximately an N fold reduction of the CPU time when compared with the pure PC. Estimating functional uncertainty using polynomial chaos and adjoint equations 25/33

26 Test problems 8. Future research directions Let us consider the following conduction model c T t ( λ T ) = 0, x x with T (0, t) = T w, T (X, t) = T R, boundary conditions and initial condition T (x, 0) = T 0 (x). T(x,t) denotes temperature, λ is the thermal conductivity and X states for the specimen thickness. The cost functional has the following form J = = T δ(x x est )δ(t t test )dxdt, with S = (0, X ) (0, T ). S Estimating functional uncertainty using polynomial chaos and adjoint equations 26/33

27 Test problems 8. Future research directions x est and t est are coordinates of pointwise temperature estimation, i.e. temperatures at important moments in a sensitive location. We look for moments and pdf of functional caused by a normally distributed error in the thermal conductivity or in the initial state. To calculate the gradient we use the adjoint function Ψ(x, t) obtained by solving adjoint problem c Psi t + λ 2 Ψ x 2 δ(t t est)δ(x x est ) = 0, with the following boundary and final conditions: Ψ Ψ x = 0, x=0 x = 0, Ψ(x, t f ) = 0. x=x Estimating functional uncertainty using polynomial chaos and adjoint equations 27/33

28 Test problems 8. Future research directions The gradient of the cost functional w.r.t. initial temperature is J = Ψ(x, 0), and regarding conductivity coefficient is ( 2 ) T Ψ x 2 dxdt. S This gradient is connected with derivatives of functional over random variables (RVs) in T 0 : and in λ: J(ξ (m) ) ξ i = J T 0,i T 0 ξ i = Ψ(x i, 0)σ i, J = J λ = ξ i λ ξ i S Ψ 2 T x 2 σ idxdt. Estimating functional uncertainty using polynomial chaos and adjoint equations 28/33

29 Test problems We consider 3 test problems 1) We consider 4 random variables in the thermal conductivity λ = λ + σ i ξ i. Here ξ i are RVs with normal distribution of unit variance. The variation of cost function is δj = J < J >= T 0 S i Ψ 2 T x 2 σ iξ i dxdt. 2) The initial data contain normally distributed error T 0 (x i ) + δt 0 (x i ) = T 0 (x i ) + σ i ξ i. In this test when λ(t ) = λ 1 = Const and J = Ψ(x, 0) does not depend on T 0 - i.e. gradient info cannot be used in PC and adjoint PC cannot be applied. Estimating functional uncertainty using polynomial chaos and adjoint equations 29/33

30 Test problems 3) Here the thermal conductivity is a nonlinear function assuming form λ(t ) = λ 0 + λ 1 (T /T s ) 3 fort < 0, λ(t ) = λ 0. and initial data contain a normally distributed error T 0 (x i ) + δt 0 (x i ) = T 0 (x i ) + σ i ξ i Estimating functional uncertainty using polynomial chaos and adjoint equations 30/33

31 Numerical results 8. Future research directions Heat transfer equations and adjoint equations were solved by conservative finite differences scheme of 2 nd order over space and time. Numerical tests: Monte - Carlo (MC) Monte - Carlo using adjoint acceleration (AMC) Nonintrusive Polynomial Chaos (PC) Polynomial Chaos using adjoint equations (APC) Estimating functional uncertainty using polynomial chaos and adjoint equations 31/33

32 Error in thermal conductivity Fig. 1. Probability density. 1 -MC, 2 -AMC, 3-PC 1.2E-02 P(T) 1.0E E E E E E T Estimating functional uncertainty using polynomial chaos and adjoint equations 32/33

33 Error in thermal conductivity Thermal conductivity contained 4 normally distributed random variables with standard deviation equal 0.1 λ. PC implemented using third order Hermite polynomials. All tests used samples. Comparison of mean values ε and standard deviation σ is shown in next table. Table 1 MC AMC PC PC MC averaged PC coeff. (4) ε std. dev. σ Also shown in this table is the computational cost (CPU normalized) by time of single problem run. Estimating functional uncertainty using polynomial chaos and adjoint equations 33/33

The estimation of functional uncertainty using polynomial chaos and adjoint equations

The estimation of functional uncertainty using polynomial chaos and adjoint equations INERNAIONAL JOURNAL FOR NUMERICAL MEHODS IN FLUIDS Int J Numer Meth Fluids (200) Published online in Wiley InterScience (wwwintersciencewileycom) DOI: 0002/fld2355 he estimation of functional uncertainty

More information

Polynomial chaos expansions for sensitivity analysis

Polynomial chaos expansions for sensitivity analysis c DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Polynomial chaos expansions for sensitivity analysis B. Sudret Chair of Risk, Safety & Uncertainty

More information

ON ESTIMATION OF TEMPERATURE UNCERTAINTY USING THE SECOND ORDER ADJOINT PROBLEM

ON ESTIMATION OF TEMPERATURE UNCERTAINTY USING THE SECOND ORDER ADJOINT PROBLEM ON ESTIMATION OF TEMPERATURE UNCERTAINTY USING THE SECOND ORDER ADJOINT PROBLEM Aleksey K. Alekseev a, Michael I. Navon b,* a Department of Aerodynamics and Heat Transfer, RSC, ENERGIA, Korolev (Kaliningrad),

More information

A Stochastic Collocation based. for Data Assimilation

A Stochastic Collocation based. for Data Assimilation A Stochastic Collocation based Kalman Filter (SCKF) for Data Assimilation Lingzao Zeng and Dongxiao Zhang University of Southern California August 11, 2009 Los Angeles Outline Introduction SCKF Algorithm

More information

Polynomial chaos expansions for structural reliability analysis

Polynomial chaos expansions for structural reliability analysis DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Polynomial chaos expansions for structural reliability analysis B. Sudret & S. Marelli Incl.

