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1 Adaptive Observations And Multilevel Optimization In Data Assimilation by S. Gratton, M. M. Rincon-Camacho and Ph. L. Toint Report NAXYS May 203 ENSEEIHT, 2, rue Camichel, 3000 Toulouse, France CERFACS, 42, avenue Gaspard Coriolis, 3057 Toulouse, France University o Namur, 6, rue de Bruxelles, B5000 Namur (Belgium)

2 Adaptive Observations And Multilevel Optimization In Data Assimilation Serge Gratton, M. Monserrat Rincon-Camacho, and Philippe L. Toint May 2, 203 Abstract We propose to use a decomposition o large-scale incremental our dimensional (4D-Var) data assimilation problems in order to make their numerical solution more eicient. This decomposition is based on exploiting an adaptive hierarchy o the observations. Starting with a low-cardinality set and the solution o its corresponding optimization problem, observations are adaptively added based on a posteriori error estimates. The particular structure o the sequence o associated linear systems allows the use o a variant o the conjugate gradient algorithm which eectively exploits the act that the number o observations is smaller than the size o the vector state in the 4D-Var model. The method proposed is justiied by deriving the relevant error estimates at dierent levels o the hierarchy and a practical computational technique is then derived. The new algorithm is tested on a D-wave equation and on the Lorenz-96 system, the latter one being o special interest because o its similarity with Numerical Weather Prediction (NWP) systems. Keywords: Data assimilation, adaptive observations, numerical algorithms, multilevel optimization, a posteriori errors. Introduction In data assimilation, a substantial body o research has been conducted in the ield o adaptive observations. The main idea in this area is to adapt the set o observations used in the model calibration to improve computational eiciency while not sacriicing accuracy. Several techniques have been considered, depending on the practical context o the relevant application. In a Kalman ilter ramework, Leutbecher (2003b) introduces a Hessian reduced rank estimate whose aim is to predict expected change in the square norm o orecast error due to intermittent modiications o the observation network; its application to the Lorenz96 model (see Lorenz, 995) is presented in Leutbecher (2003a). Lorenz and Emanuel (998) present dierent strategies or weather orecasting which target the locations where the irst-guess errors are largest, and veriy i the including additional data improve the analyses and orecasts. Berliner, Lu and Snyder (999) introduce rigorous mathematical criteria to select uture data in a dynamical process, based on the uncertainty o the current analysis, the dynamics o error evolution, the orm and errors o observations and data assimilation. In Hansen and Smith (2000) the perormance o various observation targeting strategies is compared or both direct insertion and ensemble Kalman ilter assimilation. The authors conclude that the optimal adaptive observation strategies depend on the combination o model, assimilation scheme and observational network. Daescu and Navon (2004) describe a method based on a periodic update o the adjoint sensitivity ield which takes into account the ENSEEIHT, 2, rue Camichel, 3000 Toulouse, France and CERFACS, 42, avenue Gaspard Coriolis, 3057 Toulouse, France. (serge.gratton@enseeiht.r). CERFACS, 42, avenue Gaspard Coriolis, 3057 Toulouse, France. (monserratrc@ceracs.r). Namur Center or Complex Systems, University o Namur, 6, rue de Bruxelles, B-5000 Namur, Belgium. (philippe.toint@unamur.be).

