A Computational Framework for Quantifying and Optimizing the Performance of Observational Networks in 4D-Var Data Assimilation

Transcription

1 A Computational Framework for Quantifying and Optimizing the Performance of Observational Networks in 4D-Var Data Assimilation Alexandru Cioaca Computational Science Laboratory (CSL) Department of Computer Science Virginia Tech August 9, 213 [1/64]

2 Outline 1. Introduction 2. Fusing simulations and observations through data assimilation 3. Quantifying observation contribution Sensitivity analysis Efficient computation Experimental applications 4. Optimal observational networks Formulating the optimization problem Optimizing observation values, weights and locations 5. Ending remarks [2/64]

3 Setting Scientific context Computer-generated numerical simulations of real time-evolving processes Focus area Computational Fluid Dynamics atmosphere, ocean (meteorology, hydrology, climate modeling, air-quality studies etc), industrial processes (manufacturing, energy production, hazard proliferation etc), blood flow and other biological systems (medicine), and many others. General methodology 1. Model the governing physical laws as systems of ODEs/PDEs/SDEs; 2. Solve the equations by discretizing space and time ( numerical models ); 3. Efficient computation for speed and accuracy. Numerical simulation (forecast, prediction) initial value problem [3/64]

4 Challenging problems 1. Anchor simulations in reality using observations of the real system states: surface observations, weather balloons, aircraft, ship, radar, satellite, etc 2. Mitigate the amplification of discretization errors 3. Handle the high non-linearity of fluid flow (turbulence, large eddy etc) 4. Perform model calibration and observations quality control 5. Develop adaptive strategies for simulating and observing 6. Reduce computational cost for large-scale applications [4/64]

5 Data assimilation Assimilating data means fusing measurements and numerical simulations Objective analysis of model states required for restarting forecasts Numerical techniques of considerable computational cost Various flavors of data assimilation: 1. Optimal interpolation 2. Statistical estimators 3. Ensemble-based methods (particle filter, Kalman filter) 4. Variational methods (3D, 4D) 4D-Var (four-dimensional variational) requires 1. A priori estimates of model states (background) 2. Measurements of the real system (observations) 3. Error statistics 4. Numerical models for forecast and sensitivity analysis [5/64]

6 4D-Var data assimilation PDE-constrained optimization problem: J (x ) = 1 2 ( ) T x x b B 1 (x x b ) N k=1 x a = arg min x J (x ) subject to x k = M t t k (x ). (H k (x k ) y k ) T R 1 k (H k (x k ) y k ), J D-Var cost function x k initial solution (at t k ) x b background initial solution B background error covariance y k observations R k observation error covariance H k observation selection operator M nonlinear model (forecast) x a improved initial solution [6/64]

7 4D-Var interpretation Fit model predictions to data PDE-constrained nonlinear optimization Minimize the uncertainty of the model states Inverse problem for estimating parameters of maximum likelihood Reconstruct initial conditions, boundary conditions and other parameters [7/64]

8 4D-Var solution x a is called the minimizer, optimal solution or optimum of J (x ) x a represents an improved estimate of the initial model states 4D-Var first-order optimality condition: x J (x a ) = B 1 (x x b ) + N k= M T,k H k R 1 k (H k (x k ) y k ) = M,k and H k are linearized operators corresponding to M,k and H k x a 4D-Var analysis. Obtaining x a by minimizing 4D-Var cost function assimilating the data. Difficult to compute x a directly for real problems [8/64]

9 Computing the 4D-Var solution 4D-Var optimization is solved iteratively using gradient-based solvers: Quasi-Newton Nonlinear conjugate gradients Truncated Newton Each solver iteration requires: 1. Evaluating the 4D-Var cost function at the current iterate Forecast model runs and simple algebraic operations 2. Generating the descent direction to the next iterate Computing the gradient/hessian of the objective cost function 3. Advancing along the descent direction through linesearch or trust-regions More forecast model runs and gradient/hessian evaluations 4D-Var is characterized by a large computational cost For large-scale applications like weather forecast, the solver is stopped after a few iterations, before reaching the global optimum [9/64]