More information

Enabling Advanced Automation Tools to manage Trajectory Prediction Uncertainty

Enabling Advanced Automation Tools to manage Trajectory Prediction Uncertainty Engineering, Test & Technology Boeing Research & Technology Enabling Advanced Automation Tools to manage Trajectory Prediction Uncertainty ART 12 - Automation Enrique Casado (BR&T-E) enrique.casado@boeing.com

More information

Algorithms for Uncertainty Quantification

Algorithms for Uncertainty Quantification Algorithms for Uncertainty Quantification Lecture 9: Sensitivity Analysis ST 2018 Tobias Neckel Scientific Computing in Computer Science TUM Repetition of Previous Lecture Sparse grids in Uncertainty Quantification

More information

Uncertainty Quantification in MEMS

Uncertainty Quantification in MEMS Uncertainty Quantification in MEMS N. Agarwal and N. R. Aluru Department of Mechanical Science and Engineering for Advanced Science and Technology Introduction Capacitive RF MEMS switch Comb drive Various

More information

Uncertainty Quantification in Computational Science

Uncertainty Quantification in Computational Science DTU 2010 - Lecture I Uncertainty Quantification in Computational Science Jan S Hesthaven Brown University Jan.Hesthaven@Brown.edu Objective of lectures The main objective of these lectures are To offer

More information

Performance Evaluation of Generalized Polynomial Chaos

Performance Evaluation of Generalized Polynomial Chaos Performance Evaluation of Generalized Polynomial Chaos Dongbin Xiu, Didier Lucor, C.-H. Su, and George Em Karniadakis 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA, gk@dam.brown.edu

More information

CERTAIN THOUGHTS ON UNCERTAINTY ANALYSIS FOR DYNAMICAL SYSTEMS

CERTAIN THOUGHTS ON UNCERTAINTY ANALYSIS FOR DYNAMICAL SYSTEMS CERTAIN THOUGHTS ON UNCERTAINTY ANALYSIS FOR DYNAMICAL SYSTEMS Puneet Singla Assistant Professor Department of Mechanical & Aerospace Engineering University at Buffalo, Buffalo, NY-1426 Probabilistic Analysis

More information

EFFICIENT SHAPE OPTIMIZATION USING POLYNOMIAL CHAOS EXPANSION AND LOCAL SENSITIVITIES

EFFICIENT SHAPE OPTIMIZATION USING POLYNOMIAL CHAOS EXPANSION AND LOCAL SENSITIVITIES 9 th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability EFFICIENT SHAPE OPTIMIZATION USING POLYNOMIAL CHAOS EXPANSION AND LOCAL SENSITIVITIES Nam H. Kim and Haoyu Wang University

More information

Stochastic Spectral Approaches to Bayesian Inference

Stochastic Spectral Approaches to Bayesian Inference Stochastic Spectral Approaches to Bayesian Inference Prof. Nathan L. Gibson Department of Mathematics Applied Mathematics and Computation Seminar March 4, 2011 Prof. Gibson (OSU) Spectral Approaches to

More information

Employing Model Reduction for Uncertainty Visualization in the Context of CO 2 Storage Simulation

Employing Model Reduction for Uncertainty Visualization in the Context of CO 2 Storage Simulation Employing Model Reduction for Uncertainty Visualization in the Context of CO 2 Storage Simulation Marcel Hlawatsch, Sergey Oladyshkin, Daniel Weiskopf University of Stuttgart Problem setting - underground

More information

A reduced-order stochastic finite element analysis for structures with uncertainties

A reduced-order stochastic finite element analysis for structures with uncertainties A reduced-order stochastic finite element analysis for structures with uncertainties Ji Yang 1, Béatrice Faverjon 1,2, Herwig Peters 1, icole Kessissoglou 1 1 School of Mechanical and Manufacturing Engineering,

More information

Introduction to Uncertainty Quantification in Computational Science Handout #3

Introduction to Uncertainty Quantification in Computational Science Handout #3 Introduction to Uncertainty Quantification in Computational Science Handout #3 Gianluca Iaccarino Department of Mechanical Engineering Stanford University June 29 - July 1, 2009 Scuola di Dottorato di

More information

SENSITIVITY ANALYSIS IN NUMERICAL SIMULATION OF MULTIPHASE FLOW FOR CO 2 STORAGE IN SALINE AQUIFERS USING THE PROBABILISTIC COLLOCATION APPROACH

SENSITIVITY ANALYSIS IN NUMERICAL SIMULATION OF MULTIPHASE FLOW FOR CO 2 STORAGE IN SALINE AQUIFERS USING THE PROBABILISTIC COLLOCATION APPROACH XIX International Conference on Water Resources CMWR 2012 University of Illinois at Urbana-Champaign June 17-22,2012 SENSITIVITY ANALYSIS IN NUMERICAL SIMULATION OF MULTIPHASE FLOW FOR CO 2 STORAGE IN

