809 research outputs found
Iterative algorithms for solutions of nonlinear equations in Banach spaces.
Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF
General existence and uniqueness of viscosity solutions for impulse control of jump-diffusions
General theorems for existence and uniqueness of viscosity solutions for
Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVI) with integral
term are established. Such nonlinear partial integro-differential equations
(PIDE) arise in the study of combined impulse and stochastic control for
jump-diffusion processes. The HJBQVI consists of an HJB part (for stochastic
control) combined with a nonlocal impulse intervention term.
Existence results are proved via stochastic means, whereas our uniqueness
(comparison) results adapt techniques from viscosity solution theory. This
paper is to our knowledge the first treating rigorously impulse control for
jump-diffusion processes in a general viscosity solution framework; the jump
part may have infinite activity. In the proofs, no prior continuity of the
value function is assumed, quadratic costs are allowed, and elliptic and
parabolic results are presented for solutions possibly unbounded at infinity
Controlled diffusion processes
This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis
The vanishing moment method was introduced by the authors in [37] as a
reliable methodology for computing viscosity solutions of fully nonlinear
second order partial differential equations (PDEs), in particular, using
Galerkin-type numerical methods such as finite element methods, spectral
methods, and discontinuous Galerkin methods, a task which has not been
practicable in the past. The crux of the vanishing moment method is the simple
idea of approximating a fully nonlinear second order PDE by a family
(parametrized by a small parameter \vepsi) of quasilinear higher order (in
particular, fourth order) PDEs. The primary objectives of this book are to
present a detailed convergent analysis for the method in the radial symmetric
case and to carry out a comprehensive finite element numerical analysis for the
vanishing moment equations (i.e., the regularized fourth order PDEs). Abstract
methodological and convergence analysis frameworks of conforming finite element
methods and mixed finite element methods are first developed for fully
nonlinear second order PDEs in general settings. The abstract frameworks are
then applied to three prototypical nonlinear equations, namely, the
Monge-Amp\`ere equation, the equation of prescribed Gauss curvature, and the
infinity-Laplacian equation. Numerical experiments are also presented for each
problem to validate the theoretical error estimate results and to gauge the
efficiency of the proposed numerical methods and the vanishing moment
methodology.Comment: 141 pages, 16 figure
Approximation schemes for mixed optimal stopping and control problems with nonlinear expectations and jumps
We propose a class of numerical schemes for mixed optimal stopping and
control of processes with infinite activity jumps and where the objective is
evaluated by a nonlinear expectation. Exploiting an approximation by switching
systems, piecewise constant policy timestepping reduces the problem to nonlocal
semi-linear equations with different control parameters, uncoupled over
individual time steps, which we solve by fully implicit monotone approximations
to the controlled diffusion and the nonlocal term, and specifically the
Lax-Friedrichs scheme for the nonlinearity in the gradient. We establish a
comparison principle for the switching system and demonstrate the convergence
of the schemes, which subsequently gives a constructive proof for the existence
of a solution to the switching system. Numerical experiments are presented for
a recursive utility maximization problem to demonstrate the effectiveness of
the new schemes
Iterative Methods for the Elasticity Imaging Inverse Problem
Cancers of the soft tissue reign among the deadliest diseases throughout the world and effective treatments for such cancers rely on early and accurate detection of tumors within the interior of the body. One such diagnostic tool, known as elasticity imaging or elastography, uses measurements of tissue displacement to reconstruct the variable elasticity between healthy and unhealthy tissue inside the body. This gives rise to a challenging parameter identification inverse problem, that of identifying the Lamé parameter μ in a system of partial differential equations in linear elasticity. Due to the near incompressibility of human tissue, however, common techniques for solving the direct and inverse problems are rendered ineffective due to a phenomenon known as the “locking effect”. Alternative methods, such as mixed finite element methods, must be applied to overcome this complication. Using these methods, this work reposes the problem as a generalized saddle point problem along with a presentation of several optimization formulations, including the modified output least squares (MOLS), energy output least squares (EOLS), and equation error (EE) frameworks, for solving the elasticity imaging inverse problem. Subsequently, numerous iterative optimization methods, including gradient, extragradient, and proximal point methods, are explored and applied to solve the related optimization problem. Implementations of all of the iterative techniques under consideration are applied to all of the developed optimization frameworks using a representative numerical example in elasticity imaging. A thorough analysis and comparison of the methods is subsequently presented
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