Show simple item record

dc.contributor.advisorYau, Stephenen_US
dc.contributor.authorLuo, Xueen_US
dc.date.accessioned2013-10-24T20:39:54Z
dc.date.available2013-10-24T20:39:54Z
dc.date.created2013-08en_US
dc.date.issued2013-10-24
dc.date.submitted2013-08en_US
dc.identifier.urihttp://hdl.handle.net/10027/10167
dc.description.abstractThis dissertation provides an affirmative answer to the well-known half century old engineering question raised by Office of Naval Research: “How can one solve nonlinear filtering (NLF) problems in real time without memory, if enough computational resources are provided?” Instead of the prestigious Kalman filter (KF) and its derivatives to estimate the mean and the covariance matrix of the states, we resort to solving the Duncan-Mortensen-Zakai (DMZ) equation, which is satisfied by the un-normalized probability density function of the states. In this dissertation, we develop a novel algorithm, which is applicable to the most general settings of the NLF problems and keeps two of the most important properties of KF: real-time and memory-less. Briefly speaking, in our algorithm, we split the approximation of the conditional density function into two parts: one part could be pre-computed before any on-line experiments ran (so-called off-line computation); the other part has to be sychronized the real-time data with the pre-computed data (so-called on- line computation). More precisely, the off-line computation solves a forward Kolmogorov equation (FKE) with the initial conditions, which are chosen to be a complete base functions in square-integrable function space, while the on-line part computes the projection of the conditional density function at each time step onto the basis, and then synchronize them with the off-line data to obtain the conditional density function at the next time step. First, we validate our algorithm theoretically, by estimating the convergence rate with respect to the sampling frequency. Second, we tackle some difficulties in the implementation of our algorithm and apply it to some 1-D benchmark NLF problems. Compared with the two most widely used methods nowadays, extended Kalman filter and particle filters, our algorithm surpasses both of them in the real-time manner with comparable accuracy. Last, when we investigate the application of our algorithm to the high-dimensional state NLF problems, we combine the sparse grid algorithm with the Hermite spectral method to serve as the off-line solver of FKE. The convergence rate is investigated both theoretically and numerically.en_US
dc.language.isoenen_US
dc.rightsen_US
dc.rightsCopyright 2013 Xue Luoen_US
dc.subjectNonlinear filtering problemsen_US
dc.subjectreal-time solveren_US
dc.subjectDuncan-Mortensen-Zakai equationen_US
dc.subjectHermite spectral methoden_US
dc.subjectsparse grid algorithmsen_US
dc.titleAn Novel Algorithm to Solve the Nonlinear Filtering Problems in Real-Timeen_US
thesis.degree.departmentMathematics, Statistics and Computer Scienceen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Illinois at Chicagoen_US
thesis.degree.levelDoctoralen_US
thesis.degree.namePhD, Doctor of Philosophyen_US
dc.type.genrethesisen_US
dc.contributor.committeeMemberVerschelde, Janen_US
dc.contributor.committeeMemberYang, Jieen_US
dc.contributor.committeeMemberKnessl, Charlesen_US
dc.contributor.committeeMemberJia, Lixingen_US
dc.type.materialtexten_US


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record