--'Ibis paper presents a substantial extension of the method of complementary modek for minimum variance linear estimation introduced by Weinert and Desai in their important paper [l]. Specifically, the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions. A major contribution of this paper is an application of Green's identity in denting a differential operator representation of the estimator. [l]. Weinert and Desai showed that the fixed interval smoothing problem for causal one-dirnensional' processes described by linear state equations driven by white noise could be solved by introducing the so-called complementary process. The complementary process has the property that it is orthogonal to the observations and that, when combined with the observations, contains information equivalent to the initial conditions, driving noise and measurement noise, Le., all of the underlying variables which determine the system state and observations. Here we build upon this general concept of complementation to solve estimation problems for both discrete and continuous parameter boundary value stochastic processes in one and higher dimensions. This class of processes is a generalization of the 1-D boundary value process introduced by Krener in [14] and includes processes governed by ordinary and partial linear differential equations and ordinary and partial linear difference equa-
This paper considers the smoothing problem for 2-D random fields described by stochastic nearest-neighbor models (NNMs). The class of 2-D estimation problems that can be modeled in this way is quite large since NNMs arise whenever partial differential equations are discretized with finite difference methods. The NNM smoother is obtained by using a general smoothing technique developed in [1]-[3] for boundary-value processes in one or several dimensions. In this approach, the smoother is described by a Hamiltonian system of twice the dimension of the original system. For the problem considered here, the smoother is itself in NNM form. By converting this 2-D NNM system into an equivalent 1-D two-point boundary-value descriptor system (TPBVDS) of large dimension, a recursive and stable solution technique is obtained. Under slightly restrictive assumptions, an even faster procedure can be obtained by using the FFT with respect to one of the space dimensions to convert the 1-D TPBVDS mentioned above into a set of decoupled TPBVDSs of low-order which can be solved in parallel. This fast implementation of the smoother is illustrated by two examples, corresponding respectively to the discretized Poisson and heat equations.
In this paper we discuss the problem of estimating processes by the innovations approach [6] and [7]. This boundary value processes in one or several dimensions. approach has some advantages over diagonalization. In The estimator dynamics are described, and by using particular, in constrast to diagonalization, no operaoperator transformations for these dynamics, several tor inversions are required in computing the estimate implementations are obtained which either diagonalize by triangularization. Having established the conditions or triangularize the linear least-squares estimator.for diagonalization we formulate matrix Riccati equaThese implementations enable us to compute the estimate tions which lead to stable diagonal forms for the esof the process by using two-filter type of smoothing timator for l-D boundary value processes. Finally, formulas, or more general smoothing formulas similar to questions of existence and uniqueness of diagonalizing those used for solving the smoothing problem for 1-D transformations for 2-D estimators are discussed. causal processes.2. where L is referred to as the formal adjoint differenand consequently an issue is the construction of tial operator [81, xb and Xb are elements of a Hilbert efficient methods to implement that solution. In space Hb of processes defined on Ag , and E is a mapp-[5] a detailed solution in the 1-D case is developed by ing from Hb into itself; E:HR*Hb. Yn particular, diagonalizing the dynamics of the smoother boundary these processes are defined Ehrough the action of an value differential equation (see also [2] for a brief operator A : L n() Hb, so that description). The computations involved in these reb 2 N sults relied heavily on the specific 1-D problem. In x = A x and X = abX (2) this paper we extend the ideas underlying that diagob b b nalization approach by describing the diagonalization The nature of Hb, Ab, and E all depend upon L and QN of estimator dynamics in an operator framework appliFor a discussion of Green's identity for ordinary cable to problems in several dimensions as well.differential operators see [9] and Chapter 3 of [10]; for elliptic, hyperbolic and parabolic second order After reviewing the form of the estimator solution partial differential operators see [8] and Chapter 7 in Section 2, we describe some equivalent dynamical reof [10].In this paper, we will restrict our discuspresentations for the differential operator description sions to operators L and regions QN that admit a Green's of the estimator dynamics in Section 3. The diagonal identity. form we seek, if it exists, is in the class of equivalent differential operator representations, and inThe boundary condition associated with L is deSection 4 we present the conditions which define the fined by a mapping V: class of transformation operators which lead to such forms. As an alternative to diagonalization, we out-V:Hb R(V) (3) line a method for triangularizing the dynamics which The work of these authors was supported by the National Science Foundation unde...
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