Complete Real Time Solution of the General Nonlinear Filtering Problem without Memory

It is well known that the nonlinear filtering problem has important applications in both military and civil industries. The central problem of nonlinear filtering is to solve the Duncan-Mortensen-Zakai (DMZ) equation in real time and in a memoryless manner. In this paper, we shall extend the algorithm developed previously by S.-T. Yau and the second author to the most general setting of nonlinear filterings, where the explicit time-dependence is in the drift term, observation term, and the variance of the noises could be a matrix of functions of both time and the states. To preserve the off-line virture of the algorithm, necessary modifications are illustrated clearly. Moreover, it is shown rigorously that the approximated solution obtained by the algorithm converges to the real solution in the $L^1$ sense. And the precise error has been estimated. Finally, the numerical simulation support the feasibility and efficiency of our algorithm.Comment: 15 pages, 2-column format, 2 figure

Euclidean Quantum Mechanics and Universal Nonlinear Filtering

An important problem in applied science is the continuous nonlinear filtering problem, i.e., the estimation of a Langevin state that is observed indirectly. In this paper, it is shown that Euclidean quantum mechanics is closely related to the continuous nonlinear filtering problem. The key is the configuration space Feynman path integral representation of the fundamental solution of a Fokker-Planck type of equation termed the Yau Equation of continuous-continuous filtering. A corollary is the equivalence between nonlinear filtering problem and a time-varying Schr\"odinger equation.Comment: 19 pages, LaTeX, interdisciplinar

Universal Nonlinear Filtering Using Feynman Path Integrals II: The Continuous-Continuous Model with Additive Noise

In this paper, the Feynman path integral formulation of the continuous-continuous filtering problem, a fundamental problem of applied science, is investigated for the case when the noise in the signal and measurement model is additive. It is shown that it leads to an independent and self-contained analysis and solution of the problem. A consequence of this analysis is Feynman path integral formula for the conditional probability density that manifests the underlying physics of the problem. A corollary of the path integral formula is the Yau algorithm that has been shown to be superior to all other known algorithms. The Feynman path integral formulation is shown to lead to practical and implementable algorithms. In particular, the solution of the Yau PDE is reduced to one of function computation and integration.Comment: Interdisciplinary, 41 pages, 5 figures, JHEP3 class; added more discussion and reference

A Digital Predistortion Scheme Exploiting Degrees-of-Freedom for Massive MIMO Systems

The primary source of nonlinear distortion in wireless transmitters is the power amplifier (PA). Conventional digital predistortion (DPD) schemes use high-order polynomials to accurately approximate and compensate for the nonlinearity of the PA. This is not practical for scaling to tens or hundreds of PAs in massive multiple-input multiple-output (MIMO) systems. There is more than one candidate precoding matrix in a massive MIMO system because of the excess degrees-of-freedom (DoFs), and each precoding matrix requires a different DPD polynomial order to compensate for the PA nonlinearity. This paper proposes a low-order DPD method achieved by exploiting massive DoFs of next-generation front ends. We propose a novel indirect learning structure which adapts the channel and PA distortion iteratively by cascading adaptive zero forcing precoding and DPD. Our solution uses a 3rd order polynomial to achieve the same performance as the conventional DPD using an 11th order polynomial for a 100x10 massive MIMO configuration. Experimental results show a 70% reduction in computational complexity, enabling ultra-low latency communications.Comment: IEEE International Conference on Communications 201

Parallel Self-Consistent-Field Calculations via Chebyshev-Filtered Subspace Acceleration

Solving the Kohn-Sham eigenvalue problem constitutes the most computationally expensive part in self-consistent density functional theory (DFT) calculations. In a previous paper, we have proposed a nonlinear Chebyshev-filtered subspace iteration method, which avoids computing explicit eigenvectors except at the first SCF iteration. The method may be viewed as an approach to solve the original nonlinear Kohn-Sham equation by a nonlinear subspace iteration technique, without emphasizing the intermediate linearized Kohn-Sham eigenvalue problem. It reaches self-consistency within a similar number of SCF iterations as eigensolver-based approaches. However, replacing the standard diagonalization at each SCF iteration by a Chebyshev subspace filtering step results in a significant speedup over methods based on standard diagonalization. Here, we discuss an approach for implementing this method in multi-processor, parallel environment. Numerical results are presented to show that the method enables to perform a class of highly challenging DFT calculations that were not feasible before

Computational intelligence approaches to robotics, automation, and control [Volume guest editors]

No abstract available

Best-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics

A new method of deriving reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. Given a set of resolved variables that define a model reduction, the quasi-equilibrium ensembles associated with the resolved variables are employed as a family of trial probability densities on phase space. The residual that results from submitting these trial densities to the Liouville equation is quantified by an ensemble-averaged cost function related to the information loss rate of the reduction. From an initial nonequilibrium state, the statistical state of the system at any later time is estimated by minimizing the time integral of the cost function over paths of trial densities. Statistical closure of the underresolved dynamics is obtained at the level of the value function, which equals the optimal cost of reduction with respect to the resolved variables, and the evolution of the estimated statistical state is deduced from the Hamilton-Jacobi equation satisfied by the value function. In the near-equilibrium regime, or under a local quadratic approximation in the far-from-equilibrium regime, this best-fit closure is governed by a differential equation for the estimated state vector coupled to a Riccati differential equation for the Hessian matrix of the value function. Since memory effects are not explicitly included in the trial densities, a single adjustable parameter is introduced into the cost function to capture a time-scale ratio between resolved and unresolved motions. Apart from this parameter, the closed equations for the resolved variables are completely determined by the underlying deterministic dynamics