More information

Dinesh Kumar, Mehrdad Raisee and Chris Lacor

Dinesh Kumar, Mehrdad Raisee and Chris Lacor Dinesh Kumar, Mehrdad Raisee and Chris Lacor Fluid Mechanics and Thermodynamics Research Group Vrije Universiteit Brussel, BELGIUM dkumar@vub.ac.be; m_raisee@yahoo.com; chris.lacor@vub.ac.be October, 2014

More information

NON-LINEAR APPROXIMATION OF BAYESIAN UPDATE

NON-LINEAR APPROXIMATION OF BAYESIAN UPDATE tifica NON-LINEAR APPROXIMATION OF BAYESIAN UPDATE Alexander Litvinenko 1, Hermann G. Matthies 2, Elmar Zander 2 http://sri-uq.kaust.edu.sa/ 1 Extreme Computing Research Center, KAUST, 2 Institute of Scientific

More information

Model Calibration under Uncertainty: Matching Distribution Information

Model Calibration under Uncertainty: Matching Distribution Information Model Calibration under Uncertainty: Matching Distribution Information Laura P. Swiler, Brian M. Adams, and Michael S. Eldred September 11, 008 AIAA Multidisciplinary Analysis and Optimization Conference

More information

Hyperbolic Polynomial Chaos Expansion (HPCE) and its Application to Statistical Analysis of Nonlinear Circuits

Hyperbolic Polynomial Chaos Expansion (HPCE) and its Application to Statistical Analysis of Nonlinear Circuits Hyperbolic Polynomial Chaos Expansion HPCE and its Application to Statistical Analysis of Nonlinear Circuits Majid Ahadi, Aditi Krishna Prasad, Sourajeet Roy High Speed System Simulations Laboratory Department

More information

Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos

Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 6t 7 - April 28, Schaumburg, IL AIAA 28-892 Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized

More information

Uncertainty Propagation and Global Sensitivity Analysis in Hybrid Simulation using Polynomial Chaos Expansion

Uncertainty Propagation and Global Sensitivity Analysis in Hybrid Simulation using Polynomial Chaos Expansion Uncertainty Propagation and Global Sensitivity Analysis in Hybrid Simulation using Polynomial Chaos Expansion EU-US-Asia workshop on hybrid testing Ispra, 5-6 October 2015 G. Abbiati, S. Marelli, O.S.

More information

Parametric Problems, Stochastics, and Identification

Parametric Problems, Stochastics, and Identification Parametric Problems, Stochastics, and Identification Hermann G. Matthies a B. Rosić ab, O. Pajonk ac, A. Litvinenko a a, b University of Kragujevac c SPT Group, Hamburg wire@tu-bs.de http://www.wire.tu-bs.de

More information

Uncertainty Evolution In Stochastic Dynamic Models Using Polynomial Chaos

Uncertainty Evolution In Stochastic Dynamic Models Using Polynomial Chaos Noname manuscript No. (will be inserted by the editor) Uncertainty Evolution In Stochastic Dynamic Models Using Polynomial Chaos Umamaheswara Konda Puneet Singla Tarunraj Singh Peter Scott Received: date

More information

PC EXPANSION FOR GLOBAL SENSITIVITY ANALYSIS OF NON-SMOOTH FUNCTIONALS OF UNCERTAIN STOCHASTIC DIFFERENTIAL EQUATIONS SOLUTIONS

PC EXPANSION FOR GLOBAL SENSITIVITY ANALYSIS OF NON-SMOOTH FUNCTIONALS OF UNCERTAIN STOCHASTIC DIFFERENTIAL EQUATIONS SOLUTIONS PC EXPANSION FOR GLOBAL SENSITIVITY ANALYSIS OF NON-SMOOTH FUNCTIONALS OF UNCERTAIN STOCHASTIC DIFFERENTIAL EQUATIONS SOLUTIONS M. Navarro, O.P. Le Maître,2, O.M. Knio,3 mariaisabel.navarrojimenez@kaust.edu.sa

More information

Polynomial Chaos and Karhunen-Loeve Expansion

Polynomial Chaos and Karhunen-Loeve Expansion Polynomial Chaos and Karhunen-Loeve Expansion 1) Random Variables Consider a system that is modeled by R = M(x, t, X) where X is a random variable. We are interested in determining the probability of the

More information

Beyond Wiener Askey Expansions: Handling Arbitrary PDFs

Beyond Wiener Askey Expansions: Handling Arbitrary PDFs Journal of Scientific Computing, Vol. 27, Nos. 1 3, June 2006 ( 2005) DOI: 10.1007/s10915-005-9038-8 Beyond Wiener Askey Expansions: Handling Arbitrary PDFs Xiaoliang Wan 1 and George Em Karniadakis 1

More information

Uncertainty quantification for flow in highly heterogeneous porous media

Uncertainty quantification for flow in highly heterogeneous porous media 695 Uncertainty quantification for flow in highly heterogeneous porous media D. Xiu and D.M. Tartakovsky a a Theoretical Division, Los Alamos National Laboratory, Mathematical Modeling and Analysis Group

More information

Fast Numerical Methods for Stochastic Computations

Fast Numerical Methods for Stochastic Computations Fast AreviewbyDongbinXiu May 16 th,2013 Outline Motivation 1 Motivation 2 3 4 5 Example: Burgers Equation Let us consider the Burger s equation: u t + uu x = νu xx, x [ 1, 1] u( 1) =1 u(1) = 1 Example:

More information

Accepted Manuscript. SAMBA: Sparse approximation of moment-based arbitrary polynomial chaos. R. Ahlfeld, B. Belkouchi, F.