10 General research interests on 4D-Var Experimental setting 1. Collecting observations y 2. Background estimation x b 3. Learning error statistics B, R k Efficient computation 1. Faster models 2. Better solvers 3. Preconditioning/Accelerating convergence [1/64]

11 Our research interests on 4D-Var 1. Quantify the contribution of each observation in reducing uncertainty 2. Devise efficient computation techniques (speed + accuracy) for practical use 3. Optimize the process of collecting and processing observations [11/64]

12 Quantifying the contribution of observations Theoretical frameworks: 1. Observation impact How much did each observation contribute to reducing the error? 2. Sensitivity analysis To which observations is the forecast error most sensitive? 3. Information theory How much information content does each observation carry? 4. Statistical Design How much trust can we put in each observation? 5. Observability, predictability, controllability Which system states are more difficult to be determined solely from simulations? Experimental frameworks: Observing system experiments real observations Observing simulated system experiments synthetic observations [12/64]

13 Sensitivity analysis Describes locally valid linear dependencies between inputs and outputs of a dynamical system: ( ) xf x f = x x Large sensitivities small input variations translate into large output variations Also applied to scalar cost functions defined on the output model states: ( ) T xf x E(x f ) = x E xf WANTED: Sensitivity of E to y k, R k, x b, B, H etc PROPOSED APPROACH: Chain-rule derivation of sensitivity equations [13/64]

14 Computing sensitivity values Classic approach: Finite Difference Uses just the forward (forecast, nonlinear) numerical model Requires two or more model runs from the perturbed initial solution Still used operationally but obsolete Our approach: Adjoint Models Involves building auxiliary numerical models counterpart to the forward model Solves the linear differential equations of original system by time integration Tangent linear model and its (first-order) adjoint for first-order derivatives Second-order adjoint model for curvature information (computationally expensive) [14/64]

15 Deriving the 4D-Var sensitivity equations 4D-Var optimality condition depends on model states x a AND parameters u: x J (x a, u) = Implicit dependence between x a and u: x a = x a (u) Apply the implicit function theorem to the optimality condition to obtain the sensitivity of x a to u: [ 1 u x a (u) = 2 x,uj (x a, u) 2 x,x J (x a, u)]. [15/64]

16 Deriving the 4D-Var sensitivity equations Replacing u with y k we have: [ ] 1 yk x a = 2 y k,xj (x a ) 2 x,x J (x a ) Derive the first-order optimality condition by y k : to obtain ( ) x J (x a ) = B 1 x a x b + N k=1 2 y k,x J (x a ) = R k H k M,k M T,kH T k R 1 k (H k (x a k) y k ) = The sensitivity of the 4D-Var analysis x a to observations y k is: yk x a = R 1 k H k M,k ( x,x J (x a )) 1 [16/64]

17 Deriving the 4D-Var sensitivity equations Consider the forecast score E as the 2-norm error towards a verification forecast: E(x a ) = ( x a F x v ) T F C ( x a F x v ) F x v F can be seen as a reference forecast C is a matrix used to prescribe the scaling, weighting or selecting. Using chain rule differentiation we obtain the sensitivity of the forecast score yk E(x a ) = yk x a x E(x a ) = yk x a M T,F C (x a F x v ) F [17/64]

18 Deriving the 4D-Var sensitivity equations Sensitivity to observations y k : yk E(x a ) = R 1 k H k M,k ( x,x J (x a )) 1 x E(x a ), Sensitivity to observation error covariance R k : ( ) Rk E = R 1 k [H(x) y] ye(x a ), [18/64]

19 Computing sensitivity analysis Forecast sensitivity to observations: yk E(x a ) = R 1 k H k M,k ( x,x J (x a )) 1 x E(x a ) Computation can be split in three steps: x E(x a ) = MT,F C (x a F ) xv F ) µ = ( 2 x,x J (x a 1 ), y) x E(x a ( ) 2 x,x J (x a, y) µ = x E(x a ) yk E(x a ) = R 1 k H k M,k µ Must build the required computational tools: ) 1 1. adjoint models for evaluating M, M T and ( 2 x,x J (x a, y) 2. linear iterative solvers for obtaining the supersensitivity µ 3. preconditioning, multigrid for accelerating convergence 4. spectral decomposition for low-rank approximations of sensitivity values [19/64]