3 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 2 interaction between time-distributed adaptive and routine observations. Cardinali, Pezzulli and Andersson (2004) introduce an inluence matrix whose purpose is to identiy inluential data in a 4D-Var ramework. They observe that low-inluence data occur in areas with a large number o observations while high-inluence data points occur in regions with sparse ones or in dynamically active areas. Finally, Liu, Kalnay, Miyoshi and Cardinali (2009) describe, in a Kalman ilter setting, the impact o the sel-sensitivity and the cross sensitivity (o-diagonal elements o inluence matrix). Our motivation or the present paper lies in several o the conclusions drawn in this literature. In particular, our work aims at a good numerical exploitation o the conclusion that the spatial distribution o observations may be inluential. More speciically, we introduce new techniques or adaptively selecting observations in a 4D-Var setting, based on mathematically sound a posteriori error estimates within a dynamic hierarchy. The technique used or constructing the adaptive set o observations is inspired by adaptive inite elements techniques studied in Section 2.2 (p. 00) o Rincon-Camacho (20). Related techniques or adaptive multigrid can be ound in Brandt (973) and McCormick (984). The paper is structured as ollows: Section 2 introduces the 4D-Var vocabulary and the considered hierarchy o observations. Section 3 then covers the associated error estimates and Section 4 presents the new adaptive algorithm. Section 5 discusses its application to a one-dimensional nonlinear wave equation and to the Lorenz96 model. Conclusions and perspectives are discussed in Section 6. 2 The problem and associated observation hierarchy Consider the nonlinear least-squares problems in 4D-Var data assimilation, whose objective is to ind an initial vector state at an initial time denoted as x = x(t 0 ) IR n. The structure o the problem is as ollows: min 2 x x b 2 x IR n B + 2 N t j=0 H j (x(t j )) y j 2 R j (2.) where the squared norm x 2 M is induced by the inner product xt Mx or a symmetric positive deinite matrix M IR l l and a vector x IR l. Here x b IR n is the background vector, which is an a priori estimate. The vector y j IR mj is the vector o observations at time t j and H j is the operator modeling the observation process at the same time. The state vector x(t j ) satisies the nonlinear model o evolution x(t j ) = M 0 j [x(t 0 )]. The matrix B IR n n is a symmetric positive deinite matrix representing the background-error covariance and the matrix R j IR mj mj is also a symmetric positive deinite matrix representing the observation-error covariance at time t j. The nonlinear least-squares problem (2.) is solved iteratively, and, at iteration k, the problem is simpliied by linearizing the nonlinear observation operator H j (x(t j )) at the iterate x k (which or the moment we denote only by x), leading to the optimization problem min 2 x x b + δx 2 δx IR n B + 2 H δx d 2 R (2.2) where R = diag(r 0, R,..., R Nt ) IR m m, d IR m denotes the concatenated misits over time y j H j (x(t j )) and H IR m n is the concatenated version o the linearized observation process d 0 H 0 d d =. H M 0 IRm, H =. IRn m. d Nt H Nt M 0 Nt Here H j and M 0 j are a (possibly approximate) linearization o the observation operator H j and the model M 0 j around x = x k (t 0 ) respectively.

