Lie algebraic method in nonlinear filtering with state space dimension up to 6

R. W. Brockett (1983) proposed to classify all finite dimensional estimation algebras. An affirmative solution to Brockett's problem will allow construction of all possible finite dimensional recursive filters from the Lie algebraic point of view. The concept of an estimation algebra with maximal rank was introduced by L. F. Tam et al. (1990). This is the most important general subclass of estimation algebras. S. S.-T. Yau and W.L. Chiou (1991) have already classified all maximal rank finite dimensional estimation algebras with state space dimension at most 2. Here, the case for state space dimension 3 is studied. >

This paper considers general finite-dimensional estimation algebras associated with nonlinear filtering systems. General considerations and approaches toward the classification of finite-dimensional estimation algebras are proposed. Some structural results are obtained. The properties of Euler operator and the solutions to an underdetermined partial differential equation, which inevitably arise in an estimation algebra, are studied. These tools and techniques are applied to the study of finite-dimensional estimation algebras with state dimension $2$ to obtain a complete classification result. It is shown that a finite-dimensional estimation algebra with state dimension $2$ can only have dimension less than or equal to $6$. Moreover, the Mitter conjecture and the Levine conjecture hold for finite-dimensional estimation algebras with state dimension $2$.

In this paper, we study the structure of finite dimensional estimation algebras with state dimension 3 and rank 2 arising from a nonlinear filtering system by using the theories of the Euler operator and under-determined partial differential equations. It is proved that if the estimation algebra contains a degree two polynomial, then the Wong Ω-matrix must be a constant matrix. Moreover, all functions in the estimation algebra must be linear functions.

Structure theorem for five-dimensional estimation algebras

Ever since Brockett, Clark and Mitter introduced the estimation algebra method, it becomes a powerful tool to classify the finite-dimensional filtering system. In this paper, we investigate finite-dimensional estimation algebra with non-maximal rank. The structure of Wong matrix Ω will be focused on since it plays a critical role in the classification of finite-dimensional estimation algebras. In this paper, we first consider general estimation algebra with non-maximal rank and determine the linear structure of the submatrix of Ω by using rank condition and property of Euler operator. In the second part, we proceed to consider the case of linear rank n−1 and prove the linear structure of Ω. Finally, we give the structure of finite-dimensional filters which implies the drift term must be a quadratic function plus a gradient of a smooth function.

In this paper we study the structure of finite dimensional estimation algebras with state dimension 3 and rank 2 arising from a nonlinear filtering system by using the theories of the Euler operator and underdetermined partial differential equations. The structure of the Wong $\Omega$-matrix is shown to be linear. The fundamental strategy we use in this paper to prove these results is to show that if they were not true, then infinite sequences could be constructed in the finite dimensional estimation algebra.

The authors present a new way to construct finite dimensional estimation algebra of nonmaximal rank. For a given finite dimensional estimation algebra E, one can construct a finite dimensional estimation algebra E' of nonmaximal rank which is isomorphic to E. The interesting fact is that even if E comes from a linear filtering system, there are arbitrary nonlinear filtering systems associated to E. Since every finite dimensional estimation algebra gives rise to a finite dimensional filter, we have constructed quite a large class of finite dimensional nonlinear filters.

Of central importance in nonlinear filtering theory is the classification of finite-dimensional estimation algebras, through which we can construct recursive filters. We classify all t-dimensional estimation algebras, t/spl les/4, for arbitrary state space dimension.

On a new class of finite dimensional estimation algebras

An open-loop digital receiver performing simultaneous carrier phase/frequency estimation and block (two-bit) detection is proposed for MSK signaling in Gaussian channels affected by Doppler frequency shift. Phase estimation and symbol detection are provided by a parallel bank of matched estimators (each one comprising a phase rotation block and a stochastic nonlinear filter) and by a decision algorithm based on the nonlinear filters' innovations processes. The Doppler frequency shift, estimated by a Kalman-Bucy (1968) filter coupled with a Kay (1989) estimator, is used for fine tuning of the nonlinear filters. The receiver exhibits a, fast phase/frequency acquisition capability and is able to withstand large carrier frequency deviations. These are essential features in LEO satellite burst communications.

Exact propagator of the Fokker–Planck equation with logarithmic factors in diffusion and drift terms

Affine Chevalley algebras

In this paper, we study the superconvergence properties of the LDG method for the one-dimensional linear Schrodinger equation. We build a special interpolation function by constructing a correction function, and prove the numerical solution is superclose to the interpolation function in the $$L^{2}$$ norm. The order of superconvergence is $$2k+1$$ , when the polynomials of degree at most k are used. Even though the linear Schrodinger equation involves only second order spatial derivative, it is actually a wave equation because of the coefficient i. It is not coercive and there is no control on the derivative for later time based on the initial condition of the solution itself, as for the parabolic case. In our analysis, the special correction functions and special initial conditions are required, which are the main differences from the linear parabolic equations. We also rigorously prove a $$(2k+1)$$ -th order superconvergence rate for the domain, cell averages, and the numerical fluxes at the nodes in the maximal and average norm. Furthermore, we prove the function value and the derivative approximation are superconvergent with a rate of $$(k+2)$$ -th order at the Radau points. All theoretical findings are confirmed by numerical experiments.

We consider the problem of approximating an analytic function on a compact interval from its values at $M+1$ distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best possible convergence rate of a stable method is root-exponential in M, and that any method with faster exponential convergence must also be exponentially ill conditioned at a certain rate. This result hinges on a classical theorem of Coppersmith & Rivlin concerning the maximal behavior of polynomials bounded on an equispaced grid. In this paper, we first generalize this theorem to arbitrary point distributions. We then present an extension of the impossibility theorem valid for general nonequispaced points and apply it to the case of points that are equidistributed with respect to (modified) Jacobi weight functions. This leads to a necessary sampling rate for stable approximation from such points. We prove that this rate is also sufficient, and therefore exactly quantify (up to constants) the precise sampling rate for approximating analytic functions from such node distributions with stable methods. Numerical results—based on computing the maximal polynomial via a variant of the classical Remez algorithm—confirm our main theorems. Finally, we discuss the implications of our results for polynomial least-squares approximations. In particular, we theoretically confirm the well-known heuristic that stable least-squares approximation using polynomials of degree N < M is possible only once M is sufficiently large for there to be a subset of N of the nodes that mimic the behavior of the $N$th set of Chebyshev nodes.

This chapter is a brief introduction into the structure of algebras, mostly finite dimensional, over any field k. The main contents are the Wedderburn theorems for a finite dimensional algebras A over an algebraically closed field k. If A has no nilpotent ideals ≠ 0, then A is a finite product of total matrix algebras over k. In this case, the set d (A) of degrees of the total matrix algebras is a complete set of invariants of A. Thus, 13.7 two finite dimensional semiprime algebras A and B over k are isomorphic if and only if d (A) = d (B).