Accepted Manuscript. SAMBA: Sparse approximation of moment-based arbitrary polynomial chaos. R. Ahlfeld, B. Belkouchi, F. Accepted Manuscript SAMBA: Sparse approximation of moment-based arbitrary polynomial chaos R. Ahlfeld, B. Belkouchi, F. Montomoli PII: S0021-9991(16)30151-6 DOI: http://dx.doi.org/10.1016/j.jcp.2016.05.014

More information

Winter 2019 Math 106 Topics in Applied Mathematics. Lecture 1: Introduction

Winter 2019 Math 106 Topics in Applied Mathematics. Lecture 1: Introduction Winter 2019 Math 106 Topics in Applied Mathematics Data-driven Uncertainty Quantification Yoonsang Lee (yoonsang.lee@dartmouth.edu) Lecture 1: Introduction 19 Winter M106 Class: MWF 12:50-1:55 pm @ 200

More information

A Polynomial Chaos Approach to Robust Multiobjective Optimization

A Polynomial Chaos Approach to Robust Multiobjective Optimization A Polynomial Chaos Approach to Robust Multiobjective Optimization Silvia Poles 1, Alberto Lovison 2 1 EnginSoft S.p.A., Optimization Consulting Via Giambellino, 7 35129 Padova, Italy s.poles@enginsoft.it

More information

Sampling and low-rank tensor approximation of the response surface

Sampling and low-rank tensor approximation of the response surface Sampling and low-rank tensor approximation of the response surface tifica Alexander Litvinenko 1,2 (joint work with Hermann G. Matthies 3 ) 1 Group of Raul Tempone, SRI UQ, and 2 Group of David Keyes,

More information

Robust Optimal Control using Polynomial Chaos and Adjoints for Systems with Uncertain Inputs

Robust Optimal Control using Polynomial Chaos and Adjoints for Systems with Uncertain Inputs 2th AIAA Computational Fluid Dynamics Conference 27-3 June 211, Honolulu, Hawaii AIAA 211-369 Robust Optimal Control using Polynomial Chaos and Adjoints for Systems with Uncertain Inputs Sriram General

More information

Stochastic Collocation Methods for Polynomial Chaos: Analysis and Applications

Stochastic Collocation Methods for Polynomial Chaos: Analysis and Applications Stochastic Collocation Methods for Polynomial Chaos: Analysis and Applications Dongbin Xiu Department of Mathematics, Purdue University Support: AFOSR FA955-8-1-353 (Computational Math) SF CAREER DMS-64535

More information

Sobol-Hoeffding Decomposition with Application to Global Sensitivity Analysis

Sobol-Hoeffding Decomposition with Application to Global Sensitivity Analysis Sobol-Hoeffding decomposition Application to Global SA Computation of the SI Sobol-Hoeffding Decomposition with Application to Global Sensitivity Analysis Olivier Le Maître with Colleague & Friend Omar

More information

A Non-Intrusive Polynomial Chaos Method For Uncertainty Propagation in CFD Simulations

A Non-Intrusive Polynomial Chaos Method For Uncertainty Propagation in CFD Simulations An Extended Abstract submitted for the 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada January 26 Preferred Session Topic: Uncertainty quantification and stochastic methods for CFD A Non-Intrusive

More information

An Empirical Chaos Expansion Method for Uncertainty Quantification

An Empirical Chaos Expansion Method for Uncertainty Quantification An Empirical Chaos Expansion Method for Uncertainty Quantification Melvin Leok and Gautam Wilkins Abstract. Uncertainty quantification seeks to provide a quantitative means to understand complex systems

More information

Keywords: Sonic boom analysis, Atmospheric uncertainties, Uncertainty quantification, Monte Carlo method, Polynomial chaos method.

Keywords: Sonic boom analysis, Atmospheric uncertainties, Uncertainty quantification, Monte Carlo method, Polynomial chaos method. Blucher Mechanical Engineering Proceedings May 2014, vol. 1, num. 1 www.proceedings.blucher.com.br/evento/10wccm SONIC BOOM ANALYSIS UNDER ATMOSPHERIC UNCERTAINTIES BY A NON-INTRUSIVE POLYNOMIAL CHAOS

More information

Probabilistic Collocation Method for Uncertainty Analysis of Soil Infiltration in Flood Modelling

Probabilistic Collocation Method for Uncertainty Analysis of Soil Infiltration in Flood Modelling Probabilistic Collocation Method for Uncertainty Analysis of Soil Infiltration in Flood Modelling Y. Huang 1,2, and X.S. Qin 1,2* 1 School of Civil & Environmental Engineering, Nanyang Technological University,

More information

UNCERTAINTY ASSESSMENT USING STOCHASTIC REDUCED BASIS METHOD FOR FLOW IN POROUS MEDIA

UNCERTAINTY ASSESSMENT USING STOCHASTIC REDUCED BASIS METHOD FOR FLOW IN POROUS MEDIA UNCERTAINTY ASSESSMENT USING STOCHASTIC REDUCED BASIS METHOD FOR FLOW IN POROUS MEDIA A REPORT SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT

More information

Stochastic structural dynamic analysis with random damping parameters

Stochastic structural dynamic analysis with random damping parameters Stochastic structural dynamic analysis with random damping parameters K. Sepahvand 1, F. Saati Khosroshahi, C. A. Geweth and S. Marburg Chair of Vibroacoustics of Vehicles and Machines Department of Mechanical

