Predictability of Chemical Systems
|
|
- Buddy Cannon
- 5 years ago
- Views:
Transcription
1 Predictability of Chemical Systems Habib N. Najm Sandia National Laboratories, Livermore, CA, USA Collaborators: M.T. Reagan & B.J. Debusschere O.M. Knio, A. Matta, & R.G. Ghanem O.P. Le Maitre Sandia National Labs, Livermore, CA, USA Johns Hopkins University, Baltimore, MD, USA Univ. d Evry Val d Essonne, Evry, France Acknowledgement: US Dept. of Energy, Basic Energy Sciences, Div. Chemical Sciences, Biosciences, and Geosciences Sandia National Labs, LDRD Defense Advanced Research Projects Agency (DARPA) Workshop on the Elements of Predictability, The Johns Hopkins University, November 13-14, HNN-SNL JHU Pred. 03-1
2 Preliminaries Scope Systems involving chemical reactions, in the gas, liquid, or solid phases Combustion, Atmospheric chemistry, Electrochemistry, Biochemistry, Catalysis,... Definition of predictability A system is predictable with respect to a given observable when we have a model which predicts that observable over the requisite range of operating conditions with known and acceptable accuracy and precision. Accuracy: agreement between mean model predictions and experimental measurements Precision: uncertainty in model predictions resulting from model and parametric uncertainties and numerical discretization errors HNN-SNL JHU Pred. 03-2
3 Uncertainty Quantification (UQ) in Chemically Reacting Flow Rational model validation with respect to experimental measurements requires estimates of ranges of error in each set of data Experimental error-bars are (usually) available How large are the error bars on the computational results? Sources of error/uncertainty in the prediction Model uncertainty Parametric uncertainty Numerical discretization errors Determining that numerical and experimental error bars do not overlap enables a decision on the efficacy of the model as distinct from the role of the parameters Focus on quantification of parameteric uncertainty (UQ) in chemically reacting flow computations HNN-SNL JHU Pred. 03-3
4 Computational Issues in Reacting Flow Modeling Reacting Flow Model Formulation Chemical & Transport Models Low Mach number or Compressible flow Model Spatial Discretization Large range of length scales & sharp moving fronts Adaptive Mesh Refinement (AMR) Turbulence High Order spatial discretization Time Integration Large range of time scales & Stiffness Implicit chemistry Diffusive stability implicit or stabilized explicit Diffusion constructions Convection accuracy, CFL, explicit convection Operator-Splitting splitting errors HNN-SNL JHU Pred. 03-4
5 Governing Dimensionless Low Mach Number Equations (ρv) t ρ t + (ρv) = 0 + (ρvv) = p + 1 Re [ µ[( v) + ( v) T ] 2 3 µ( v)u ] ρc p DT Dt (γ 1) dp o = γ dt + 1 ReP r (λ T ) ρ ReSc N c p,i V i T Da i=1 N h i w i i=1 (ρy i ) t + (ρvy i ) = 1 ReSc (ρy iv i ) + Da w i p o = ρt W i = 1,, N Low Mach No., no body forces, no radiation, mixture-averaged transport Neglect Soret and Dufour effects HNN-SNL JHU Pred. 03-5
6 Dominant Reaction Pathways, GRImech1.2, Atm. CH 4 +Air C 2 H 6 H, OH C 2 H 5 C 2 H 4 C 2 H 3 C 2 H 2 C 2 H H, M H, OH H, M OH O CH 3 + M OH, H, O H + M OH + M H O O OH OH, H CH 3 CH * 2 N 2, CO 2, H 2 O HO 2 O 2 O, O 2 O 2 CH 2 H, OH CH* M CH H 2 O O 2 O 2 CH 3 OH CH 3 O O OH* H M O CO 2 * M CH 2 OH O 2, M CH 2 O HCO H, OH, O, CH 3 H, OH, M, O 2, H 2 O CO OH CO 2 HNN-SNL JHU Pred. 03-6
7 General Approach for Forward Parametric UQ Define the physical model and associated parameters Determine which parameters are uncertain, and by how much Identify the set of important parameters Define a representation for uncertainty Interval Arithmetic, Fuzzy Sets, Stochastic,... Propagate the uncertainty through the model Non-intrusive, Intrusive Error estimation/propagation, Stochastic Sensitivity analysis Evaluate resulting uncertainty in model predictions HNN-SNL JHU Pred. 03-7
8 Sensitivity Analysis For an ODE system: dx = w(x, t; A) dt where X = {X i } is the solution vector, and A = {A k } is the vector of model parameters, the first order sensitivity coefficients, ζ ik = X i A k are integrated in time using the ODE system dζ ik dt = w i A k + j w i X j ζ jk obtained by differentiation of dx i /dt w.r.t. A k. The normalized sensitivity coefficients are given by: S ik = ln X i ln A k = A k X i ζ ik HNN-SNL JHU Pred. 03-8