20 Research objectives 1. Test problem 2D Shallow Water Equations 2. Build adjoint models for computing model state derivatives 3. Validate the benefits of second-order adjoint models 4. Interface the numerical tools according to the sensitivity analysis methodology 5. Efficient computation through preconditioning, multigrid and low-rank approximations 6. Use sensitivities to optimize the parameters of the observational network [2/64]

21 Test model: 2D shallow water equations Shallow water equations (Saint-Venant) describe fluid movement: t h + x (uh) + y (vh) = t (uh) + (u 2 h + 12 ) x gh2 + y (uvh) = t (vh) + x (uvh) + (v 2 h + 12 ) y gh2 =. Simulating the following physical variables: h(t, x, y) is the fluid layer thickness (height), u(t, x, y) and v(t, x, y) are the components of the velocity field, Simplified version of the primitive equations of the atmosphere Conservation of mass and momentum No terms for pressure, friction, Coriolis or viscosity Used to model gravity waves in atmosphere and ocean [21/64]

22 SWE numerical models 1. SWE_EXP Time discretization: explicit, 4th order Runge-Kutta Space discretization: finite volume method Adjoint models: Automatic differentiation CPU time ratios for FWD : TLM : FOA : SOA 1 : 4.5 : 4.5 : 14 (slower models) 2. SWE_IMP Time discretization: implicit, Crank-Nicholson Space discretization: 3rd order upwind finite differences Adjoint models: Manual differentiation CPU time ratios for FWD : TLM : FOA : SOA 1 :.1 :.1 :.15 (faster models) Boundary conditions periodic Cartesian grid of size 4 x 4 48 variables Number of time steps 1; Time step size [22/64]

23 Implicit timestepping discretization schemes Forward model (FWD) x n+1 t 2 f (xn+1 ) = x n + t 2 f (xn ) Tangent linear model (TLM) ( I t ) ( 2 f (x n+1 ) δx n+1 = I + t ) 2 f (x n ) δx n First-order adjoint model (FOA) ( I t ) ( 2 f (x n+1 ) T λ n = I + t ) 2 f (x n ) T λ n+1 Second-order adjoint model (SOA) ( I t ) 2 f (x n+1 ) T σ n = + t 2 + t 2 ( I + t ) 2 f (x n ) T σ n+1 ( f (x n ) δx n) T λ n+1 ( f (x n+1 ) δx n+1) T λ n. [23/64]

24 Data assimilation testing scenario Circular dam problem Reference solution: Build h as Gaussian pulse, u and v as constant fields Can propagate solution through model to make variables consistent (a) Initial time (b) Final time Figure: Model solution for height variable. Note: Solving the 4D-Var problem using information provided by adjoint models is equivalent to using them for sensitivity analysis. [24/64]

25 Crafting the data assimilation scenario Background solution: B correlation matrix scaled with reference solution x b reference solution plus noise of standard deviation 8% Observations: Available at each grid point at the final time (t 1 ) R k diagonal error covariance matrix (observations uncorrelated) y k reference solution run plus white noise Models: Interface models with data Forcing terms into adjoint Comparison different nonlinear solvers: First-order adjoints: L-BFGS, TNSOA, HYBRID, NLCG, CGDESC Second-order adjoints: DANCG, TNFD [25/64]

26 Adjoint sensitivity analysis for minimizing 4D-Var 1 Relative reduction in cost function 1 1 BFGS HYBRID TNFD TNSOA NLCG CGDESC DANCG 1 Relative reduction in cost function BFGS HYBRID TNFD TNSOA NLCG CGDESC DANCG Model Runs Model Runs Figure: Solver convergence for SWE_EXP (left) and SWE_IMP (right) versus time scaled by forward model runs SWE_EXP: Solvers using SOAs can converge as fast as those using FOAs SWE_IMP: Solvers using SOAs can converge faster than those using FOAs [26/64]

27 4D-Var data assimilation results (a) Observations (final time) (b) 4D-Var analysis (initial time) Figure: Observations assimilated with 4D-Var and the resulting initial condition. Now available adjoint models to solve 4D-Var sensitivity equations We can proceed to computing 4D-Var forecast sensitivity to observations [27/64]

