We study the nonlinear stochastic heat equation in the spatial domain R, driven by space–time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on R, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a r...

We consider preconditioned iterative methods applied to discretizations of the linearized Navier–Stokes equations in twoand three-dimensional bounded domains. Both unsteady and steady flows are considered. The equations are linearized by Picard iteration. We make use of the rotation form of the momentum equations, which has several advantages from the linear algebra point of view. We focus on a...

Iterative Closest Point (ICP) is a widely used method for performing scan-matching and registration. Being simple and robust method, it is still computationally expensive and may be challenging to use in real-time applications with limited resources on mobile platforms. In this paper we propose novel effective method for acceleration of ICP which does not require substantial modifications to th...

In the recently developed Krylov deferred correction (KDC) methods for ordinary differential equation initial value problems [11], a Picard-type collocation formulation is preconditioned using low-order time integration schemes based on spectral deferred correction (SDC), and the resulting system is solved efficiently using a Newton-Krylov method. Existing analyses show that these KDC methods a...

A computationally simple, numerical algorithm capable of solving a wide variety of twodimensional, variably saturated flow problems is developed. Recent advances in modeling variably saturated flow are incorporated into the algorithm. A physically based form of the general, variably saturated flow equation is solved using finite differences (centered in space, fully implicit in time) employing ...

Many scientific problems are posed as Ordinary Differential Equations (ODEs). A large subset of these are initial value problems, which are typically solved numerically. The solution starts by using a known state-space of the ODE system to determine the state at a subsequent point in time. This process is repeated several times. When the computational demand is high due to large state space, pa...

In this paper, we presented new and important existence theorems of solution for quasi-equilibrium problems, and we show the uniqueness of its solution which is also a fixed point of some mapping. We show that this unique solution can be obtained by Picard's iteration method. We also get new minimax theorem, and existence theorems for common solution of fixed point and optimization problem on c...

The Carathéodory approximation scheme was introduced by the Greek mathematician named Constantine Carathéodory in the early part of 20th century for ordinary differential equations (Chapter 2 of [1]). Later this was extended by Bell and Mohammad to stochastic differential equations [2] and then by Mao [3, 4]. Generally, the solutions of stochastic differential equations (SDEs) do not have expli...

Stochastic differential equations in R with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W β 2 for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution p...

Abstract This article introduces the basic qualitative and basic quantitative theory of Volterra integral equations on time scales and thus may be considered as a foundation for future advanced studies in the field. New sufficient conditions are introduced that guarantee: existence; uniqueness; approximation; boundedness and certain growth rates of solutions to both linear and nonlinear problem...