4 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 3 Diverse techniques or solving problem (2.) have been proposed, see or instance Courtier, Thépaut and Hollingsworth (994) or the so called incremental method which is equivalent to applying a truncated Gauss-Newton iteration to problem (2.) (see Gratton, Lawless and Nichols, 2007). The general algorithm is the ollowing: Algorithm 2. Incremental 4D-Var. Initialize x 0 = x b IR n and set k = Compute x k (t j ) rom x k (t 0 ) = x k by running the nonlinear model M rom time t 0 to t Nt. 3. Calculate the vectors d k,j = y j H j (x k (t j )) or j =,..., N t. 4. Find an approximate solution δx k o the minimization problem (2.2), where x = x k, H = H k and d = d k. 5. Update the current solution as x k+ = x k + δx k. 6. Set k := k +. I convergence is not achieved return to Step 2. This algorithm is known as the outer loop and the minimization step (Step 4) as the inner loop. Our interest is to reduce the cost o this optimization problem. Termination criteria or the outer and inner loops o this algorithm are discussed in Gratton et al. (2007). It is most natural to solve the subproblem in Step 4 o Algorithm 2. directly in the space o dimension n, the size o x. This technique is reerred to as the primal approach. At variance, the quadratic optimization problem (2.2) may be rewritten in a space o dimension m, the number o observations. This is known as the dual approach and it is especially useul when m is signiicantly smaller than n, in which case the second term in (2.2) is o relatively low rank compared to the irst. A irst approach o this type is the Physical-space Statistical Analysis System (PSAS) method (see Courtier, 997), where the low rank observation term in (2.2) is handled by using a Sherman-Morrison ormula and applying the standard conjugate-gradient algorithm (see Hestenes and Stieel, 952, or Golub and Van Loan, 996, Section 0.2, p. 520) to the reduced system. This method has the drawback o not maintaining the inherent monotonicity o the conjugate-gradient algorithm in IR n, thereby making any stopping rule o the inner minimization diicult to deine (see El Akkroui, Gauthier, Pellerin and Buis, 2008, or Gratton, Gürol and Toint, 203). Fortunately, a better alternative is available, where the conjugate-gradient algorithm is reormulated in IR m using the inner product deined by the metric HBH T in order to guarantee the desired monotonicity properties without incurring additional cost. This technique, known as the Restricted Preconditioned Conjugate Gradient method (RPCG, see Gratton and Tshimanga, 2009 and Gratton et al., 203), provides an eicient numerical procedure where substantial computing gains are obtained when m n. We reer the reader to the cited publications or details. Our aim in this work is to make the best possible use o this dual technique and to propose an adaptive observations strategy or solving the data assimilation problem (2.). Suppose that we have a large set o m observations O which can be decomposed into a hierarchical collection o sets {O i } r i=0, each with cardinality m i, such that O i O i+ and m i < m i+ (i = 0,..., r ) where, by convention, O r = O and m r = m. To each set o observations O i we associate a misit vector y i IR mi (by selecting the relevant components in the vector y), and the corresponding observation-error covariance matrix R i. Given {O i } r i=0, we may thereore consider the hierarchical collection o minimization problems min 2 x x b 2 x IR n B + 2 H i (x) y i 2, i = 0,..., r, (2.3) R i where the vector H i (x) denotes the nonlinear observation operator concatenated over time. Thus our objective is to construct the collection {O i } r i=0 such that the sequential solution o the problems (2.3) is obtained at signiicantly lower cost compared to solving (2.2) directly, while at the same time maintaining equivalent accuracy requirements.

5 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 4 3 Mathematical analysis Our adaptive method is based on the exploitation o a posteriori bounds or the error between the solutions obtained using ew observations or many. In particular, we are interested in comparing the accuracy o the solutions o problems (2.3) or O i and O i+. For simplicity o notation, we denote them as O c a set with m c observations and O a set containing m observations such that m c < m and O c O, where the indices c and stand or coarse and ine, respectively. The ine optimization problem (2.3) is given, or the starting point x, by min 2 x + δx x b 2 δx IR n B + 2 H δx d 2 R (3.4) where d = y H (x) and H is the linearized version o the observation process at x involving the set o observations O. We may reormulate (3.4) as a convex quadratic problem with linear equality constraints given by min 2 x + δx x b 2 δx IR n, v IR m B + 2 v 2, subject to v R = H δx d, (3.5) see Gratton et al. (203). The Lagrange unction corresponding to this reormulated problem is given by L(δx, v, λ ) de = 2 x + δx x b 2 B + 2 v 2 λ T R (H δx v d ). Using this unction, the optimality conditions or problem (3.5) can then be expressed as which leads to the system δx L(δx, v, λ ) = B (x + δx x b ) H T λ = 0, v L(δx, v, λ ) = R (v ) + λ = 0, λ L(δx, v, λ ) = v H δx + d = 0, H T λ = B (x + δx x b ), λ = R (H δx d ). Gratton et al. (203) and Gratton and Tshimanga (2009) show that the solution o this system can be obtained by solving (3.6) (R H BH T + I m )λ = R (d H (x b x)), (3.7) or λ and then substituting in the second equation o (3.6) or δx and v. We now compare the solution (δx, λ ) to the solution o the coarse level minimization problem min 2 x + δx c x b 2 δx c IR n B + 2 Γ (H δx c d ) 2, (3.8) Rc where Γ is the restriction operator Γ : IR m IR mc rom the ine observation space to the coarse one. As above, we reormulate (3.8) as a convex quadratic problem with linear equality constraints, and obtain min 2 x + δx c x b 2 δx c IR n, v c IR mc B + 2 v c 2, subject to v Rc c = Γ (H δx c d ), (3.9) whose Lagrangian unction is given by L(δx c, v c, λ c ) de = 2 x + δx c x b 2 B + 2 v c 2 λ T Rc c (Γ H δx c v c Γ d ). The optimality conditions now become H T ΓT λ c = B (x + δx c x b ), λ c = R c Γ (H δx c d ) (3.0)