9 Error Propagation Let the standard deviation in each ln A k be σ k, then the total contribution of each reaction k to the variance in the mole fraction of species i is ( ) 2 zki 2 X i = A k σk 2 A k Variances are additive, and the total variance in the mole fraction of each species i, is the sum of the N reaction contributions: N [σ(x i )] 2 = Issues: Linear... no higher order information or coupling between parameters Intrusive... involves recoding k=1 Concerns about overprediction of output uncertainties z 2 ki HNN-SNL JHU Pred. 03-9
10 Spectral Stochastic UQ Formulation Model uncertain parameters as random variables A stochastic process u(x, t, θ) can be described by : a Polynomial Chaos (PC) expansion in terms of Hermite polynomials Ψ k, their associated Gaussian basis ξ(θ), and spectral mode strengths u k (x, t) P u(x, t, θ) = u k (x, t)ψ k (ξ(θ)) u k (x, t)ψ k (ξ(θ)) k=0 Literature: Wiener : 1938 : Homog. Chaos span of Hermite pol. functionals of a Gaussian process k=0 Cameron & Martin : 1947 : L 2 Convergence for any L 2 stochastic process Ghanem & Spanos : 1991 : Application to UQ in Stochastic Finite Element Method Le Maître et al. : 2001,2002 : Application to Fluid Flow Xiu & Karniadakis : 2002 : Conv. rate for Gaussian/non-Gaussian processes Debusschere et al. : 2003 : Application to electrochemistry in microfluid flow Reagan et al. : 2003 : Application to reacting thermofluid flow J. Comp. Phys. 2001,2002; Phys. Fluids 2003; Comb. Flame 2003 HNN-SNL JHU Pred
11 Intrusive Spectral Stochastic UQ Formulation: ODE Example du Sample ODE with parameter λ: dt = λu, u(0) = u 0 Let λ be uncertain : Represent it as a stochastic quantity Introduce a new dimension ξ, where ξ is a Normal random variable Use P -th order Polynomial Chaos (PC) expansions: λ = P λ k Ψ k (ξ), u = k=0 P u k Ψ k (ξ), k=0 The Ψ k s are orthogonal < Ψ i Ψ j >=< Ψ 2 i > δ ij (λ k known, u k (t) unknown) Substitute PC expansions in the ODE, and apply Galerkin projection: du i dt = λuψ i P P Ψ 2 i λu i = λ p u q C pqi, i = 0,, P p=0 q=0 where the C pqi = Ψ p Ψ q i are known coefficients. HNN-SNL JHU Pred
12 Pseudo-Spectral Implementation Spectral Product : w = uv w = u v w i = uv i, i = 0,, P Psuedo-spectral higher-order polynomial terms : w = λu 2 v w = λ (u (u v)) Division : w = u v vw i = u i, i = 0,, P solve linear equation system for w i Arbitrary functions u = f(x) where u = u(x, u), and u i are algebraically found from x i, u i, i = 0,, P : P (xb ) j P u k (x b ) u k (x a ) = C ijk ( u) i dx j j=0 (x a ) j i=0 HNN-SNL JHU Pred
13 Pseudo-Spectral UQ Formulation: low M 2D Reacting Flow ρv q t T q t + v T q = ρ q t + ρv q = 0 + ρvv q = p q + 1 Re µ[( v) + ( v) T ] 2 3 µ( v)u (γ 1) γρc p dp o + 1 dt q ReP r (λ T ) ρc p q 1 N c p,i V i T Da ReSc c i=1 p q q 1 ρc p N h i w i i=1 q ρy i q t + ρvy i q = 1 ReSc ρy iv i q + Da w i q i = 1,, N Time Integration: Operator-Split reaction-diffusion integration of (P + 1)(N + 1) species and energy/density equations Stochastic Projection Method integration of (P + 1) momentum equations HNN-SNL JHU Pred
14 0D Intrusive H 2 -Air Ignition : Uncertainty in [H 2 O 2 ] 1.5e 10 1e 22 all modes 1.0e 10 c H2 O 2 σ 2 H 2 O 2 5e 23 reaction 5.0e 11 enthalpy 0.0e t (s) H 2 -Air (Yetter et al., 1991), 9 species, 19 reactions 3 rd -order PC, 2 uncertain parameters 0e t (s) Rxn. 11 pre-exponential: A 11 ]A m 11/3, 3 A m 11[ Lognormal distribution H enthalpy: σ hh = 1% Normal distribution Fast rise in the mean [H 2 O 2 ], little amplification in its uncertainty HNN-SNL JHU Pred
15 0D Intrusive H 2 -Air Ignition : Uncertainty in [HO 2 ] Very fast rate of growth of the mean [HO 2 ] Followed by a similarly fast rise in the standard deviation Much larger uncertainty than H 2 O 2 COV of HO 2 is about 40% persists near equilibrium Amplification of enthalpy uncertainty 5e 10 4e 10 3e 10 c HO2 2e 10 1e 10 0e t (s) HNN-SNL JHU Pred
16 Experience with Instabilities When integrating a chemical system, e.g. ignition, regions of explosive mode growth (positive eigenvalues) can lead to instabilities. Instability manifested in the fast growth of higher order modes, and fast drift of the solution towards unphysical values Typically occurs when the standard deviation increases significantly, becoming a sizeable fraction of the mean. Consider a model problem du = u(u + 10)(1 u) dt Attractors at u = 10, u = 1, and a repulsive fixed point at u = 0. Let the initial condition u(t = 0) = U be stochastic, U = P k=0 U kψ k. Integrate the reformulated chaos system for the time evolution of u k, k = 0,..., P HNN-SNL JHU Pred
