Feynman Path Integral Discretization And Its Applications To Nonlinear Filtering Pdf
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Nonlinear Dynamics of Path Integrals
Metrics details. 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 Gaussian and 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 the configuration space 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 have excellent numerical properties. The Feynman path integral formulation is shown to lead to practical and implementable algorithms. In particular, the solution of the Yau partial differential equation is reduced to one of function computation and integration. The fundamental dynamical laws of physics, both classical and quantum mechanical, are described in terms of variables continuous in time.
The continuous nature of the dynamical variables has been verified at all length scales probed so far, even though the relevant dynamical variables, and the fundamental laws of physics, are very different in the microscopic and macroscopic realms. In practical situations, one often deals with macroscopic objects whose state is phenomenologically well-described by classical deterministic laws modified by external disturbances that can be modelled as random noise, or Langevin equations.
Even when there is no underlying fundamental dynamical law, the Langevin equation provides an effective description of the state variables in many applications. It is therefore natural to consider the problem of the evolution of a state of a signal of interest described by a Langevin equation called the state process. When the state model noise is Gaussian or more generally multiplicatively Gaussian the state process is a Markov process.
Since the process is stochastic, the state process is completely characterized by a probability density function. The Fokker-Planck-Kolmogorov foward equation FPKfe describes the evolution of this probability density function or equivalently, the transition probability density function and is the complete solution of the state evolution problem.
However, in many applications the signal, or state variables, cannot be directly observed. Instead, what is measured is a nonlinearly related stochastic process called the measurement process. The measurement process can often be modelled by yet another continuous stochastic dynamical system called the measurement model. In other words, the observations, or measurements, are discrete-time samples drawn from a different Langevin equation called the measurement process.
The conditional probability density function of the state variables, given the observations, is the complete solution of the filtering problem. This is because it contains all the probabilistic information about the state process that is in the measurements and in the initial condition [ 1 ].
This is the Bayesian approach, i. Given the conditional probability density, optimality may be defined under various criterion. Usually, the conditional mean, which is the least mean-squares estimate, is studied due to its richness in results and mathematical elegance.
The solution of the optimal nonlinear filtering problem is termed universal, if the initial distribution can be arbitrary. When the state and measurement processes are linear, the linear filtering problem was solved by Kalman and Bucy [ 2 , 3 ]. The celebrated Kalman filter has been successfully applied to a large number of problems in many different areas.
Nevertheless, the Kalman filter suffers from some major limitations. The Kalman filter is not optimal even for the linear filtering case if the initial distribution is not Gaussian. It may still be optimal for a linear system under certain criteria, such as minimum mean square error, but not a general criterion.
In other words, the Kalman filter is not a universal optimal filter, even when the filtering problem is linear. Secondly, the Kalman filter cannot be an optimal solution for the general nonlinear filtering problem since it assumes that the signal and measurement models are linear. The extended Kalman filter EKF , obtained by applying the Kalman filter to a linearized model, cannot be a reliable solution, in general. Thirdly, even when the EKF estimates the state well in some cases, it gives no reliable indication of the accuracy of the state estimate, i.
Finally, the Kalman filter assumes that the conditional probability distribution is Gaussian, which is a very restrictive assumption; for instance, it rules out the possibility of a multi-modal conditional probability distribution.
The continuous-continous nonlinear filtering problem i. This led to a stochastic differential equation, called the Kushner equation, for the conditional probability density in the continuous-continuous filtering problem. It was noted in [ 8 , 9 ], and [ 10 ] that the stochastic differential equation satisfied by the unnormalized conditional probability density, called the Duncan-Mortensen-Zakai DMZ equation, is linear and hence considerably simpler than the Kushner equation.
A disadvantage of the robust DMZ equation is that the coefficients depend on the measurements. Thus, one does not know the PDE to solve prior to the measurements. As a result, real-time solution is impossible. A fundamental advance was made in tackling the general nonlinear filtering problem by S-T.
Yau and Stephen Yau. In [ 13 ], it was proved that the robust DMZ equation is equivalent to a partial differential equation that is independent of the measurements, which is referred to as the Yau Equation YYe in this paper. Specifically, the measurements only enter as initial condition at each measurement step. However, numerical solution of partial differential equations presents several challenges. Alternatively, when the discretization spacing is decreased, it may tend to a different equation, i.
Furthermore, the numerical method may be unstable. Finally, since the solution of the YYe is a probability density, it must be positive which may not be guaranteed by the discretization. A different approach to solving the PDE was taken in [ 14 ] and [ 15 ]. An explicit expression for the fundamental solution of the YYe as an ordinary integral was derived. It was shown that the formal solution to the YYe may be written down as an ordinary, but somewhat complicated, multi-dimensional integral, with an infinite series as the integrand.
In addition, an estimate of the time needed for the solution to converge to the true solution was presented. In this paper, the Euclidean Feynman path integral FPI formulation is employed to tackle the continuous-continuous nonlinear filtering problem.
Specifically, phrasing the stochastic filtering problem in a language common in physics, the solution of the stochastic filtering problem is presented. The path integral formulation leads to a path integral formula for the transition probability density for the general additive noise case.