6 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 5 These conditions may again be solved by computing λ c such that (Rc Γ H BH T Γ T + I mc )λ c = Rc Γ (d H (x b x)). (3.) Since the dimension o the system (3.) is smaller than that o the system (3.7), solving problem (3.8) is (oten much) cheaper than solving problem (3.4). The RPCG algorithm mentioned in the previous section derives its eiciency by using the ormulation (3.) rather than (3.7), see Gratton et al. (203) and Gratton and Tshimanga (2009). Ater obtaining the solution (δx c, λ c ), our objective is now to compute the dierence between the (still unknown) λ and Π c λ c in order to identiy the set o observations where this dierence is large. Here Π c is the prolongation operator given by Π c de = σ Γ T (3.2) or some σ > 0. Our intention is then to construct the ine problem rom the coarse one by adding to the coarse the observations that are singled out by this comparison. We start by computing the dierence between λ and Π c λ c in an appropriate norm. Using (3.6) and (3.0), we deine and E 2 E de = λ Π c λ c 2 R = R (λ Π c λ c ), R (H δx d ) Π c λ c = λ Π c λ c, H δx + d R Π c λ c, de = λ Π c λ c 2 H BH T = H T λ H T Π cλ c 2 B By adding errors E and E 2 we obtain that = B(H T λ H T Π cλ c ), B (x + δx x b ) H T c Π c λ c = λ Π c λ c, H (x x b ) + H δx H BH T Π cλ c. E + E 2 = λ Π c λ c, H δx + d R Π c λ c As a consequence, we deduce that + λ Π c λ c, H (x x b ) + H δx H BH T Π cλ c + λ Π c λ c, H δx c H δx c = λ Π c λ c, d R Π c λ c H δx c + λ Π c λ c, H (x x b + δx c ) H BH T Π cλ c. E + E 2 = R /2 (λ Π c λ c ), R /2 (d H δx c R Π c λ c ) Thus, we obtain that + H BH T (λ Π c λ c ), (H BH T ) (H (x x b + δx c ) H BH T Π cλ c ) = R /2 (λ Π c λ c ), R /2 (d H δx c R Π c λ c ) + B /2 H T (λ Π c λ c ), B /2 H T (H BH T ) (H (x x b + δx c ) H BH T Π cλ c ). E + E 2 λ Π c λ c R d H δx c R Π c λ c R + H T λ H T Π cλ c B H T (H BH T ) (H (x x b + δx c ) H BH T Π cλ c ) B λ Π c λ c R d H δx c R Π c λ c R + λ Π c λ c H BH T H (x x b + δx c ) H BH T Π cλ c (H BH T ).