17 Model Problem: Consequence of Initial PDF tail zero crossing 6.0 U 0 =0.2,0.3; U 1 = U 0 =0.2,0.3; U 1 = PDF(U) u U time HNN-SNL JHU Pred
18 Non-intrusive Spectral Projection (NISP) UQ Formulation Construct spectral stochastic descriptions of uncertain parameters λ Sample parameter space and compute Monte-Carlo (MC) realizations of the deterministic model u i (t), i = 1,..., N Project MC statistics on the spectral mode strengths u k (t) u k = uψ k Ψ 2 k = 1 Ψ 2 k uψ k (ξ)ρ(ξ)dξ, k = 0,..., P Evaluate integral numerically Sacrifices efficiency for reduced complexity and improved stability May require excessively large number of samples to converge Retains spectral sensitivity information Allows designing UQ wrappers around legacy code HNN-SNL JHU Pred
19 Sampling Issues in NISP UQ Need to minimize the number of samples required for evaluating spectral mode strengths Collocation techniques (DEMM, SRSM) Minimize errors at sample points High efficiency : number of samples number of unknowns Galerkin projection (NISP) Minimize RMS error Less efficient but potentially more robust to nonlinearities Projection is a Quadrature operation - samples are quadrature points Latin Hypercube Sampling Gauss-Hermite Quadrature Sparse Quadrature / Cubature HNN-SNL JHU Pred
20 NISP UQ Application: Premixed H 2 -O 2 Chemistry at Super-Critical Water Oxidation (SCWO) Conditions Allow uncertainties in reaction rate constants and thermodynamic properties, per published experimental data Wrap NISP processing around a deterministic reacting flow code Using 8-step simplified SCWO Hydrogen mechanism (McRae) Reaction A n E a /R UF 1. OH + H H 2 O 1.620E H 2 + OH H 2 O + H 1.024E H + O 2 HO E HO 2 + HO 2 H 2 O 2 + O E H 2 O 2 + OH H 2 O + HO E H 2 O 2 + H HO 2 + H E H 2 O 2 OH + OH E OH + HO 2 H 2 O + O E Species µ 0 2σ H OH H 2 O H 2 O HO HNN-SNL JHU Pred
21 0D H 2 -O 2 Isothermal Isobaric SCWO Ignition 2.0e e 12 +σ 6.00e 13 All 1st order c OH (mol/cm 3 ) 1.0e e 13 σ 0.0e t (s) σ (OH) 4.00e e 13 Rxns 7,5 Rxns 8,7,5 Rxn e t (s) Mean and standard deviation predictions validated against published data Initial fast growth in uncertainty followed by a slower approach to a steady-state with large OH uncertainty Reactions 7 & 8 have dominant roles in the OH uncertainty HNN-SNL JHU Pred
22 Convergence with Spectral Order 6.00e 25 3rd order σ 2 (OH) 4.00e e 25 2nd order Nord=3 Nord=2 1st order 0.00e t (s) HNN-SNL JHU Pred
23 1D H 2 -O 2 SCWO Flame NISP UQ/Chemkin-Premix σ Rxns 1,6 Rxns 5,8,1,6 Y OH 0.10 σ σ (OH) Rxns 8,1, All 1st order x (cm) x (cm) 1D freely propagating H 2 -O 2 flame at SCWO conditions Fast growth in OH uncertainty in the primary reaction zone Steady level of uncertainty and mean of OH in the post-flame region Uncertainty in pre-exponential of Rxn.5 (H 2 O 2 +OH=H 2 O+HO 2 ) has largest contribution to uncertainty in the predicted OH field HNN-SNL JHU Pred
24 Higher-Order Terms Rxns 1,6 All modes Rxns 5,8,1,6 σ (OH) Rxns 8,1,6 All 1st order x (cm) HNN-SNL JHU Pred
25 1D H 2 -O 2 SCWO Flame NISP UQ/Chemkin-Premix Y H2 O σ σ (H 2 O 2 ) All 1st order Rxns 5,8,7,6 Rxns 8,7,6 Rxns 7, Rxn 6 σ x (cm) x (cm) Very large uncertainty in H 2 O 2 prediction, COV=σ/µ=100%!! Dominant source is again Rxn.5 (H 2 O 2 +OH=H 2 O+HO 2 ) Not a robust model for predicting H 2 O 2 under SCWO conditions Results highlight the utility of additional experimental measurements of A 5 HNN-SNL JHU Pred
26 Higher-Order Terms σ (H 2 O 2 ) All modes All 1st order Rxns 5,8,1,6 Rxns 8,1,6 Rxns 1,6 Rxn x (cm) HNN-SNL JHU Pred
27 CPU-time Savings with Intrusive Spectral Strategy 0D H 2 -O 2 SCWO ignition NISP standard deviation tends to that from the intrusive construction NISP comes to within 50% of the intrusive value after 6000 Latin- Hypercube realizations 6000 sample runs 48 CPU hrs 1 intrusive run 2 hrs max σ(h 2 O) MC cycles HNN-SNL JHU Pred
28 Discussion State of predictability awareness Useful definitions of predictability, examples Obstacles in the way of predictability What can realistically be done to mitigate these obstacles Benefits of improved predictability, potential sci/tech achievements Elements of a predictability aware meta-model HNN-SNL JHU Pred