28 Computing 4D-Var forecast sensitivity to observations [28/64]

29 Computing 4D-Var forecast sensitivity to observations Computational cost dominated by the solution of the linear system ( ) 2 x,x J (x a ) µ = x E(x a ) Need Hessian-vector products which can be evaluated via: Second-order adjoint models Finite difference from adjoint (gradient) runs: 2 x,x J (x a ) u x J (xa + ɛ u)t x J (x a )T ɛ Gauss-Newton approximation obtained from deriving 4D-Var twice: 2 x,x J (x a ) = B 1 + N k= M T,k HT k R 1 k H k M,k Convergence information generated during the minimization of 4D-Var Inverse of error covariance matrix for the 4D-Var analysis:. 2 x,x J (x a ) = A 1 [29/64]

30 Solving the linear system Working in a matrix-free environment restricts the numerical set of tools Linear solvers Krylov solvers Preconditioners Knowledge of the problem Reuse computation from minimizing 4D-Var Few Hessian-vector evaluations Multigrid Numerical models with coarser space discretizations Krylov solvers for smoothing operators Transfer operators from error correlation As shown during the preliminary exam, preconditioning and multigrid improve the convergence of the iterative linear solvers. [3/64]

31 Figure: Forecast sensitivity to h observations and approximation errors y x (a) perfect observations via second- (b) noisy observations via secondorder adjoint models. order adjoint models (error) y y x x (c) perfect observations via finite dif- (d) perfect observations via Gaussference (error). [31/64] Newton (error).

32 Data pruning 4 y x RMSE [log scale] 1 1 FULL HIGH LOW BFGS iterations [log scale] (a) Location of HIGH (red) and LOW impact observations. (b) h RMS error decrease versus the number of L-BFGS iterations. RMSE [log scale] FULL HIGH LOW RMSE [log scale] FULL HIGH LOW BFGS iterations [log scale] BFGS iterations [log scale] (c) u RMS error decrease versus the number of L-BFGS iterations. (d) v RMS error decrease versus the number of L-BFGS iterations. [32/64]

33 Sensor malfunctioning y x (a) 4D-Var increment y x (b) Supersensitivity field (c) Sensitivity to observations Figure: Observation sensitivity field when the assimilated data is corrupted at two locations with coordinates (1,1) and (2,2). The location of the faulty sensors is unknown to the data assimilation system, but is retrieved via the observation impact methodology. [33/64]

34 Computing low-rank approximations for the 4D-Var forecast sensitivity to observations [34/64]

35 Singular value decomposition for the observation sensitivity matrix Consider the sensitivity matrix mapping from model space to observation space [ ] T = yx a x a = = R 1 k H k M,k A y Singular Value Decomposition (SVD) is a popular technique used in image reconstruction, information retrieval, data analysis etc for: Principal Component Analysis Reduced Order Modeling Error Estimation For a given matrix T (not necessarily square) SVD represents a factorization: where the factor matrices are: T = U S V T U right-singular vectors as matrix columns S singular values as matrix diagonal entries V left-singular vectors as matrix columns [35/64]

36 Low-rank approximations for sensitivity to observations Each triplet U i, S i, V i captures one singular mode of T action Leading (dominant) singular vectors are associated with the largest singular values T low-rank approximations (rank p) from dominant modes in the truncated SVD: (T) (p) = U (p) S (p) V T (p) Computationally challenging to perform SVD on large matrices Even more challenging for a chain sequence of matrix-free operators (T) We propose two approaches for matrix-free SVD: 1. Iterative: Computing the leading singular pairs one at a time 2. Parallel: Computing the leading singular pairs all at once [36/64]

37 Deriving the serial algorithm Compute the leading eigenvectors of the product between the observation impact matrix and its transpose, T T = US 2 U, where T T = A N M,kM,k A = k=1 N T k T k, k=1 M,k = R 1 k H k M,k, T k = M,k A. Start from the truncated eigenvalue decomposition for A : ( ) A = (V D V ) 1 = V D 1 V V (p) D 1 (p) V (p) = (A ) (p). Eigenvalue decomposition for A (or A 1 ) can be obtained via: Jacobi-Davidson (JDQZ) Lanczos Arnoldi other Krylov-based approaches [37/64]