7 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 6 The Cauchy-Schwarz inequality then yields that [ ] /2 E + E 2 λ Π c λ c 2 R + λ Π c λ c 2 H BH T [ ] /2 d H δx c R Π c λ c 2 + H R (x x b + δx c ) H BH T Π cλ c 2 (H BH T ) = [ E + E 2 ] /2 [ ] /2 d H δx c R Π c λ c 2 + H R (x x b + δx c ) H BH T Π cλ c 2 (H. BH T ) This inequality leads to the a posteriori error bound E + E 2 d H δx c R Π c λ c 2 R + H (x x b + δx c ) H BH T Π c λ c 2 (H BH T ). (3.3) The question then arises o how to compute the second term in the right hand side o this inequality, i.e. H (x x b + δx c ) H BH T Π cλ c 2 (H BH T ) = H (x x b + δx c ) H BH T Π cλ c, (H BH T ) ( H (x x b + δx c ) H BH T Π cλ c ) = H (x x b + δx c ) H BH T Π cλ c, (H BH T ) H (x x b + δx c ) Π c λ c. But the irst equation o (3.0) multiplied by H B gives that (3.4) H BH T Γ T λ c = H (x x b + δx c ), (3.5) and hence that Thereore Γ T λ c = (H BH T ) H (x x b + δx c ). (3.6) H (x x b + δx c ) H BH T Π cλ c 2 (H BH T ) because o (3.2). We note also that = H (x x b + δx c ) H BH T Π cλ c, Γ T λ c Π c λ c = ( σ ) H (x x b + δx c ) H BH T Π cλ c, Π c λ c, (3.7) λ Π c λ c 2 R +H BH T = λ Π c λ c 2 R + λ Π c λ c 2 H BH T = E + E 2. (3.8) As a consequence, the let-hand side o this inequality (the sought error estimate) can be bounded above using inequality (3.3), giving the ollowing a posteriori error. Theorem 3. Let δx be the solution to the problem (3.5) and λ the corresponding Lagrange multiplier to the constraint in (3.5) such that the couple (δx, λ ) satisies (3.6). Analogously, let δx c be the solution to (3.9) and λ c the corresponding Lagrange multiplier such that (δx c, λ c ) satisies (3.0). Then the a posteriori error bound satisies the inequality λ Π c λ c 2 R +H BH T d H δx c R Π c λ c 2 R + ( σ ) H (x x b + δx c ) H BH T Π cλ c, Π c λ c. (3.9) Note that the bound (3.9) does not involve the computation o λ or δx. The use o an a posteriori error bound depending on a primal-dual problem in adaptive inite elements was studied in Rincon-Camacho (20). In the next section we describe how to make use o the bound (3.9) in order to algorithmically construct the ine problem rom the coarse one.

8 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 7 4 A new adaptive algorithm Starting rom a small set o observations O c, our goal is to add only signiicant observations to produce O so that the a posteriori error (3.9) is reduced. Our strategy is to deine an auxiliary set o potential ine observations Õ rom which the observations in O are selected. However, describing our strategy (and algorithm) requires additional assumptions on the hierarchy o (potential) observations. More speciically, we complete our assumptions as ollows. The observations correspond to localizations in some underlying continuous measurable observation space. The coarse observation set partitions the observation space in a inite number o cells {c j } pc j= o measures {w j } pc j=. The auxiliary set Õ is constructed by considering all observations in O c with the addition o a single additional potential observation point in the interior o each cell. The cell is said to be associated with this additional potential observation. There exists a prolongation operator Π c rom O c to Õ such that, or each potential observation o j in Õ \ O c, Π c deines the value o this observation only in terms o the observations o the associated cell c j. As expected, we deine Π c = σ ΓT. We illustrate these assumptions by an example: suppose that the observation space is the plane and the coarse observation set O c is the rectangular grid shown in Figure 4. (a): the cells are elementary rectangles in this grid, whose measure is given by their surace. We may then deine Õ as the grid shown in Figure 4. (b), which we obtained by locally adding a new potential observation in the center o each rectangle and our additional ones on its boundary (eectively doubling the mesh in every direction). The observations in O (as shown in Figure 4. (c)) can then be extracted rom Õ. (a) Coarse observation set O c (b) Auxiliary observation set Õ (c) Fine observation set O Figure 4.: Auxiliary observations In this example, the restriction operator Γ can be deined as the usual ull weighting operator associated with bilinear interpolation prolongations (which justiies the introduction o the

