Graduation date: 1995

The standard way of representing numbers on computers gives rise to errors which increase as computations progress. Using p-adic valuations can reduce error accumulation. Valuation theory tells us that p-adic and standard valuations cannot be directly compared. The p-adic valuation can, however, be used in an indirect way. This gives a method of doing arithmetic on a subset of the rational numbers without any error. This exactness is highly desirable, and can be used to solve certain kinds of problems which the standard valuation cannot conveniently handle. Programming a computer to use these p-adic numbers is not difficult, and in fact uses computer resources similar to the standard floating-point representation for real numbers. This thesis develops the theory of p-adic valuations, discusses their implementation, and gives some examples where p-adic numbers achieve better results than normal computer computation. / Graduation date: 1994

We present a method for Poisson’s equation that computes guaranteed upper and lower bounds for the values of linear functional outputs of the exact weak solution of the infinite dimensional continuum problem using traditional finite element approximations. The guarantee holds uniformly for any level of refinement, not just in the asymptotic limit of refinement. Given a finite element solution and its output adjoint solution, the method can be used to provide a certificate of precision for the output with an asymptotic complexity which is linear in the number of elements in the finite element discretization. / Singapore-MIT Alliance (SMA)

We introduce a new method for computing a posteriori bounds on engineering outputs from finite element discretizations of the incompressible Stokes equations. The method results from recasting the output problem as a minimization statement without resorting to an error formulation. The minimization statement engenders a duality relationship which we solve approximately by Lagrangian relaxation. We demonstrate the method for a stabilized equal-order approximation of Stokes flow, a problem to which previous output bounding methods do not apply. The conceptual framework for the method is quite general and shows promise for application to stabilized nonlinear problems, such as Burger's equation and the incompressible Navier-Stokes equations, as well as potential for compressible flow problems. / Singapore-MIT Alliance (SMA)

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations -- Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation -- relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures -- methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage -- in which, given a new parameter value, we calculate the output of interest and associated error bound -- depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. / Singapore-MIT Alliance (SMA)

We present predictions for the statistical error due to finite sampling in the presence of thermal fluctuations in molecular simulation algorithms. Expressions for the fluid velocity, density and temperature are derived using equilibrium statistical mechanics. The results show that the number of samples needed to adequately resolve the flow-field scales as the inverse square of the Mach number. The theoretical results are verified for a dilute gas using direct Monte Carlo simulations. The agreement between theory and simulation verifies that the use of equilibrium theory is justified. / Singapore-MIT Alliance (SMA)

This thesis presents there important results in visual object recognition based on shape. (1) A new algorithm (RAST; Recognition by Adaptive Sudivisions of Tranformation space) is presented that has lower average-case complexity than any known recognition algorithm. (2) It is shown, both theoretically and empirically, that representing 3D objects as collections of 2D views (the "View-Based Approximation") is feasible and affects the reliability of 3D recognition systems no more than other commonly made approximations. (3) The problem of recognition in cluttered scenes is considered from a Bayesian perspective; the commonly-used "bounded-error errorsmeasure" is demonstrated to correspond to an independence assumption. It is shown that by modeling the statistical properties of real-scenes better, objects can be recognized more reliably.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 135-139).

Current-controlled logic and single electron logic processors have been investigated with respect to thermal-induced bit error. A maximal error rate for both logic processors is regarded as one bit-error/year/chip. A maximal clock frequency and an information channel capacity at a given operation current are derived when a current-controlled logic processor works without error. An available operation range in a current-controlled processor with 100 million elements is discussed. The dependence of an error-free condition on temperature in single electron logic processors is derived. The size of the quantum dot of single electron transistor is predicted when a single electron logic processor with the a billion single electron transistors works without error at room temperature.

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