38 Deriving the iterative algorithm Plug the low-rank approximation (A ) (p) into the expression of T T k T k ( ) ( ) V (p) D 1 (p) V (p) M,k M,k V (p) D 1 (p) V (p) = V (p) D 1 (p) W k W k D 1 (p) V (p), Efficient computation Perform economy SVD on the small product matrix: D 1 (p) W D 1 (p) = VredDredV red, Low-rank approximation for T is obtained as: T T ( V (p) V red ) Dred ( V(p) V red ). D red is the matrix of dominant singular values V p V red the matrix of left singular vectors. [38/64]

39 Iterative algorithm 1. Solve iteratively the eigenvalue problem for the 4D-Var Hessian; 2. Map newly generated eigenvectors through the tangent linear model; 3. Compute the truncated SVD of the resulting (small-size) matrix; 4. Project the left singular vectors onto the eigenvector base of the 4D-Var Hessian; 5. Build the low-rank approximation of T. [39/64]

40 Deriving the parallel algorithm Random sampling techniques exhibit trivial parallelism 1. Draw p random vectors and form a matrix Ω. 2. Compute the product Y = A 1 Ω using Hessian-vector multiplications, i.e., running the second order order adjoint model for each column. 3. Construct the QR decomposition Y = QR. Q is an orthonormal basis for the range of Y, but also the orthonormal factor in: A 1 = Q B, B = Q A 1, B = A 1 Q. Efficient computation Perform economy SVD of B to obtain: A 1 = Q U B Σ B V B = U A Σ B V B. Obtain approximate T from TLM runs with the columns of the pseudoinverse of A : T p M,k (A 1 )+ p = M,k V B Σ + B U A. [4/64]

41 Parallel algorithm 1. Build the matrix B, through parallel second-adjoint runs; 2. Compute a full SVD of B; 3. Project the left singular vectors of B in Q and form the SVD of A 1 ; 4. Compute the Hessian pseudoinverse A + ; 5. Build the impact matrix T through tangent linear runs. [41/64]

42 Low-rank observation impact 1 5 Value RMS Mode (a) Singular value spectrum Rank (b) Truncation error Figure: Singular value decay for the observation impact matrix T and the corresponding truncation error norm. [42/64]

43 Low-rank observation impact y y x x.2.4 (a) Low-rank estimate (b) Truncation error field Figure: Low-rank approximation of observation sensitivity for h data and the associated truncation error field for 16 modes. [43/64]

44 Low-rank observation impact y 2 y x x.5 (a) Full-rank impact for single center obs. (b) Low-rank impact for single center obs y 2 y x 2 4 x (c) Full-rank impact for single corner obs. (d) Low-rank impact for single corner obs. [44/64]

45 Minimizing the forecast error with respect to parameters of the observational network [45/64]

46 Optimizing the observational network Consider again the forecast score E defined on the data assimilation output: E(x a ) = ( x a F x v F ) T C ( x a F x v F ). We can now compute the sensitivity of E to parameters u: ue(x a ) = T x a. Sensitivities describe directions of descent for E in u parameter space Proposed approach nonlinear optimization for parameter values: 1. Solve the 4D-Var optimization problem to obtain improved initial condition x a 2. Evaluate the cost function E 3. Compute sensitivity to observations yk E(x a ) 4. Change parameter values along descent directions y new k 5. Repeat = y prev k α yk E(x a ) [46/64]

47 Nonlinear optimization problem Optimization constrained by optimization (data assimilation) Tuning parameters of an optimization problem Meta-optimization Objective cost function: u opt = arg min E ( x a ) v u subject to { x a = arg min x J ( x, u ), x a v = M t t v (x a ). First-order optimality condition: ( ) 1 ( ) ue(x a v(u)) = 2 u,x J 2 x,x J M T,v C x a v x verif v =. [47/64]