29 Predictability Awareness in the Modeling of Chemical Systems Awareness of uncertainty in chemical models, thermodynamic properties, & kinetic rate constants Significant focus on prediction accuracy; only a few studies on precision/uncertainty. Pan et al. Atmospheric Chemistry J. Geophys. Res.-Atmos., Phenix et al. H 2 -O 2 ignition Combustion & Flame, DEMM, Polynomial Chaos, sampling, collocation Isukapalli et al. Environmental & Bio. systems Risk Analysis, Stoch. Resp. Surf. Meth., Polynomial Chaos, Collocation, Regression Turanyi et al. CH 4 -Air flame Phys. Chem. Chem. Phys., Sensitivity analysis, Error propagation Debusschere et al. Electrochemistry & Biodetection Phys. Fluids, Polynomial Chaos, Galerkin, intrusive Reagan et al. H 2 -O 2 flame Combustion & Flame, Polynomial Chaos, Latin Hypercube Sampling, non-intrusive HNN-SNL JHU Pred
30 Useful Definitions of Predictability in Chemical Systems A system is predictable with respect to a given observable when we have a model which predicts that observable over the requisite range of operating conditions with known and acceptable accuracy and precision. Accuracy: agreement between mean model predictions and experimental measurements Precision: uncertainty in model predictions resulting from model and parametric uncertainties and numerical discretization errors Some systems are simply not predictable w.r.t. a given observable, independent of the model, e.g. predictions of local weather patterns. HNN-SNL JHU Pred
31 Obstacles in the way of Predictability of Chemical Systems Identification of chemical kinetic network/mechanism structure Robust model inference under noise and uncertainty Measurement/Computation of chemical rate constants and thermo properties Estimation of uncertainty in chemical/thermo parameters Availability of uncertainties in published/online chemical mechanisms Strong nonlinearities, sensitivity to IC s, high amplification of uncertainties Need for accurate reduced-order/subgrid turbulent reacting flow models CPU intensive reacting flow computations sampling limitations Large size and complexity of chemical models Large ranges of length and time scales (turbulence, thin flames, stiffness) Multiphysics, Heterogeneous Multiscale Models HNN-SNL JHU Pred
32 What can realistically be done to mitigate these Obstacles More experimental/ab-initio data on chemical model structure, chemical rate constants, and thermodynamic properties Development of techniques for robust model inference under noise and uncertainty Development of techniques for identifying important parameters w.r.t. specific observables in complex reacting flow models Development of accurate/efficient formulations and numerical techniques for UQ in higb-dimensional models with strong non-linearities and/or bifurcations Public dissemination of uncertainties in thermochemical model parameters, along with the typically published mean values Development of accurate reduced order models for turbulent combustion and other multiscale reactive systems Development of efficient direct numerical simulation constructions for turbulent reacting flow in laboratory-scale settings HNN-SNL JHU Pred
33 Benefits of improved predictability, new S & T Enhanced understanding of physical, chemical, and biological systems More efficient engineering development of chemically reacting systems... engines, chemical processing plants, etc. HNN-SNL JHU Pred
34 Elements of a predictability-aware model A predictive model has to predict requisite observables with acceptable accuracy and precision. To do this, it is necessary for such meta-models to include accurate models of the physical and chemical phenomena at hand data on nominal values and uncertainties of model parameters means of error/uncertainty estimation in predicted results means of identification of selective parametric contributions to uncertainties in model predictions HNN-SNL JHU Pred
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@ Sandia National laboratories. - - hanem, and O.P. Le Maitre. eacting Flow SANDIA REPORT SAND
SANDA REPORT SAND20034598 eacting Flow - - hanem, and O.P. Le Maitre a is a multir d States Dei 85OM ~1 moratory operated by Sandia Corporation, a Lockheed Martin Company, for the mt of Energy s National
More informationDinesh 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 informationUQ in Reacting Flows
UQ in Reacting Flows Planetary Entry Simulations High-Temperature Reactive Flow During descent in the atmosphere vehicles experience extreme heating loads The design of the thermal protection system (TPS)
More informationBeyond 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 informationPolynomial Chaos Based Uncertainty Propagation
