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Marta D’Elia Emory University, Math&CS Dept. joint work with M. Perego and A. Veneziani A DATA ASSIMILATION TECHNIQUE FOR INCLUDING NOISY VELOCITY MEASURES INTO NAVIER-STOKES SIMULATIONS UNCERTAINTY QUANTIFICATION International Centre for Mathematical Sciences Edinburgh, May 24-28 2010 motivation input: biomedical imaging provides efficient techniques for the collection of a large amount of data goal: use these data to have an accurate approximation of the blood flow in vessels ultimate goal: estimation of reliability of our results prediction of the occurrence of diseases in the patient how: including the data in the numerical simulations combining measurements and dynamic principles governing the system introduction linear case nonlinear case numerical results conclusions motivation input: biomedical imaging provides efficient techniques for the collection of a large amount of data goal: use these data to have an accurate approximation of the blood flow in vessels ultimate goal: estimation of reliability of our results prediction of the occurrence of diseases in the patient how: including the data in the numerical simulations combining measurements and dynamic principles governing the system observed data + mathematical model = Data Assimilation (DA) Source: Dr. Brummer, Emory CHOA introduction linear case nonlinear case numerical results conclusions state of the art history: firstly introduced in the 50’s in fluid geophysics data: sparse, irregularly distributed and noisy governing principles: known and described by a mathematical model “Analysis” Estimation Theory Dynamic Control Theory (CT) time Interpolation Kalman-type techniques relaxation Stochastic techniques line introduction linear case nonlinear case numerical results conclusions state of the art history: firstly introduced in the 50’s in fluid geophysics data: sparse, irregularly distributed and noisy governing principles: known and described by a mathematical model “Analysis” Estimation Theory Dynamic Control Theory (CT) time Interpolation Kalman-type techniques relaxation Stochastic techniques line proposed approach: availability of efficient PDE’s solvers and numerical techniques for the solution of optimization problems: choice of CT approaches among those, on the basis of preliminary comparison results, we propose a Discretize-then-Optimize (DO) technique introduction linear case nonlinear case numerical results conclusions formulation notations : vessel domain with boundaries , , variables u,p data , vector of measured velocites on sites source: Dr. Brummer introduction linear case nonlinear case numerical results conclusions formulation notations : vessel domain with boundaries , , variables u,p data , vector of measured velocites on sites state equations: (S) source: Dr. Brummer introduction linear case nonlinear case numerical results conclusions formulation notations : vessel domain with boundaries , , variables u,p data , vector of measured velocites on sites state equations: (S) data assimilation: find s.t. source: Dr. Brummer under the constraint of (S) introduction linear case nonlinear case numerical results conclusions Numerical solution of the linear pb: Oseen discretize using the Finite Element (FE) method the state eq.s and the functional discretized u and p restriction matrix selection matrix boundary mass matrix Oseen matrix introduction linear case nonlinear case numerical results conclusions Numerical solution of the linear pb: Oseen discretize using the Finite Element (FE) method the state eq.s and the functional discretized u and p restriction matrix selection matrix boundary mass matrix Oseen matrix optimize solving the KKT system induced by the Lagrangian with the Reduced Hessian method: Reduced Hessian Hr Sensitivity matrix introduction linear case nonlinear case numerical results conclusions DA procedure for Navier-Stokes eq.s nonlinear constraint: the nonlinearity introduced by the advection term makes the forward problem challenging introduction linear case nonlinear case numerical results conclusions DA procedure for Navier-Stokes eq.s nonlinear constraint: the nonlinearity introduced by the advection term makes the forward problem challenging idea: exploit Picard/Newton method for treating the nonlinearity use the DA procedure proposed for linear problems Newton-like iterative procedure introduction linear case nonlinear case numerical results conclusions well-posedness note: the regularized formulation is always well-posed sufficient condition for an equality PDE constrained opt. pb: Hr positive definite goal: find conditions on the selection matrix sufficient for the well-posedness introduction linear case nonlinear case numerical results conclusions well-posedness note: the regularized formulation is always well-posed sufficient condition for an equality PDE constrained opt. pb: Hr positive definite goal: find conditions