48 Application #1: optimal observation values Objective cost function for u = y k (y k ) opt = arg min E ( x a ) v y k subject to { x J ( x a, y k ) =, x a v = M t t v (x a ). Already have the gradient of E to y k : 1 ( ) yk E = R 1 k H k M,k ( 2 x,x J (x)) a M T,v C x a v x verif v. Reconstructs observation values which lead to the verification forecast, once assimilated with 4D-Var Useful for detecting observational errors or crafting simulated experiments [48/64]

49 Application #1: optimal observation values Reference model trajectory height field aligned along the South-North direction y y x (a) Initial Time x (b) Observation Time 99.9 Figure: The reference height field h at the initial and final (observation) times. [49/64]

50 Application #1: optimal observation values Assimilating the wrong observations leads to an initial solution aligned along the East-West direction y 2 1 y x 99.9 (a) Faulty Observations 2 4 x (b) 4D-Var Analysis 1 Figure: The faulty (unoptimized) observations of the height field h and the corresponding 4D-Var analysis. [5/64]

51 Application #1: optimal observation values The forecast score quantifies the error between the 4D-Var forecast and the verification forecast By adjusting the values of the argument (observations to be assimilated), the 4D-Var forecast is getting closer and closer to the verification Eventually, the optimized observation values correspond to what we would have expected to observe based on the reference trajectory Cost function 1 1 y Outer iterations (a) L-BFGS convergence 2 4 x (b) Optimized observation values Figure: The minimization of the verification cost function E and the optimized observations at assimilation time t 1. [51/64]

52 Application #2: optimal observation weights Objective cost function for u = R k (R k ) opt = arg min E ( x a ) v R k subject to { x J ( x a, R k ) =, x a v = M t t v (x a ). Can compute gradient of E to R k from y k : Rk E = ( ) R 1 k [H(x) y] ye(r k ), In 4D-Var, each observation is associated with a weight representing a measure of trust The weighting coefficients are prescribed through the observation error covariance R k Previous research in this field only succeedeed at tuning the global weighting of B vs R k Our approach aims at dynamically tuning the observation weights based on reducing the 4D-Var forecast error [52/64]

53 Application #2: optimal observation weights Observations on a subdomain contain a significant level of noise 4D-Var analysis does not clearly reflect noise presence y 2 11 y 2 y x x 2 4 x 9 (a) Perfect Observations (b) Prescribed Noise (physical variable units) (c) 4D-Var Analysis Figure: The h observations, the prescribed observation noise, and the resulting 4D-Var analysis using the initial specification of the error covariances. [53/64]

54 Application #2: optimal observation weights The forecast score quantifies the error between the 4D-Var forecast and the verification forecast By adjusting the values of the argument (observation weights), the 4D-Var forecast is getting closer and closer to the verification Eventually, the optimized observation weights reflect the noise present in the observations Cost function y y Outer iterations 2 4 x 2 4 x 9 (a) L-BFGS Convergence (b) Optimized Covariances (c) 4D-Var Analysis Figure: The minimization of the verification cost function, the optimized h observation error covariances, and the resulting 4D-Var analysis using the improved values. [54/64]

55 Application #3: optimal sensor locations For this application, we use sparse observations Must specify the observation selection operator H interpolation scheme Inverse Distance Weighting interpolation is widely used in geographic information systems: H k (l x, l y ; z) = i d 1 z i i, if d i, d 1 i i z i, if d i =, [ (lx ) where d i = l 2 xi + ( ) ] l y l 2 1/2 yi. Gradient of H with respect to 2D locations (l x, l y ): (d i ) 2 (d j ) 1 ( ) ( ) l x l xi zi zi z j i j lx H k (l x, l y ; z) =, 2 (d i ) 1 i [55/64]

56 Application #3: optimal sensor locations Objective cost function for u = (l x, l y) (l x, l y) opt = arg min l x,l y E ( (l x, l y)) subject to { x J ( x a, L ) =, x a v = M t t v (x a ). Gradient of E with respect to locations (l x, l y) ( 1 ( ) lx E = 2 l x,x J 2 x,x J (x)) a M T,v C x a v x verif v, Derive 4D-Var first-order optimality condition with respect to H k : 2 l x,x J = lx H k (l x, l y; x a k) + M T,k ( lx H k (l x, l y; x a k)) T R 1 k (H k (l x, l y; x a k) y k ) Second-term containing the innovation vector is negligible and can be ignored [56/64]