9 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 8 boundary points). This ull-weighting restriction operator is given, on every grid node, by the stencil (4.20) 6 2 In order to achieve our goal to select important observations rom Õ, we need to compute localized error indicators (3.9). We deine them as η j ν j de = w j ( d H δx c R Πc λ c ) j, R ( d H δx c R Πc λ c ) j, de = ( σ )w j ( H (x x b + δx c ) H B H T Π c λ c ) j, Π c λ c j, where d, H, R are constructed rom the set o observations Õ and where the symbol j denotes the restriction o the associated quantity to the cell c j. Note that the η s correspond to the irst term on the right-hand side o (3.9) while the ν s correspond the the second term. In order to decide which cells will be chosen to include a new interior potential observation, we use the bulk criterion (also known as Dörler marking, see, or instance, Dörler, 996, Morin, Nochetto and Siebert, 2000, Logg, Mardal and Wells, 202). For some constants θ, θ 2 (0, ), we construct the sets S η and S ν such that p θ η j p η k and θ 2 ν j ν k. (4.2) j= k S η j= k S ν In practice this construction is carried out by progressively constructing each set using a greedy heuristic which includes irst the non-included cell with maximal indicator value. Once S η and S ν are constructed, we decide that a cell k o O c is reined i k S η or k S ν, meaning that the observations associated with the corresponding cell in Õ are added to the set O c to construct the new set O. More ormally, de O = O c. (4.22) k S η S ν o k Thus, starting with a small set o observations O 0, we progressively add observations using the method just described, resulting in the ollowing algorithm. Algorithm 4. An algorithm using adaptive observations. Set i = 0, initialize x and the coarse observation set O Given the set o observations O i, construct the auxiliary set Õ i+ such that the conditions described at the beginning o this section hold. 3. Find the solution (δx i, λ i ) to the problem min 2 x i + δx i x b 2 δx i IR n B + 2 Γ i+ ( H i+ δx i d i+ ) 2, (4.23) R i by approximately solving the system (R i Γ i Hi+ B H i+ Γ T T i+ + I mi )λ i = R i Γ i+ ( d i+ H i+ (x b x i )) using RPCG and then setting δx i = x b x i + B H T i+ Γ T i λ i. 4. For each cell c j o observation set O i compute the error indicators η j = w j ( d i+ H i+ δx i R i+ Πi λ i ) j, R i+ ( d i+ H i+ δx i R i+ Πi λ i ) j, ν j = ( σ i+ )w j ( H i+ (x i x b + δx i ) H i+ B H T i+ Π i λ i ) j, Π i λ i j.

10 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 9 5. Build the sets S η and S ν such that p i θ p i η k, and θ 2 k S η using the bulk criterion. η j ν j j= j= k S ν η k. 6. Construct the set O i+ as. O i+ := O i k S η S ν o k 7. Update x x + δx i. 8. Increment i and return to Step 2. We note that the computation o (δx i, λ i ) in Step 3 corresponds to applying the RCPG algorithm to (4.23), thereby making this computation essentially dependent on m, the number o observations, which the algorithm maintains as small as necessary by design. We also note that the bulk o the remaining work is in the computation o the error indicators in Step 4, as this requires the product o H i+ with δx i. Observe that, because o (3.2), HT i+ Πi λ i = σ i+ HT i+ ΓT i+ λ i and that both this quantity and the product H i+ (x i x b ) are already available as by-products o Step 3. Similar methods or constructing adaptive grids are the multi-level adaptive technique (MLAT) studied in Brandt (973) and the ast adaptive composite grid (FAC) presented in McCormick (984). 5 Numerical experiments 5. Two test problems This section is devoted to showing the perormance o our new adaptive algorithm on two test cases. The irst one is a one-dimensional wave equation system, which we reer to as the D-Wave model rom now on. The dynamics on this model are governed by the ollowing nonlinear wave equations: u(z, t) 2 u(z, t) + (u) = 0, t z u(0, t) = u(, t) = 0, (5.24) u(z, 0) = u 0 (z), u(z, 0) = 0, t 0 t T, 0 z, where we have chosen (u) = µe ηu. In this case we look or the initial unction u 0 (z), which corresponds to x in the data assimilation problem (2.). We illustrate in Figure 5.2 (a) the initial state vector u 0 (x = u 0 ) and the evolution o the system in Figure 5.2 (b) (view rom the top) where the space domain corresponds to the horizontal axis and the time domain to the vertical axis. Our second example is the model reerred as Lorenz96 presented in Lorenz (995). The variable ū is a vector o N-equally spaced entries around a circle o constant latitude, i.e. ū(t) = (u (t), u 2 (t),..., u N (t)). The N-dimensional system is determined by the ollowing N equations du j dt = κ ( u j 2u j + u j u j+ u j + F ), j =,... N, (5.25)