More information

Optimization Tools in an Uncertain Environment

Optimization Tools in an Uncertain Environment Optimization Tools in an Uncertain Environment Michael C. Ferris University of Wisconsin, Madison Uncertainty Workshop, Chicago: July 21, 2008 Michael Ferris (University of Wisconsin) Stochastic optimization

More information

A Spectral Approach to Linear Bayesian Updating

A Spectral Approach to Linear Bayesian Updating A Spectral Approach to Linear Bayesian Updating Oliver Pajonk 1,2, Bojana V. Rosic 1, Alexander Litvinenko 1, and Hermann G. Matthies 1 1 Institute of Scientific Computing, TU Braunschweig, Germany 2 SPT

More information

Kinematic Analysis and Inverse Dynamics-based Control of Nondeterministic Multibody Systems

Kinematic Analysis and Inverse Dynamics-based Control of Nondeterministic Multibody Systems Kinematic Analysis and Inverse Dynamics-based Control of Nondeterministic Multibody Systems Item Type text; Electronic Thesis Authors Sabet, Sahand Publisher The University of Arizona. Rights Copyright

More information

Risk Assessment for CO 2 Sequestration- Uncertainty Quantification based on Surrogate Models of Detailed Simulations

Risk Assessment for CO 2 Sequestration- Uncertainty Quantification based on Surrogate Models of Detailed Simulations Risk Assessment for CO 2 Sequestration- Uncertainty Quantification based on Surrogate Models of Detailed Simulations Yan Zhang Advisor: Nick Sahinidis Department of Chemical Engineering Carnegie Mellon

More information

Uncertainty Quantification in Computational Models

Uncertainty Quantification in Computational Models Uncertainty Quantification in Computational Models Habib N. Najm Sandia National Laboratories, Livermore, CA, USA Workshop on Understanding Climate Change from Data (UCC11) University of Minnesota, Minneapolis,

More information

A Note on the Particle Filter with Posterior Gaussian Resampling

A Note on the Particle Filter with Posterior Gaussian Resampling Tellus (6), 8A, 46 46 Copyright C Blackwell Munksgaard, 6 Printed in Singapore. All rights reserved TELLUS A Note on the Particle Filter with Posterior Gaussian Resampling By X. XIONG 1,I.M.NAVON 1,2 and

More information

Stochastic Dimension Reduction

Stochastic Dimension Reduction Stochastic Dimension Reduction Roger Ghanem University of Southern California Los Angeles, CA, USA Computational and Theoretical Challenges in Interdisciplinary Predictive Modeling Over Random Fields 12th

More information

Spectral methods for fuzzy structural dynamics: modal vs direct approach

Spectral methods for fuzzy structural dynamics: modal vs direct approach Spectral methods for fuzzy structural dynamics: modal vs direct approach S Adhikari Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Wales, UK IUTAM Symposium

More information

PARALLEL COMPUTATION OF 3D WAVE PROPAGATION BY SPECTRAL STOCHASTIC FINITE ELEMENT METHOD

PARALLEL COMPUTATION OF 3D WAVE PROPAGATION BY SPECTRAL STOCHASTIC FINITE ELEMENT METHOD 13 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 24 Paper No. 569 PARALLEL COMPUTATION OF 3D WAVE PROPAGATION BY SPECTRAL STOCHASTIC FINITE ELEMENT METHOD Riki Honda

More information

Predictive Engineering and Computational Sciences. Local Sensitivity Derivative Enhanced Monte Carlo Methods. Roy H. Stogner, Vikram Garg

Predictive Engineering and Computational Sciences. Local Sensitivity Derivative Enhanced Monte Carlo Methods. Roy H. Stogner, Vikram Garg PECOS Predictive Engineering and Computational Sciences Local Sensitivity Derivative Enhanced Monte Carlo Methods Roy H. Stogner, Vikram Garg Institute for Computational Engineering and Sciences The University

More information

Spectral Propagation of Parameter Uncertainties in Water Distribution Networks

Spectral Propagation of Parameter Uncertainties in Water Distribution Networks Spectral Propagation of Parameter Uncertainties in Water Distribution Networks M. Braun, O. Piller, J. Deuerlein, I. Mortazavi To cite this version: M. Braun, O. Piller, J. Deuerlein, I. Mortazavi. Spectral

More information

Uncertainty Quantification of Radionuclide Release Models using Non-Intrusive Polynomial Chaos. Casper Hoogwerf

Uncertainty Quantification of Radionuclide Release Models using Non-Intrusive Polynomial Chaos. Casper Hoogwerf Uncertainty Quantification of Radionuclide Release Models using Non-Intrusive Polynomial Chaos. Casper Hoogwerf 1 Foreword This report presents the final thesis of the Master of Science programme in Applied

More information

Sparse polynomial chaos expansions in engineering applications

Sparse polynomial chaos expansions in engineering applications DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Sparse polynomial chaos expansions in engineering applications B. Sudret G. Blatman (EDF R&D,

More information

Intelligent Embedded Systems Uncertainty, Information and Learning Mechanisms (Part 1)

Intelligent Embedded Systems Uncertainty, Information and Learning Mechanisms (Part 1) Advanced Research Intelligent Embedded Systems Uncertainty, Information and Learning Mechanisms (Part 1) Intelligence for Embedded Systems Ph. D. and Master Course Manuel Roveri Politecnico di Milano,

More information

On the Nature of Random System Matrices in Structural Dynamics

On the Nature of Random System Matrices in Structural Dynamics On the Nature of Random System Matrices in Structural Dynamics S. ADHIKARI AND R. S. LANGLEY Cambridge University Engineering Department Cambridge, U.K. Nature of Random System Matrices p.1/20 Outline

More information

Overview. Bayesian assimilation of experimental data into simulation (for Goland wing flutter) Why not uncertainty quantification?