57 Application #3: optimal sensor locations Choose the initial configuration for the location of 3 observations By relocating the observations, the 4D-Var forecast is getting closer and closer to the verification Newly obtained locations provide superior information for the data assimilation process 2 y x 4 (a) Initial locations Cost function Outer iterations (b) L-BFGS convergence y x 4 (c) Optimized locations Figure: The optimization of sensor locations for the third testing scenario: initial locations, numerical solver convergence, and optimal locations. [57/64]

58 Application #3: optimal sensor locations Choose the initial configuration for the location of 3 observations By relocating the observations, the 4D-Var forecast is getting closer and closer to the verification Newly obtained locations provide superior information for the data assimilation process y x 4 (a) Initial locations Cost function Outer iterations (b) L-BFGS convergence y x 4 (c) Optimized locations Figure: The optimization of sensor locations for the second testing scenario: initial locations, numerical solver convergence, and optimal locations. [58/64]

59 Application #3: optimal sensor locations Choose the initial configuration for the location of 3 observations By relocating the observations, the 4D-Var forecast is getting closer and closer to the verification Newly obtained locations provide superior information for the data assimilation process 25 y x 4 (a) Initial locations Cost function Outer iterations (b) L-BFGS convergence y x 4 (c) Optimized locations Figure: The optimization of sensor locations for the first testing scenario: initial locations, numerical solver convergence, and optimal locations. [59/64]

60 Research achievements Built a computational framework for observation impact via sensitivity analysis Developed adjoint models competitive for practical use Efficient computation via preconditioning, multigrid and low-rank approximations Optimized parameters of the observational networks based on sensitivity results [6/64]

61 Future Work Apply the computational methodology to large-scale problems (WRF) Develop new algorithms for efficient computation Compare the sensitivity approach to other observation impact measures Optimize sensor locations using different interpolation schemes Combinatorial and mixed-integer programming for observational networks [61/64]

62 Bibliography Alexe, M., Cioaca, A., & Sandu, A. (21, April). Obtaining and using second order derivative information in the solution of large scale inverse problems. In Proceedings of the 21 Spring Simulation Multiconference (p. 85). ACM. Cioaca, A., Zavala, V., & Constantinescu, E. (211, November). Adjoint sensitivity analysis for numerical weather prediction: Applications to power grid optimization. In Proceedings of the first international workshop on High performance computing, networking and analytics for the power grid (pp ). IEEE/ACM 24th International Conference for High Performance Computing, Networking, Storage and Analysis. Cioaca, A., Alexe, M., & Sandu, A. (212). Second-order adjoints for solving PDE-constrained optimization problems. Optimization Methods and Software, 27(4-5), Cioaca, A., Sandu, A., De Sturler, E., & Constantinescu, E. (212). Efficient computation of observation impact in 4D-Var data assimilation. Uncertainty Quantification in Scientific Computing. IFIP Advances in Information and Communication Technology, 377, Rao V., Cioaca, A., & Sandu. A. (212, November). A highly scalable approach for time parallelization of long range forecasts. In Proceedings of the Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems (ScalA). IEEE/ACM 25th International Conference for High Performance Computing, Networking, Storage and Analysis. [62/64]

63 Bibliography Sandu A., Cioaca. A, Rao. V. (213). Dynamic sensor network configuration in InfoSymbiotic systems using model singular vectors Cioaca A., Sandu, A., & De Sturler E. (213). Efficient methods for computing observation impact in 4D-Var data assimilation. ACCEPTED at Journal of Computational Geosciences. Cioaca A., Sandu, A., (213). Low-Rank Approximations for Observation Impact in 4D-Var Data Assimilation. Cioaca A., Sandu, A., (213). An Optimization Approach for Observational Networks in 4D-Var Data Assimilation. [63/64]

64 Many thanks to the following: Dr. Adrian Sandu (academic advisor) Dr. Cal Ribbens Dr. Clifford Shaffer Dr. Traian Iliescu Dr. Eric de Sturler Computational Science Laboratory Department of Computer Science [64/64]