Squared Extrapolation Methods (SQUAREM): A New Class of Simple and Efficient Numerical Schemes for Accelerating the Convergence of the EM Algorithm

We derive a new class of iterative schemes for accelerating the convergence of the EM algorithm, by exploiting the connection between fixed point iterations and extrapolation methods. First, we present a general formulation of one-step iterative schemes, which are obtained by cycling with the extrapolation methods. We, then square the one-step schemes to obtain the new class of methods, which we call SQUAREM. Squaring a one-step iterative scheme is simply applying it twice within each cycle of the extrapolation method. Here we focus on the first order or rank-one extrapolation methods for two reasons, (1) simplicity, and (2) computational efficiency. In particular, we study two first order extrapolation methods, the reduced rank extrapolation (RRE1) and minimal polynomial extrapolation (MPE1). The convergence of the new schemes, both one-step and squared, is non-monotonic with respect to the residual norm. The first order one-step and SQUAREM schemes are linearly convergent, like the EM algorithm but they have a faster rate of convergence. We demonstrate, through five different examples, the effectiveness of the first order SQUAREM schemes, SqRRE1 and SqMPE1, in accelerating the EM algorithm. The SQUAREM schemes are also shown to be vastly superior to their one-step counterparts, RRE1 and MPE1, in terms of computational efficiency. The proposed extrapolation schemes can fail due to the numerical problems of stagnation and near breakdown. We have developed a new hybrid iterative scheme that combines the RRE1 and MPE1 schemes in such a manner that it overcomes both stagnation and near breakdown. The squared first order hybrid scheme, SqHyb1, emerges as the iterative scheme of choice based on our numerical experiments. It combines the fast convergence of the SqMPE1, while avoiding near breakdowns, with the stability of SqRRE1, while avoiding stagnations. The SQUAREM methods can be incorporated very easily into an existing EM algorithm. They only require the basic EM step for their implementation and do not require any other auxiliary quantities such as the complete data log likelihood, and its gradient or hessian. They are an attractive option in problems with a very large number of parameters, and in problems where the statistical model is complex, the EM algorithm is slow and each EM step is computationally demanding.

[On quality of life after surgical therapy of oral cancer - a retrospective multi-center study: the connection between dedentition, denture, quality of life, and dysphagia, and the resulting rehabilitation schemes]

PURPOSE: The surgical treatment of oral cancer results in functional and aesthetical impairments. Patients' quality of life is considerably impaired by oral symptoms resulting from therapy of oral cancer. In many cases the inevitable resection of the tumor, as well as the adjuvant radiochemotherapy will cause the destruction of physiologically and anatomically important structures. One focus of research was the specific rehabilitation of dental loss by functional dentures. Another was the course of 19 impairments (comprehension of speech for unknown others, comprehension of speech for familiar others, eating/swallowing, mobility of the tongue, opening range of the mouth, mobility of lower jaw, mobility of neck, mobility of arms and shoulders, sense of taste, sense of smell, appearance, strength, appetite, respiration, pain, swelling, xerostomia, halitosis). METHODS: Commissioned by the German, Austrian and Swiss cooperative group on tumors of the maxillofacial region (DOSAK), data were collected in 3.894 questionnaires at 43 hospitals in Germany, Austria and Switzerland. The catalogue comprised 147 items in 9 chapters. At the end of the enquiry, 1.761 anonymous questionnaires were returned by 38 hospitals. 1.652 of these could be evaluated regarding the question. RESULTS: The sum score of the 19 impairments was highly increased immediately after the operation and recovered over the next 6 months, without, however, reaching the pre-surgery level. Of 1.652 patients, only 35% did not lose any teeth during therapy. 23% lost up to 5, 17% up to 10 teeth. A quarter of the patients lost more than 10 teeth. The more teeth were lost, the greater the decline of quality of life (p < or = 0.001), although this could be allayed by the functionality of the dentures (p < or = 0.001). There is a reciprocal dependence between the functionality of dental prosthetics and impairment by eating/swallowing (p < or = 0.001). CONCLUSIONS: Patients' quality of life after radical surgery of a carcinoma of the oral cavity depends not only on the functionality of dentures and the specificity of rehabilitation, but also from the initial findings, the extent and location of the resection, the chosen therapy, the general circumstances of the patient's life as well as their strategies of coping. These factors, however, unlike those of functionality of dental prosthesis and rehabilitation, are not modifiable.

Maximum principle preserving high order schemes for convection-dominated diffusion equations

The maximum principle is an important property of solutions to PDE. Correspondingly, it's of great interest for people to design a high order numerical scheme solving PDE with this property maintained. In this thesis, our particular interest is solving convection-dominated diffusion equation. We first review a nonconventional maximum principle preserving(MPP) high order finite volume(FV) WENO scheme, and then propose a new parametrized MPP high order finite difference(FD) WENO framework, which is generalized from the one solving hyperbolic conservation laws. A formal analysis is presented to show that a third order finite difference scheme with this parametrized MPP flux limiters maintains the third order accuracy without extra CFL constraint when the low order monotone flux is chosen appropriately. Numerical tests in both one and two dimensional cases are performed on the simulation of the incompressible Navier-Stokes equations in vorticity stream-function formulation and several other problems to show the effectiveness of the proposed method.