A numerical technique for geophysical data assimilation problems using Pontryagin’s principle and splitting-up method

Formulations of the observational data assimilation problem are given and methods for its solution that constructively use conjugate equations are described. On the basis of Pontryagin's maximum principle, the initial problem of minimizing the quadratic functional, which is the difference between the solution and observational data, reduces to the spatialtemporal problem. The solution of a maximum-principle boundary-value problem is constructed on the basis of the splitting-up method, which reduces a complex multidimensional problem to the sequence of problems with simpler operators. One of the urgent issues in current geophysical hydrodynamics is an observational data assimilation problem in mathematical numerical models describing geophysical processes. The problem of finding the correspondence between observational data and simulation results is important and interesting both theoretically and practically. Here we may mention the following aspects of this problem: the updating (replenishment) of observational data arrays in space and time and the four-dimensional data analysis; the construction of 'initial' geophysical fields with the use of historical archive information and short-term observational data over the preceding period; and the processing of observational and calculated data and the compilation of unified observational-experimental data arrays. The observational data assimilation problem is gaining in importance in dynamic oceanology. This is due to high costs of gathering data of oceanic observations and an involved character of ocean variability. Mathematical methods based on the theory of conjugate equations [2,5,8] and the optimal control theory [3,9,12] have recently become very popular as an effective tool of assimilating and analysing observational data. In solving complex nonlinear data assimilation problems in the framework of the theory of conjugate equations and the optimal control theory a number of problems arise: their physical aspects are often connected with the physical problem formulation and the choice of informative optimization functionals [11]; their mathematical aspects are associated with the study of the equivalence of initial variational problems and differential problems obtained after transformations, as well as with the study of their solvability [1,12]; their computational aspects concern the choice of highly accurate discrete approximations and the development of efficient algorithms for solving discrete problems [7], This work deals with constructing numerical methods for solving the data assimilation problem. We are mainly aiming at ocean dynamics problems, though the methods suggested here may be used in various fields of geophysical hydrodynamics as 312 G. I. Marchuk and V.B. Zalesny well. The peculiarities of large-scale ocean dynamics problems are due to the following features of the ocean. The ocean is in many respects a dependent system exposed to the atmosphere. Large-scale oceanic processes have a distinct linear character. There are only meagre synchronous data on the ocean, but the ocean is a system with a 'long' memory. These features of the ocean define the way the problems are posed and data assimilation methods to be considered in this paper. With the above statements we allow the system control by the right-hand side of equations and by the initial condition, in addition we have a chance of illustrating numerical algorithms on linear problems. 1. POSING OBSERVATIONAL DATA ASSIMILATION PROBLEMS 1.1. One of the most frequently used formulations of an observational data assimilation problem is the following. Suppose we have an equation written in the operator form: Λφ = ί (1.1) where Λ is the linear operator, acting in a Hilbert space //, with the domain D(A)\ 0? ,/e//. For simplicity we assume that Λ is either a matrix or an operator corresponding to a certain system of ordinary differential equations (ODE). Equation (1.1) appears, for example, in approximating with respect to space variables, either some linear boundary-value problem (in this case Λ is a matrix) or some evolutionary system of partial differential equations (then Λ is an ODE operator). Now we assume that there is some arbitrariness in prescribing the right-hand side in equation (1.1). For instance, some components of the vector / are unknown or they are prescribed to within a certain error. Following Lions [4], we may say that we deal with an incomplete-information problem. This problem is quite often encountered in practice, including dynamic oceanology. In solving ocean dynamics problems there is scanty information, for example, on the true wind field above the ocean or the heat flux into the ocean, which define source functions in the right-hand side in (1.1). The problems with incomplete information about the ocean steam from the fact that there is practically no information about oceanic fields at the initial instant of time. In this case, provided (1.1) is an ODE system, its right-hand side includes, along with the sources, initial conditions for the sought-for solution vector φ. Thus, we shall assume that we do not have at our disposal complete information about the vector / in the right-hand side. However, we have some measuring data, i.e. we know some components of the solution vector φ. Now let φ denote a somehow redefined solution vector, which has already incorporated the known components. Now let us pose the problem as follows. Find such a solution to equation (1.1) that would minimize the functional J(<p) = (Β(φ φ\φ φ) (1.2) where Β is a certain linear positive definite operator acting in Η and ( · , · ) is the scalar product in H. Data assimilation problem 313 It should be noted that when posing problem (1.1), (1.2) we use the arbitrariness in prescribing the right-hand side /, namely we select the right-hand side so that the corresponding solution should minimize the functional in (1.2). This is the problem of finding the conditional minimum of the functional J(q>) subject to the constraints in (1.1). There are many methods for solving this problem [6,12]. Note that the problem of finding the conditional minimum of the functional in (1.2) frequently reduces to that of finding the absolute minimum of an extended functional L(<p,<p*) L(p,p*) = (Β(φ φ\φ φ) + (Αφ -/,? *) (1.3) by using the method of Lagrange multipliers. Then one of the numerous methods for finding the absolute minimum of L(0>,0>*) is employed. 1.2. Sometimes it is convenient to pose the observational data assimilation problem in a different manner [6]. Suppose we have equation (1.1), where Λ is a matrix and φ and / are vectors of the real Euclidean space R. Let us assume that we have no complete information about the right-hand side / but we know Κ values of the functional Jpk(y) Jpk(<P) = (9,Pk}, *=! ,.,* (1.4) where pk(<p) is the prescribed Veight' vector characterizing some measurements in the field φ, and ( · , · ) is the scalar product in R. Using (1.1) and (1.4), we restore / and then find φ. To solve this problem let us consider Κ equations, conjugate to (1.1), of the form: *<? kPk> *=! ,. ,*, Α=Λ (1.5) where A* is the Lagrange conjugate operator (in this case the transpose of the matrix A\ i.e. (4ζ9η) = (ζ9Α*η), ζ,η e R . (1.6) Now taking the scalar products of both sides of (1.1) by φ£ and (1.5) by φ and subtracting one from the other, we arrive at the identity: (/.P**) = (pk,V), k=l,. ,K. (1.7) We know the right-hand side in (1.7) by virtue of condition (1.4), and φ£ can be found by solving (1.5). Here we assume beforehand that the conjugate operator A* is invertible. Then we can find / from (1.7) if we solve this system in a certain manner. System (1.7) for the unknown vector / (or its individual components) can be rewritten in a matrix form: Lf=F, Ρ^(ρ^φ)9 £=! ,. ,# (1.8) where L is the NxK rectangular matrix, N is the dimension of the vector /, and K is the dimension of the vector F. As a rule, in practical problems we have N>K. The solution of (1.8) is understood to be the pseudo-solution f^ [13] satisfying the equation: 314 G. I. Marchuk and V.B. Zalesny 1.3. We give the formulation used in the optimal control theory [12] as the third way of posing the problem. Now let Λ be the operator of a system of ordinary differential equations. In this case we have