Polynomial Chaos Based Uncertainty Propagation Lecture 2: Forward Propagation: Intrusive and Non-Intrusive Bert Debusschere bjdebus@sandia.gov Sandia National Laboratories, Livermore, CA Aug 12, 2013 Summer
More informationEfficient 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 informationPolynomial 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 informationKeywords: 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 informationA 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 informationStochastic 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 informationEstimating functional uncertainty using polynomial chaos and adjoint equations
0. Estimating functional uncertainty using polynomial chaos and adjoint equations February 24, 2011 1 Florida State University, Tallahassee, Florida, Usa 2 Moscow Institute of Physics and Technology, Moscow,
More informationUncertainty 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 informationMultiscale stochastic preconditioners in non-intrusive spectral projection
Multiscale stochastic preconditioners in non-intrusive spectral projection Alen Alenxanderian a, Olivier P. Le Maître b, Habib N. Najm c, Mohamed Iskandarani d, Omar M. Knio a, a Department of Mechanical
More informationStochastic 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 informationPerformance 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 informationSparse 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 informationAnalysis and Simulation of Blood Flow in the Portal Vein with Uncertainty Quantification
Analysis and Simulation of Blood Flow in the Portal Vein with Uncertainty Quantification João Pedro Carvalho Rêgo de Serra e Moura Instituto Superior Técnico Abstract Blood flow simulations in CFD are
More informationUncertainty 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 informationNumerical Methods for Problems with Moving Fronts Orthogonal Collocation on Finite Elements
Electronic Text Provided with the Book Numerical Methods for Problems with Moving Fronts by Bruce A. Finlayson Ravenna Park Publishing, Inc., 635 22nd Ave. N. E., Seattle, WA 985-699 26-524-3375; ravenna@halcyon.com;www.halcyon.com/ravenna
More informationEFFICIENT 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 informationSENSITIVITY 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 informationA Stochastic Projection Method for Fluid Flow
Journal of Computational Physics 8, 9 44 (22) doi:.6/jcph.22.74 A Stochastic Projection Method for Fluid Flow II. Random Process Olivier P. Le Maître, Matthew T. Reagan, Habib N. Najm, Roger G. Ghanem,
More informationIntro BCS/Low Rank Model Inference/Comparison Summary References. UQTk. A Flexible Python/C++ Toolkit for Uncertainty Quantification
A Flexible Python/C++ Toolkit for Uncertainty Quantification Bert Debusschere, Khachik Sargsyan, Cosmin Safta, Prashant Rai, Kenny Chowdhary bjdebus@sandia.gov Sandia National Laboratories, Livermore,
More informationIntroduction 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 informationError Budgets: A Path from Uncertainty Quantification to Model Validation
Error Budgets: A Path from Uncertainty Quantification to Model Validation Roger Ghanem Aerospace and Mechanical Engineering Civil Engineering University of Southern California Los Angeles Advanced Simulation
More informationUncertainty Quantification for multiscale kinetic equations with high dimensional random inputs with sparse grids
Uncertainty Quantification for multiscale kinetic equations with high dimensional random inputs with sparse grids Shi Jin University of Wisconsin-Madison, USA Kinetic equations Different Q Boltmann Landau
More informationA 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 informationUncertainty Quantification for multiscale kinetic equations with random inputs. Shi Jin. University of Wisconsin-Madison, USA
Uncertainty Quantification for multiscale kinetic equations with random inputs Shi Jin University of Wisconsin-Madison, USA Where do kinetic equations sit in physics Kinetic equations with applications
More informationFast 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 informationStochastic 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 informationUncertainty Quantification and hypocoercivity based sensitivity analysis for multiscale kinetic equations with random inputs.