11 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 0 (a) Initial x 0 := u 0 (z) (b) Dynamical system over space and time Figure 5.2: Nonlinear D Wave equation where F and κ are constants independent o j. To orm a cyclic chain, we set u N = u 0, u = u N and u N+ = u. This system is known to have a chaotic behaviour over time depending on the parameters N, F and κ (see or instance Karimi and Paul, 2009) For a given set o parameters N, F and κ or which a stable behavior is observed, i.e. where data assimilation can be perormed, we consider then the ollowing dynamical system: du j+θ dt = κ ( u j+θ 2u j+θ + u j+θ u j+θ+ u j+θ + F ), j =,... N, θ =,..., Θ, (5.26) where θ and Θ are integers. Thus, the new size o the vector ū(t) is N Θ, which may be speciied as large as needed in our numerical experiments. The dynamical system is plotted or N = 40, F = 8, κ = 20 and Θ = 0, using the coordinate graph described in Figure 5.3 (a). For an initial state x = ū(0) as that shown in Figure 5.3 (b) the system develops over time as described in Figure 5.3 (c) (view rom the top). As we observe that the system becomes chaotic ater a certain time, we consider a reduced window o assimilation plotted in Figure 5.3 (d). 5.2 Results We now provide numerical results using Algorithm 4., which uses the RPCG algorithm in Step 3, as was mentioned in Section 2. When tuning the accuracy parameter or stopping the inner iterations, we noticed that it was suitable to choose a value in the middle o the possible range. In the case o the DWave equation the range o the parameter is more or less deined by [0, 0 4 ], thus we choose 0 2, while the range is approximately [0, 0 6 ] in the case o the Lorenz96 model, and we chose the value 0 4 in our experiment. For the DWave problem, we depict the background vector in a black dashed line together with the true solution as a red line, in Figure 6.4 (a). The result given by our algorithm is plotted as a blue line in Figure 6.4 (b). The background vector and the true solution are plotted in Figure 6.5 (a) and the result given by our algorithm is presented in Figure 6.5 (b) or the Lorenz96 model. In order to observe the adaptive nature o the algorithm, or an intermediate iteration i, we display two consecutive sets o observations O i and O i+ in Figures 6.6 (a)-(b) and 6.7 (a)-(b) or the DWave and the Lorenz96 respectively. In order to appreciate the impact o the local error indicators deined in Step 4, we also illustrate, in Figures 6.6 (e) and 6.7 (e), the local behaviour o the error between the prolongation o the current λ i to the set O i+ and the true λ i+, as given by ɛ j = ( λ i+ Π i λ i ) j, [( R i+ + H i+ B H i+)( λ T i+ Π i λ i )] j, together with that o the local error indicators themselves (η j and ν j are displayed Figures 6.6 (c)-(d) and 6.7 (c)-(d), respectively).