Overview. Bayesian assimilation of experimental data into simulation (for Goland wing flutter) Why not uncertainty quantification? Delft University of Technology Overview Bayesian assimilation of experimental data into simulation (for Goland wing flutter), Simao Marques 1. Why not uncertainty quantification? 2. Why uncertainty quantification?

More information

Benjamin L. Pence 1, Hosam K. Fathy 2, and Jeffrey L. Stein 3

Benjamin L. Pence 1, Hosam K. Fathy 2, and Jeffrey L. Stein 3 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 WeC17.1 Benjamin L. Pence 1, Hosam K. Fathy 2, and Jeffrey L. Stein 3 (1) Graduate Student, (2) Assistant

More information

Quantifying conformation fluctuation induced uncertainty in bio-molecular systems

Quantifying conformation fluctuation induced uncertainty in bio-molecular systems Quantifying conformation fluctuation induced uncertainty in bio-molecular systems Guang Lin, Dept. of Mathematics & School of Mechanical Engineering, Purdue University Collaborative work with Huan Lei,

More information

Uncertainty analysis of large-scale systems using domain decomposition

Uncertainty analysis of large-scale systems using domain decomposition Center for Turbulence Research Annual Research Briefs 2007 143 Uncertainty analysis of large-scale systems using domain decomposition By D. Ghosh, C. Farhat AND P. Avery 1. Motivation and objectives A

More information

At A Glance. UQ16 Mobile App.

At A Glance. UQ16 Mobile App. At A Glance UQ16 Mobile App Scan the QR code with any QR reader and download the TripBuilder EventMobile app to your iphone, ipad, itouch or Android mobile device. To access the app or the HTML 5 version,

More information

Computational methods for uncertainty quantification and sensitivity analysis of complex systems

Computational methods for uncertainty quantification and sensitivity analysis of complex systems DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Computational methods for uncertainty quantification and sensitivity analysis of complex systems

More information

Dynamic response of structures with uncertain properties

Dynamic response of structures with uncertain properties Dynamic response of structures with uncertain properties S. Adhikari 1 1 Chair of Aerospace Engineering, College of Engineering, Swansea University, Bay Campus, Fabian Way, Swansea, SA1 8EN, UK International

More information

Uncertainty Management and Quantification in Industrial Analysis and Design

Uncertainty Management and Quantification in Industrial Analysis and Design Uncertainty Management and Quantification in Industrial Analysis and Design www.numeca.com Charles Hirsch Professor, em. Vrije Universiteit Brussel President, NUMECA International The Role of Uncertainties

More information

TECHNISCHE UNIVERSITÄT MÜNCHEN. Uncertainty Quantification in Fluid Flows via Polynomial Chaos Methodologies

TECHNISCHE UNIVERSITÄT MÜNCHEN. Uncertainty Quantification in Fluid Flows via Polynomial Chaos Methodologies TECHNISCHE UNIVERSITÄT MÜNCHEN Bachelor s Thesis in Engineering Science Uncertainty Quantification in Fluid Flows via Polynomial Chaos Methodologies Jan Sültemeyer DEPARTMENT OF INFORMATICS MUNICH SCHOOL

More information

Simulating with uncertainty : the rough surface scattering problem

Simulating with uncertainty : the rough surface scattering problem Simulating with uncertainty : the rough surface scattering problem Uday Khankhoje Assistant Professor, Electrical Engineering Indian Institute of Technology Madras Uday Khankhoje (EE, IITM) Simulating

More information

A Framework for Statistical Timing Analysis using Non-Linear Delay and Slew Models

A Framework for Statistical Timing Analysis using Non-Linear Delay and Slew Models A Framework for Statistical Timing Analysis using Non-Linear Delay and Slew Models Sarvesh Bhardwaj, Praveen Ghanta, Sarma Vrudhula Department of Electrical Engineering, Department of Computer Science

More information

NEW ALGORITHMS FOR UNCERTAINTY QUANTIFICATION AND NONLINEAR ESTIMATION OF STOCHASTIC DYNAMICAL SYSTEMS. A Dissertation PARIKSHIT DUTTA

NEW ALGORITHMS FOR UNCERTAINTY QUANTIFICATION AND NONLINEAR ESTIMATION OF STOCHASTIC DYNAMICAL SYSTEMS. A Dissertation PARIKSHIT DUTTA NEW ALGORITHMS FOR UNCERTAINTY QUANTIFICATION AND NONLINEAR ESTIMATION OF STOCHASTIC DYNAMICAL SYSTEMS A Dissertation by PARIKSHIT DUTTA Submitted to the Office of Graduate Studies of Texas A&M University

More information

On a Data Assimilation Method coupling Kalman Filtering, MCRE Concept and PGD Model Reduction for Real-Time Updating of Structural Mechanics Model

On a Data Assimilation Method coupling Kalman Filtering, MCRE Concept and PGD Model Reduction for Real-Time Updating of Structural Mechanics Model On a Data Assimilation Method coupling, MCRE Concept and PGD Model Reduction for Real-Time Updating of Structural Mechanics Model 2016 SIAM Conference on Uncertainty Quantification Basile Marchand 1, Ludovic

More information

Organization. I MCMC discussion. I project talks. I Lecture.