Uncertainty Quantification and hypocoercivity based sensitivity analysis for multiscale kinetic equations with random inputs Shi Jin University of Wisconsin-Madison, USA Shanghai Jiao Tong University,
More informationFinal Report: DE-FG02-95ER25239 Spectral Representations of Uncertainty: Algorithms and Applications
Final Report: DE-FG02-95ER25239 Spectral Representations of Uncertainty: Algorithms and Applications PI: George Em Karniadakis Division of Applied Mathematics, Brown University April 25, 2005 1 Objectives
More informationStochastic Solvers for the Euler Equations
43rd AIAA Aerospace Sciences Meeting and Exhibit 1-13 January 5, Reno, Nevada 5-873 Stochastic Solvers for the Euler Equations G. Lin, C.-H. Su and G.E. Karniadakis Division of Applied Mathematics Brown
More informationComputer Science Technical Report TR May 08, 2007
Computer Science Technical Report TR-7-8 May 8, 27 Haiyan Cheng and Adrian Sandu Efficient Uncertainty Quantification with Polynomial Chaos for Implicit Stiff Systems Computer Science Department Virginia
More informationLaminar Premixed Flames: Flame Structure
Laminar Premixed Flames: Flame Structure Combustion Summer School 2018 Prof. Dr.-Ing. Heinz Pitsch Course Overview Part I: Fundamentals and Laminar Flames Introduction Fundamentals and mass balances of
More informationOverview. 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 informationUtilizing 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 informationPARALLEL 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 informationModeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos
Modeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos Dongbin Xiu and George Em Karniadakis Division of Applied Mathematics Brown University Providence, RI 9 Submitted to Journal of
More informationEFFICIENT STOCHASTIC GALERKIN METHODS FOR RANDOM DIFFUSION EQUATIONS
EFFICIENT STOCHASTIC GALERKIN METHODS FOR RANDOM DIFFUSION EQUATIONS DONGBIN XIU AND JIE SHEN Abstract. We discuss in this paper efficient solvers for stochastic diffusion equations in random media. We
More informationICES REPORT A posteriori error control for partial differential equations with random data
ICES REPORT 13-8 April 213 A posteriori error control for partial differential equations with random data by Corey M. Bryant, Serge Prudhomme, and Timothy Wildey The Institute for Computational Engineering
More informationNONLOCALITY 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 informationPC 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 informationA Non-Intrusive Polynomial Chaos Method for Uncertainty Propagation in CFD Simulations
Missouri University of Science and Technology Scholars' Mine Mechanical and Aerospace Engineering Faculty Research & Creative Works Mechanical and Aerospace Engineering --6 A Non-Intrusive Polynomial Chaos
More informationOriginal Research. Sensitivity Analysis and Variance Reduction in a Stochastic NDT Problem
To appear in the International Journal of Computer Mathematics Vol. xx, No. xx, xx 2014, 1 9 Original Research Sensitivity Analysis and Variance Reduction in a Stochastic NDT Problem R. H. De Staelen a
More informationc 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 informationQuadrature for Uncertainty Analysis Stochastic Collocation. What does quadrature have to do with uncertainty?
Quadrature for Uncertainty Analysis Stochastic Collocation What does quadrature have to do with uncertainty? Quadrature for Uncertainty Analysis Stochastic Collocation What does quadrature have to do with
More informationUNCERTAINTY 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 informationSolving the steady state diffusion equation with uncertainty Final Presentation
Solving the steady state diffusion equation with uncertainty Final Presentation Virginia Forstall vhfors@gmail.com Advisor: Howard Elman elman@cs.umd.edu Department of Computer Science May 6, 2012 Problem
More informationUncertainty Quantification in Multiscale Models
Uncertainty Quantification in Multiscale Models H.N. Najm Sandia National Laboratories Livermore, CA Workshop on Stochastic Multiscale Methods: Mathematical Analysis and Algorithms Univ. Southern California,
More informationBenjamin 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 informationA 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 informationarxiv: v1 [math.na] 14 Sep 2017
Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis arxiv:1709.04584v1 [math.na] 14 Sep 2017 Anindya Bhaduri a, Yanyan He 1b, Michael D. Shields a, Lori Graham-Brady
More informationAlgorithms 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 informationA stochastic Galerkin method for general system of quasilinear hyperbolic conservation laws with uncertainty
A stochastic Galerkin method for general system of quasilinear hyperbolic conservation laws with uncertainty Kailiang Wu, Huazhong Tang arxiv:6.4v [math.na] 6 Jan 6 HEDPS, CAPT & LMAM, School of Mathematical
More informationSobol-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 informationDynamic 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 informationStochastic Spectral Methods for Uncertainty Quantification
Stochastic Spectral Methods for Uncertainty Quantification Olivier Le Maître 1,2,3, Omar Knio 1,2 1- Duke University, Durham, North Carolina 2- KAUST, Saudi-Arabia 3- LIMSI CNRS, Orsay, France 38th Conf.