12 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA (a) Coordinate system (b) Initial x 0 := (u (0), u 2 (0),..., u N (0)) (c) Dynamical system over space and time (d) Window o assimilation Figure 5.3: Lorenz96 problem In the case o the DWave equation we observe in Figure 5.2 (a) that the peak in the middle o the signal produces a dynamical reaction over space and time, shown in part (b) o the same igure. The errors η j and ν j in Figure 6.6 (c) and (d) are larger in the regions o strong dynamical activity, whose identiication is clear in Figure 6.6 (a)-(b) which shows the evolution o the observation sets. In Figure 6.6 (e), we also observe that the regions where the dierence ɛ j is large coincide with those where η j and ν j are also large in Figure 6.6 (c)-(d). In the case o the Lorenz96 model where the initial signal has also a peak in the middle (Figure 5.3 (b)), the high dynamics are on the let part o the space and time graph (Figure 5.3 (d)). In this case the quantities η j and ν j also provide correct indication o where more observations are needed, see Figure 6.7 (c)-(d). We also note that the distance ɛ j in Figure 6.7 (c) is consistent with η j and ν j. Indeed, the sets o observations in Figure 6.7 (a)-(b) show that the observations concentrate on the let part o the graph where the highly dynamic nature o the model is most evident. In Figures 6.8 and 6.9, we compare the perormance o the new algorithm with the simple use o uniorm observations and a benchmark method where a pre-established hierarchy o uniorm distributed and progressively denser observations sets is used in Algorithm 4. (skipping Steps 4-6). This last method bypasses the new adaptive eatures o the method and its associated computing cost. For this comparison, we deine, or a given iterate x resulting rom Step 7 o Algorithm 4., the cost unction as the value o the unction in (2.3) when i = r, i.e. when all the possible observations are used. We then plot the evolution o this cost unction (in logarithmic scale) against the number o observations used and the associated lop (loating-point operations) counts or the three algorithms. (The curves or the uniorm case do not correspond to an algorithm, but show the accuracy and computational costs associated with directly solving the problem on a uniorm grid or each speciic size.) We observe in both cases that the new method achieves a smaller error (plotted in red) than

13 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 2 that obtained by using uniormly distributed observations (plotted in blue) or the benchmark (plotted in black), which is explained by the act that most observations used by the adaptive algorithm are in regions o space where their contribution to accuracy is largest. Computational experience not reported here also indicates that the eect o the choice o starting value o x does not aect signiicantly the hierarchy o observation sets beyond the act that observations concentrate in regions o high dynamical activity. We also ound that [0.4, 0.7] appears to be an adequate range or the parameters θ and θ 2 in (4.2), yielding a satisactory rate o inclusion o new observations at each iteration. Both results on accuracy and computational costs are thereore highly encouraging. 6 Conclusions and perspectives A method which identiies the inluential data on a 4D-Var data assimilation problem is proposed, as well as an algorithm that exploits this identiication to improve on computing eiciency. The cpu-time gains are obtained or two reasons, the irst being that the available number o observations is used very eectively, and the second the act that the cost o the subproblem solution is signicantly reduced by the use o dual-space conjugate-gradient techniques like RPCG. Numerical experience has been presented on two nonlinear test problems, and the results are so ar encouraging. Further reinements o the algorithm could be considered, such as the use o adaptive preconditioners in the subproblem solver, and continued experience with the method is o course desirable to assess its true potential. Extensions o these ideas in other domains are also possible: we think in particular o data assimilation problems in rameworks where each state o the system is itsel an image on which adaptive reconstruction techniques could be applied. Acknowledgments. The authors would like to thank S. Gürol and X. Vasseur at CERFACS or helpul scientiic discussions.

14 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 3 (a) Background vector and true initial u(0) (b) True u(0) and algorithm solution Figure 6.4: Results: Nonlinear D Wave equation (a) Background vector and true initial u(0) (b) True u(0) and algorithm solution Figure 6.5: Results: Lorenz96

15 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 4 (a) O i (b) O i+ (c) η j (d) ν j (e) ɛ j Figure 6.6: Observations set and adaptive errors (a) O i (b) O i+ (c) η j (d) ν j (e) ɛ j Figure 6.7: Observations set and adaptive errors

16 Gratton, Rincon-Camacho, Toint: Adaptive observations and optimization in DA 5 (a) Cost unction versus number o observations (b) Cost unction versus lops Figure 6.8: Perormance o the algorithm on the nonlinear wave equation (a) Cost unction versus number o observations (b) Cost unction versus lops Figure 6.9: Perormance o the algorithm on the Lorenz96 problem

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