on the selection matrix sufficient for the well-posedness T result: for a=0, if Null(D) ∩ Range(S-1Rin Min)={0} then Hr is positive definite and the the problem is well-posed. introduction linear case nonlinear case numerical results conclusions well-posedness note: the regularized formulation is always well-posed sufficient condition for an equality PDE constrained opt. pb: Hr positive definite goal: find conditions on the selection matrix sufficient for the well-posedness T result: for a=0, if Null(D) ∩ Range(S-1Rin Min)={0} then Hr is positive definite and the the problem is well-posed. This condition is satisfied by choosing Q s.t. its restriction to rows corresponding to sites on Γin has rank Nin (degrees of freedom, DOF, of U on Γin) introduction linear case nonlinear case numerical results conclusions well-posedness note: the regularized formulation is always well-posed sufficient condition for an equality PDE constrained opt. pb: Hr positive definite goal: find conditions on the selection matrix sufficient for the well-posedness T result: for a=0, if Null(D) ∩ Range(S-1Rin Min)={0} then Hr is positive definite and the the problem is well-posed. This condition is satisfied by choosing Q s.t. its restriction to rows corresponding to sites on Γin has rank Nin (degrees of freedom, DOF, of U on Γin) practically: sites on grid nodes sparse sites on (P1bubble-P1) inflow boundary (P1bubble-P1) introduction linear case nonlinear case numerical results conclusions well-posedness idea: given sparse measurement on inflow (not satisfying assumptions for well-posedness) recover the well-posedness with approximated data on grid nodes on inflow introduction linear case nonlinear case numerical results conclusions well-posedness idea: given sparse measurement on inflow (not satisfying assumptions for well-posedness) recover the well-posedness with approximated data on grid nodes on inflow approximation: interpolation of each velocity component of the given data dj = Πkd(xj) where Πkd is the interpolator recovered from k data. introduction linear case nonlinear case numerical results conclusions well-posedness idea: given sparse measurement on inflow (not satisfying assumptions for well-posedness) recover the well-posedness with approximated data on grid nodes on inflow approximation: interpolation of each velocity component of the given data dj = Πkd(xj) where Πkd is the interpolator recovered from k data. result: if the original data is s.t. di = uex(yi) + εi where εi ≈ N (0, σ2) then, the new data on inflow grid nodes is s.t dj = uex(yj) + ηj where η j ≈ N (μj, ν2j) introduction linear case nonlinear case numerical results conclusions well-posedness idea: given sparse measurement on inflow (not satisfying assumptions for well-posedness) recover the well-posedness with approximated data on grid nodes on inflow approximation: interpolation of each velocity component of the given data dj = Πkd(xj) where Πkd is the interpolator recovered from k data. result: if the original data is s.t. di = uex(yi) + εi where εi ≈ N (0, σ2) then, the new data on inflow grid nodes is s.t dj = uex(yj) + ηj where η j ≈ N (μj, ν2j) mean: affected by discretization error of the interpolation process variance: affected by the error caused by interpolation of noisy data introduction linear case nonlinear case numerical results conclusions analytic test case - setting test case: the advection vector corresponds to uex introduction linear case nonlinear case numerical results conclusions analytic test case - setting test case: the advection vector corresponds to uex mesh generation: software - FreeFem++ (1.5, 2) (-.5, 0) introduction linear case nonlinear case numerical results conclusions analytic test case - setting test case: the advection vector corresponds to uex mesh generation: software - FreeFem++ (1.5, 2) (-.5, 0) data generation: di = uex(yi) + εi simplified case: εi ≈ σ U(-0.5, 0.5) variance determined s.t. the signal-noise ratio (SNR) is fixed a priori according to experimental results estimated ratio SNR = 8.3 introduction linear case nonlinear case numerical results conclusions analytic test case - setting test case: the advection vector corresponds to uex mesh generation: software - FreeFem++ (1.5, 2) (-.5, 0) data generation: implementation details: di = uex(yi) + εi DA solver – lifeV external library – AZTEC simplified case: εi ≈ σ U(-0.5, 0.5) postprocessing – PARAVIEW FE spaces – P1bubble-P1 variance determined s.t. the signal-noise ratio (SNR) Oseen solver – Monolithic PCD is fixed a priori according to experimental results ν – 0.035 estimated ratio SNR = 8.3 introduction linear case nonlinear case numerical results conclusions analytic test case – error behavior sanity check: with noise-free data the FE convergence rate is recovered introduction linear case nonlinear case numerical results conclusions analytic test case – error behavior noisy data: error behavior for the Oseen