More informationQuantifying 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 informationApplication of a Stochastic Finite Element Procedure to Reliability Analysis
Application of a Stochastic Finite Element Procedure to Reliability Analysis Bruno Sudret & Marc Berveiller Electricité de France, R&D Division, Site des Renardières, F-77818 Moret-sur-Loing Maurice Lemaire
More informationPolynomial 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 informationApplication and validation of polynomial chaos methods to quantify uncertainties in simulating the Gulf of Mexico circulation using HYCOM.
Application and validation of polynomial chaos methods to quantify uncertainties in simulating the Gulf of Mexico circulation using HYCOM. Mohamed Iskandarani Matthieu Le Hénaff Carlisle Thacker University
More informationChaospy: A modular implementation of Polynomial Chaos expansions and Monte Carlo methods
Chaospy: A modular implementation of Polynomial Chaos expansions and Monte Carlo methods Simen Tennøe Supervisors: Jonathan Feinberg Hans Petter Langtangen Gaute Einevoll Geir Halnes University of Oslo,
More informationSpectral Representation of Random Processes
Spectral Representation of Random Processes Example: Represent u(t,x,q) by! u K (t, x, Q) = u k (t, x) k(q) where k(q) are orthogonal polynomials. Single Random Variable:! Let k (Q) be orthogonal with
More informationCombustion Behind Shock Waves
Paper 3F-29 Fall 23 Western States Section/Combustion Institute 1 Abstract Combustion Behind Shock Waves Sandeep Singh, Daniel Lieberman, and Joseph E. Shepherd 1 Graduate Aeronautical Laboratories, California
More informationPolynomial 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 informationCERTAIN 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 informationLiquid-Rocket Transverse Triggered Combustion Instability: Deterministic and Stochastic Analyses
Liquid-Rocket Transverse Triggered Combustion Instability: Deterministic and Stochastic Analyses by W. A. Sirignano Mechanical and Aerospace Engineering University of California, Irvine Collaborators:
More informationIdentification of multi-modal random variables through mixtures of polynomial chaos expansions
Identification of multi-modal random variables through mixtures of polynomial chaos expansions Anthony Nouy To cite this version: Anthony Nouy. Identification of multi-modal random variables through mixtures
More informationIntroduction to multiscale modeling and simulation. Explicit methods for ODEs : forward Euler. y n+1 = y n + tf(y n ) dy dt = f(y), y(0) = y 0
Introduction to multiscale modeling and simulation Lecture 5 Numerical methods for ODEs, SDEs and PDEs The need for multiscale methods Two generic frameworks for multiscale computation Explicit methods
More informationarxiv: v1 [math.na] 3 Apr 2019
arxiv:1904.02017v1 [math.na] 3 Apr 2019 Poly-Sinc Solution of Stochastic Elliptic Differential Equations Maha Youssef and Roland Pulch Institute of Mathematics and Computer Science, University of Greifswald,
More informationFluid Dynamics and Balance Equations for Reacting Flows
Fluid Dynamics and Balance Equations for Reacting Flows Combustion Summer School 2018 Prof. Dr.-Ing. Heinz Pitsch Balance Equations Basics: equations of continuum mechanics balance equations for mass and
More informationUncertainty 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 informationSchwarz Preconditioner for the Stochastic Finite Element Method
Schwarz Preconditioner for the Stochastic Finite Element Method Waad Subber 1 and Sébastien Loisel 2 Preprint submitted to DD22 conference 1 Introduction The intrusive polynomial chaos approach for uncertainty
More informationAdjoint based multi-objective shape optimization of a transonic airfoil under uncertainties
EngOpt 2016-5 th International Conference on Engineering Optimization Iguassu Falls, Brazil, 19-23 June 2016. Adjoint based multi-objective shape optimization of a transonic airfoil under uncertainties
More informationExperiences with Model Reduction and Interpolation
Experiences with Model Reduction and Interpolation Paul Constantine Stanford University, Sandia National Laboratories Qiqi Wang (MIT) David Gleich (Purdue) Emory University March 7, 2012 Joe Ruthruff (SNL)
More informationSTOCHASTIC FINITE ELEMENTS WITH MULTIPLE RANDOM NON-GAUSSIAN PROPERTIES
STOCHASTIC FIITE ELEMETS WITH MULTIPLE RADOM O-GAUSSIA PROPERTIES By Roger Ghanem, 1 Member, ASCE ABSTRACT: The spectral formulation of the stochastic finite-element method is applied to the problem of
More informationStochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations Boris Jeremić 1 Kallol Sett 2 1 University of California, Davis 2 University of Akron, Ohio ICASP Zürich, Switzerland August
More informationSpectral Polynomial Chaos Solutions of the Stochastic Advection Equation
Spectral Polynomial Chaos Solutions of the Stochastic Advection Equation M. Jardak, C.-H. Su and G.E. Karniadakis Division of Applied Mathematics Brown University October 29, 21 Abstract We present a new
More informationSpectral Methods for Uncertainty Quantification
Spectral Methods for Uncertainty Quantification Scientific Computation Editorial Board J.-J.Chattot,Davis,CA,USA P. Colella, Berkeley, CA, USA W. E, Princeton, NJ, USA R. Glowinski, Houston, TX, USA Y.