Organization. I MCMC discussion. I project talks. I Lecture. Organization I MCMC discussion I project talks. I Lecture. Content I Uncertainty Propagation Overview I Forward-Backward with an Ensemble I Model Reduction (Intro) Uncertainty Propagation in Causal Systems

More information

Collocation based high dimensional model representation for stochastic partial differential equations

Collocation based high dimensional model representation for stochastic partial differential equations Collocation based high dimensional model representation for stochastic partial differential equations S Adhikari 1 1 Swansea University, UK ECCM 2010: IV European Conference on Computational Mechanics,

More information

Quantifying Uncertainty: Modern Computational Representation of Probability and Applications

Quantifying Uncertainty: Modern Computational Representation of Probability and Applications Quantifying Uncertainty: Modern Computational Representation of Probability and Applications Hermann G. Matthies with Andreas Keese Technische Universität Braunschweig wire@tu-bs.de http://www.wire.tu-bs.de

More information

c 2004 Society for Industrial and Applied Mathematics

c 2004 Society for Industrial and Applied Mathematics SIAM J. SCI. COMPUT. Vol. 6, No., pp. 578 59 c Society for Industrial and Applied Mathematics STOCHASTIC SOLUTIONS FOR THE TWO-DIMENSIONAL ADVECTION-DIFFUSION EQUATION XIAOLIANG WAN, DONGBIN XIU, AND GEORGE

More information

Paradigms of Probabilistic Modelling

Paradigms of Probabilistic Modelling Paradigms of Probabilistic Modelling Hermann G. Matthies Brunswick, Germany wire@tu-bs.de http://www.wire.tu-bs.de abstract RV-measure.tex,v 4.5 2017/07/06 01:56:46 hgm Exp Overview 2 1. Motivation challenges

More information

Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs

Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs Roman Andreev ETH ZÜRICH / 29 JAN 29 TOC of the Talk Motivation & Set-Up Model Problem Stochastic Galerkin FEM Conclusions & Outlook Motivation

More information

Utilizing Adjoint-Based Techniques to Improve the Accuracy and Reliability in Uncertainty Quantification

Utilizing Adjoint-Based Techniques to Improve the Accuracy and Reliability in Uncertainty Quantification Utilizing Adjoint-Based Techniques to Improve the Accuracy and Reliability in Uncertainty Quantification Tim Wildey Sandia National Laboratories Center for Computing Research (CCR) Collaborators: E. Cyr,

More information

Efficient Sampling for Non-Intrusive Polynomial Chaos Applications with Multiple Uncertain Input Variables

Efficient Sampling for Non-Intrusive Polynomial Chaos Applications with Multiple Uncertain Input Variables Missouri University of Science and Technology Scholars' Mine Mechanical and Aerospace Engineering Faculty Research & Creative Works Mechanical and Aerospace Engineering 4-1-2007 Efficient Sampling for

More information

DATA ASSIMILATION FOR FLOOD FORECASTING

DATA ASSIMILATION FOR FLOOD FORECASTING DATA ASSIMILATION FOR FLOOD FORECASTING Arnold Heemin Delft University of Technology 09/16/14 1 Data assimilation is the incorporation of measurement into a numerical model to improve the model results

More information

NONLOCALITY AND STOCHASTICITY TWO EMERGENT DIRECTIONS FOR APPLIED MATHEMATICS. Max Gunzburger

NONLOCALITY AND STOCHASTICITY TWO EMERGENT DIRECTIONS FOR APPLIED MATHEMATICS. Max Gunzburger NONLOCALITY AND STOCHASTICITY TWO EMERGENT DIRECTIONS FOR APPLIED MATHEMATICS Max Gunzburger Department of Scientific Computing Florida State University North Carolina State University, March 10, 2011

More information

NUMERICAL SOLUTIONS FOR OPTIMAL CONTROL PROBLEMS UNDER SPDE CONSTRAINTS

NUMERICAL SOLUTIONS FOR OPTIMAL CONTROL PROBLEMS UNDER SPDE CONSTRAINTS NUMERICAL SOLUTIONS FOR OPTIMAL CONTROL PROBLEMS UNDER SPDE CONSTRAINTS AFOSR grant number: FA9550-06-1-0234 Yanzhao Cao Department of Mathematics Florida A & M University Abstract The primary source of

More information

Dimension-adaptive sparse grid for industrial applications using Sobol variances

Dimension-adaptive sparse grid for industrial applications using Sobol variances Master of Science Thesis Dimension-adaptive sparse grid for industrial applications using Sobol variances Heavy gas flow over a barrier March 11, 2015 Ad Dimension-adaptive sparse grid for industrial

More information

Random Matrix Eigenvalue Problems in Probabilistic Structural Mechanics

Random Matrix Eigenvalue Problems in Probabilistic Structural Mechanics Random Matrix Eigenvalue Problems in Probabilistic Structural Mechanics S Adhikari Department of Aerospace Engineering, University of Bristol, Bristol, U.K. URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html

More information

Hierarchical Parallel Solution of Stochastic Systems

Hierarchical Parallel Solution of Stochastic Systems Hierarchical Parallel Solution of Stochastic Systems Second M.I.T. Conference on Computational Fluid and Solid Mechanics Contents: Simple Model of Stochastic Flow Stochastic Galerkin Scheme Resulting Equations