A corollary of the FPI formalism is the path integral formula for the fundamental solution of the YYe and the Yau algorithm — a fundamental result of nonlinear filtering theory. It is noted that this paper provides a detailed derivation of results that were used in [ 16 ]. However, in spite of considerable effort it has not been proven to be directly useful in the development of reliable practical algorithms with desirable numerical properties. It also obscures the physics of the problem.
In contrast, it is shown that the FPI leads to formulas that are eminently suitable for numerical implementation.
It also provides a simple and clear physical picture. Finally, the theoretical insights provided by the FPI are highly valuable, as evidenced by numerous examples in modern theoretical physics see, for instance, [ 17 ] , and shall be employed in subsequent papers.
The outline of this paper is as follows. In the following section, the filtering problem is reformulated in a language common in physics. In Section 3, the path integral formula for the transition probability density is derived for the general additive noise case. The Yau algorithm is then derived in the following section.
In Sections 5 and 6 some conceptual remarks and numerical examples are presented. The conclusions are presented in Section 7. In Appendix 1, aspects of continuous-continuous filtering are reviewed. For more details on the path integral methods, see any modern text on quantum field theory, such as [ 17 ], and especially [ 18 ] which discusses application of FPI to the study of stochastic processes. In this section, the filtering problem is stated in a language commonly used in theoretical physics.
When the diffusion vielbein is independent of the state x t , the noise is termed additive. It is the additive noise that is studied here, since it enables the use of functional methods common in quantum field theory. Due to the random noise, each system leads to a different vector x t that depends on time. Although only one realization of the stochastic process is ever observed, it is meaningful to speak about an ensemble average.
The complete information on the random vector x t is contained in the infinite hierarchy of such probability densities. The quantity of interest here is the conditional probability density. It can be shown that the transition probability satisfies the Fokker-Planck-Kolmogorov forward equation FPKfe see, for instance, [ 19 ]. The path integral formula for the fundamental solution for the FPkfe is applied to the continuous-discrete filtering problem with additive state model noise in [ 20 , 21 ].
Thus, in continuous-continuous filtering, the continuous-time measurement stochastic process needs to be incorporated as well. Consider another ensemble of systems with state variables whose time evolution is governed by the measurement process.
The measurement noise means that each system in the ensemble leads to a different time-dependent vector y t. Thus, even though only one realization of the measurement stochastic process is observed, it is still meaningful to talk about an ensemble average of the measurement process in addition to one over the state process. Thus, the quantity of interest in continuous-continuous filtering is the conditional probability density. A crucial difference between the state and measurement stochastic process is that, unlike the state, the measurement samples are known.
Note that the conditional transition probability density is the complete solution to the continuous-continuous filtering problem, since if the initial distribution is u t i -1 , x' Y i -1 , where Y i -1 is the set of all measurements prior to t i -1 , then the evolved conditional probability distribution is. In the following sections, the path integral formulas for P t 2 , x 2 ; y 2 t 1 , x 1 ; y 1 are derived. In Section 4 it shall be shown that it leads to the Yau algorithm.
It shall be shown that the YYe plays the same role here that the FPKfe does in continuous-discrete filtering. The path integral formula for the conditional transition probability density shall now be derived using functional methods.
Note that implicit in the use of these formal functional methods is the use of the Feynman convention, or symmetric discretization for the drift. As noted in Section 2, the transition probability density is computed by averaging over the signal and measurement ensembles, i. The Jacobian J y is trivial as the measurement model drift is y -independent and can be absorbed into the measure.
It is noteworthy that J is not trivial. In quantum field theory, nontrivial Jacobians usually imply that there is an anomaly, as in the case of chiral anomalies in gauge theories. However, there is no reason for an anomaly here; after all, this is not even a quantum field theoretical system. The puzzle is resolved by noting that path integral anomalies in quantum field theory arise from the "multiplicative" part in the change of variables i. In contrast, the nontrivial Jacobian term here arises from the additive term; the multiplicative term does not contribute to the Jacobian, in accordance with expectations.
Nonlinear Dynamics of Path Integrals
Manuscript received September 13, ; final manuscript received November 6, ; published online January 27, Editor: John B. Morzfeld, M. January 27, May ; 5 : Implicit sampling is a recently developed variationally enhanced sampling method that guides its samples to regions of high probability, so that each sample carries information. Implicit sampling may thus improve the performance of algorithms that rely on Monte Carlo MC methods.
Kolmogorov forward equation (FPKfe) and the Feynman path integral The complete solution of the filtering problem is the conditional probability density function (pdf) of to a different PDE in the limit that the discretization spacing vanishes. accurate solution of the nonlinear continuous-discrete filtering.
Information field theory
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Information field theory IFT is a Bayesian statistical field theory relating to signal reconstruction , cosmography , and other related areas. It uses computational techniques developed for quantum field theory and statistical field theory to handle the infinite number of degrees of freedom of a field and to derive algorithms for the calculation of field expectation values. For example, the posterior expectation value of a field generated by a known Gaussian process and measured by a linear device with known Gaussian noise statistics is given by a generalized Wiener filter applied to the measured data. IFT extends such known filter formula to situations with nonlinear physics , nonlinear devices , non-Gaussian field or noise statistics, dependence of the noise statistics on the field values, and partly unknown parameters of measurement. For this it uses Feynman diagrams , renormalisation flow equations, and other methods from mathematical physics. Fields play an important role in science, technology, and economy. They describe the spatial variations of a quantity, like the air temperature, as a function of position.