More informationarxiv: v2 [physics.comp-ph] 15 Sep 2015
On Uncertainty Quantification of Lithium-ion Batteries: Application to an LiC 6 /LiCoO 2 cell arxiv:1505.07776v2 [physics.comp-ph] 15 Sep 2015 Mohammad Hadigol a, Kurt Maute a, Alireza Doostan a, a Aerospace
More informationResearch Article Multiresolution Analysis for Stochastic Finite Element Problems with Wavelet-Based Karhunen-Loève Expansion
Mathematical Problems in Engineering Volume 2012, Article ID 215109, 15 pages doi:10.1155/2012/215109 Research Article Multiresolution Analysis for Stochastic Finite Element Problems with Wavelet-Based
More informationSteps in Uncertainty Quantification
Steps in Uncertainty Quantification Challenge: How do we do uncertainty quantification for computationally expensive models? Example: We have a computational budget of 5 model evaluations. Bayesian inference
More informationUncertainty 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 information5. Coupling of Chemical Kinetics & Thermodynamics
5. Coupling of Chemical Kinetics & Thermodynamics Objectives of this section: Thermodynamics: Initial and final states are considered: - Adiabatic flame temperature - Equilibrium composition of products
More informationUncertainty 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 informationMultilevel stochastic collocations with dimensionality reduction
Multilevel stochastic collocations with dimensionality reduction Ionut Farcas TUM, Chair of Scientific Computing in Computer Science (I5) 27.01.2017 Outline 1 Motivation 2 Theoretical background Uncertainty
More informationThe 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 informationSafety 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 informationStochastic 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 informationFast Numerical Methods for Stochastic Computations: A Review
COMMUNICATIONS IN COMPUTATIONAL PHYSICS Vol. 5, No. 2-4, pp. 242-272 Commun. Comput. Phys. February 2009 REVIEW ARTICLE Fast Numerical Methods for Stochastic Computations: A Review Dongbin Xiu Department
More informationSpectral 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 informationStrain 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 informationUncertainty Quantification of an ORC turbine blade under a low quantile constrain
Available online at www.sciencedirect.com ScienceDirect Energy Procedia 129 (2017) 1149 1155 www.elsevier.com/locate/procedia IV International Seminar on ORC Power Systems, ORC2017 13-15 September 2017,
More informationStochastic simulation of Advection-Diffusion equation considering uncertainty in input variables
INTERNATIONAL JOURNAL OF MATEMATICAL MODELS AND METODS IN APPLIED SCIENCES Volume, 7 Stochastic simulation of Advection-Diffusion equation considering uncertainty in input variables ossein Khorshidi, Gholamreza
More informationEfficient Solvers for Stochastic Finite Element Saddle Point Problems
Efficient Solvers for Stochastic Finite Element Saddle Point Problems Catherine E. Powell c.powell@manchester.ac.uk School of Mathematics University of Manchester, UK Efficient Solvers for Stochastic Finite
More informationModel 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 informationPolynomial Chaos Quantification of the Growth of Uncertainty Investigated with a Lorenz Model
JUNE 2010 S H E N E T A L. 1059 Polynomial Chaos Quantification of the Growth of Uncertainty Investigated with a Lorenz Model COLIN Y. SHEN, THOMAS E. EVANS, AND STEVEN FINETTE Naval Research Laboratory,
More informationUncertainty 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