53,204 research outputs found
Deterministic Versus Randomized Kaczmarz Iterative Projection
Kaczmarz's alternating projection method has been widely used for solving a
consistent (mostly over-determined) linear system of equations Ax=b. Because of
its simple iterative nature with light computation, this method was
successfully applied in computerized tomography. Since tomography generates a
matrix A with highly coherent rows, randomized Kaczmarz algorithm is expected
to provide faster convergence as it picks a row for each iteration at random,
based on a certain probability distribution. It was recently shown that picking
a row at random, proportional with its norm, makes the iteration converge
exponentially in expectation with a decay constant that depends on the scaled
condition number of A and not the number of equations. Since Kaczmarz's method
is a subspace projection method, the convergence rate for simple Kaczmarz
algorithm was developed in terms of subspace angles. This paper provides
analyses of simple and randomized Kaczmarz algorithms and explain the link
between them. It also propose new versions of randomization that may speed up
convergence
A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance
We present a simple and easy to implement method for the numerical solution
of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many
cases, the considered problems have only a viscosity solution, to which,
fortunately, many intuitive (e.g. finite difference based) discretisations can
be shown to converge. However, especially when using fully implicit time
stepping schemes with their desirable stability properties, one is still faced
with the considerable task of solving the resulting nonlinear discrete system.
In this paper, we introduce a penalty method which approximates the nonlinear
discrete system to first order in the penalty parameter, and we show that an
iterative scheme can be used to solve the penalised discrete problem in
finitely many steps. We include a number of examples from mathematical finance
for which the described approach yields a rigorous numerical scheme and present
numerical results.Comment: 18 Pages, 4 Figures. This updated version has a slightly more
detailed introduction. In the current form, the paper will appear in SIAM
Journal on Numerical Analysi
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