More information

Safety Envelope for Load Tolerance and Its Application to Fatigue Reliability Design

Safety Envelope for Load Tolerance and Its Application to Fatigue Reliability Design Safety Envelope for Load Tolerance and Its Application to Fatigue Reliability Design Haoyu Wang * and Nam H. Kim University of Florida, Gainesville, FL 32611 Yoon-Jun Kim Caterpillar Inc., Peoria, IL 61656

More information

Dynamic System Identification using HDMR-Bayesian Technique

Dynamic System Identification using HDMR-Bayesian Technique Dynamic System Identification using HDMR-Bayesian Technique *Shereena O A 1) and Dr. B N Rao 2) 1), 2) Department of Civil Engineering, IIT Madras, Chennai 600036, Tamil Nadu, India 1) ce14d020@smail.iitm.ac.in

More information

Modelling Under Risk and Uncertainty

Modelling Under Risk and Uncertainty Modelling Under Risk and Uncertainty An Introduction to Statistical, Phenomenological and Computational Methods Etienne de Rocquigny Ecole Centrale Paris, Universite Paris-Saclay, France WILEY A John Wiley

More information

A stochastic collocation approach for efficient integrated gear health prognosis

A stochastic collocation approach for efficient integrated gear health prognosis A stochastic collocation approach for efficient integrated gear health prognosis Fuqiong Zhao a, Zhigang Tian b,, Yong Zeng b a Department of Mechanical and Industrial Engineering, Concordia University,

More information

Spacecraft Uncertainty Propagation using. Gaussian Mixture Models and Polynomial Chaos. Expansions

Spacecraft Uncertainty Propagation using. Gaussian Mixture Models and Polynomial Chaos. Expansions Spacecraft Uncertainty Propagation using Gaussian Mixture Models and Polynomial Chaos Expansions Vivek Vittaldev 1 and Ryan P. Russell 2 The University of Texas at Austin, Austin, Texas, 78712 Richard

More information

Introduction to uncertainty quantification An example of application in medicine

Introduction to uncertainty quantification An example of application in medicine Introduction to uncertainty quantification An example of application in medicine Laurent Dumas Laboratoire de Mathématiques de Versailles (LMV) Versailles University Short course, University of Mauritius,

More information

Strain and stress computations in stochastic finite element. methods

Strain and stress computations in stochastic finite element. methods INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2007; 00:1 6 [Version: 2002/09/18 v2.02] Strain and stress computations in stochastic finite element methods Debraj

More information

Sparse polynomial chaos expansions as a machine learning regression technique

Sparse polynomial chaos expansions as a machine learning regression technique Research Collection Other Conference Item Sparse polynomial chaos expansions as a machine learning regression technique Author(s): Sudret, Bruno; Marelli, Stefano; Lataniotis, Christos Publication Date:

More information

Kernel principal component analysis for stochastic input model generation

Kernel principal component analysis for stochastic input model generation Kernel principal component analysis for stochastic input model generation Xiang Ma, Nicholas Zabaras Materials Process Design and Control Laboratory, Sibley School of Mechanical and Aerospace Engineering,

More information

Probabilistic Structural Dynamics: Parametric vs. Nonparametric Approach

Probabilistic Structural Dynamics: Parametric vs. Nonparametric Approach Probabilistic Structural Dynamics: Parametric vs. Nonparametric Approach S Adhikari School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/

More information

Polynomial Chaos Based Design of Robust Input Shapers

Polynomial Chaos Based Design of Robust Input Shapers Tarunraj Singh Professor e-mail: tsingh@buffalo.edu Puneet Singla Assistant Professor e-mail: psingla@buffalo.edu Umamaheswara Konda e-mail: venkatar@buffalo.edu Department of Mechanical and Aerospace

More information

Numerical Approximation of Stochastic Elliptic Partial Differential Equations

Numerical Approximation of Stochastic Elliptic Partial Differential Equations Numerical Approximation of Stochastic Elliptic Partial Differential Equations Hermann G. Matthies, Andreas Keese Institut für Wissenschaftliches Rechnen Technische Universität Braunschweig wire@tu-bs.de

More information

UNIVERSITY OF CALIFORNIA, SAN DIEGO. An Empirical Chaos Expansion Method for Uncertainty Quantification

UNIVERSITY OF CALIFORNIA, SAN DIEGO. An Empirical Chaos Expansion Method for Uncertainty Quantification UNIVERSITY OF CALIFORNIA, SAN DIEGO An Empirical Chaos Expansion Method for Uncertainty Quantification A Dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy

More information

Research Collection. Basics of structural reliability and links with structural design codes FBH Herbsttagung November 22nd, 2013.

Research Collection. Basics of structural reliability and links with structural design codes FBH Herbsttagung November 22nd, 2013. Research Collection Presentation Basics of structural reliability and links with structural design codes FBH Herbsttagung November 22nd, 2013 Author(s): Sudret, Bruno Publication Date: 2013 Permanent Link:

More information

Stochastic Capacitance Extraction Considering Process Variation with Spatial Correlation

Stochastic Capacitance Extraction Considering Process Variation with Spatial Correlation Stochastic Capacitance Extraction Considering Process Variation with Spatial Correlation Tuck Chan EE Department UCLA, CA 90095 tuckie@ucla.edu Fang Gong EE Department UCLA, CA 90095 gongfang@ucla.edu

More information