test case 1. error of the sample mean over Nr realizations, UmNr, vs number of realizations, Nr : ||UmNr - Uex||2 E∆x as Nr , O(Nr-0.5) 8 since Uex = Um+ E∆x where E∆x is the noise-free discretization error and Um is the mean value for the variable UmNr introduction linear case nonlinear case numerical results conclusions analytic test case – error behavior noisy data: error behavior for the Oseen test case 1. error of the sample mean over Nr realizations, UmNr, vs number of realizations, Nr : ||UmNr - Uex||2 E∆x as Nr , O(Nr-0.5) 8 since Uex = Um+ E∆x where E∆x is the noise-free discretization error and Um is the mean value for the variable UmNr ∆x = 0.16 introduction linear case nonlinear case numerical results conclusions analytic test case – error behavior noisy data: error behavior for the Oseen test case 1. error of the sample mean over Nr realizations, UmNr, vs number of realizations, Nr : ||UmNr - Uex||2 E∆x as Nr , O(Nr-0.5) 8 since Uex = Um+ E∆x where E∆x is the noise-free discretization error and Um is the mean value for the variable UmNr ∆x = 0.072 introduction linear case nonlinear case numerical results conclusions analytic test case – error behavior noisy data: error behavior for the Oseen test case 1. error of the sample mean over Nr realizations, UmNr, vs number of realizations, Nr : ||UmNr - Uex||2 E∆x as Nr , O(Nr-0.5) 8 since Uex = Um+ E∆x where E∆x is the noise-free discretization error and Um is the mean value for the variable UmNr ∆x = 0.05 introduction linear case nonlinear case numerical results conclusions analytic test case – error behavior noisy data: error behavior for the Oseen test case 2. average error vs Ns , O(Ns-0.5) with ∆x = 0.16, SNR = 10 introduction linear case nonlinear case numerical results conclusions analytic test case – interpolation validation comparison: regularization vs interpolation parameters: SNR = 20; ∆x = 0.072 introduction linear case nonlinear case numerical results conclusions towards real geometries – velocity & vorticity curved domain: approximation of a section of the carotid data generation: reference solution (FE solution on very fine grid) with addition of white noise introduction linear case nonlinear case numerical results conclusions towards real geometries – velocity & vorticity curved domain: approximation of a section of the carotid data generation: reference solution (FE solution on very fine grid) with addition of white noise introduction linear case nonlinear case numerical results conclusions final remarks conclusion 1. the DA procedure proved to be consisitent in case of non-noisy data 2. in presence of noise it is a tool for filtering the unavoidable uncertainties in the measured data: the assimilation of boundary and internal measures enhances the accuracy and reliability of the recovered variables 3. the combination of the DA procedure with the interpolation technique is a competitive tool with respect to common regularization approaches state of the art approaches linear case nonlinear case conclusions final remarks conclusion 1. the DA procedure proved to be consisitent in case of non-noisy data 2. in presence of noise it is a tool for filtering the unavoidable uncertainties in the measured data: the assimilation of boundary and internal measures enhances the accuracy and reliability of the recovered variables 3. the combination of the DA procedure with the interpolation technique is a competitive tool with respect to common regularization approaches future guidelines 1. precise estimation of statistics related to assimilated variables via more sophisticated stochastic tools 2. assimilation of velocities from real geometries and data state of the art approaches linear case nonlinear case conclusions thank you for your attention questions? … ... special thanks to Alessandro Veneziani Mauro Perego Michele Benzi Max Gunzburger state of the art approaches linear case nonlinear case conclusions more... comparison: (accuracy and conditioning) vs (data location) introduction linear case nonlinear case numerical results conclusions more... outer iterations: GMRES system in S to be solved twice (transpose) decomposed using exact factorization 1st inner iteration: system in C: GMRES 2nd inner iteration: system in Σ (*) : GMRES preconditioner needed (Elman, Silverster, Wathen) (*) state of the art approaches linear case nonlinear case conclusions more... results with Stream line Diffusion stabilization for noise/free Oseen state of the art approaches linear case nonlinear case conclusions

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Uncertainty Quantification, Uncertainty quantiﬁcation, Polynomial Chaos, uncertainty analysis, model uncertainty, evidence theory, mathematical model, model validation, stochastic PDEs, Monte Carlo

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posted: | 3/26/2011 |

language: | Basque |

pages: | 40 |

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