A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with user-defined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels. By assigning each pixel to the label for which the greatest probability is calculated, a high-quality image segmentation may be obtained. Theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits. This algorithm is formulated in discrete space (i.e., on a graph) using combinatorial analogues of standard operators and principles from continuous potential theory, allowing it to be applied in arbitrary dimension on arbitrary graphs

/pdf/random-walks-for-image-segmentation-4ppfcpqq4u.pdf

Random Walks for Image Segmentation

We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision).

Spectral Deferred Correction Methods for Ordinary Differential Equations

Recently, a number of closely related techniques for error estimation and iterative improvement in discretization algorithms have been proposed. In this article, we expose the common structural principle of all these techniques and exhibit the principal modes of its implementation in a discretization context.

The defect correction principle and discretization methods

We consider multiphysics applications from algorithmic and architectural perspectives, where âalgorithmicâ includes both mathematical analysis and computational complexity, and âarchitecturalâ includes both software and hardware environments. Many diverse multiphysics applications can be reduced, en route to their computational simulation, to a common algebraic coupling paradigm. Mathematical analysis of multiphysics coupling in this form is not always practical for realistic applications, but model problems representative of applications discussed herein can provide insight. A variety of software frameworks for multiphysics applications have been constructed and refined within disciplinary communities and executed on leading-edge computer systems. We examine several of these, expose some commonalities among them, and attempt to extrapolate best practices to future systems. From our study, we summarize challenges and forecast opportunities.

https://journals.sagepub.com/doi/pdf/10.1177/1094342012468181

Multiphysics simulations: Challenges and opportunities

Stiff differential equations solved by Radau methods

A useful concept of convergence for nonlinear stiff ODE’s will be developed, which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem. This concept will be discussed for three simple methods for stiff systems, the implicit Euler scheme, the implicit midpoint rule and the implicit trapezoidal rule.

The Concept of B-Convergence

An extensive order analysis is given for the local and global errors of certain classes of Runge–Kutta methods applied to general nonlinear, stiff initial value problems. It turns out that there are three order levels: for linear problems $y' = Ay$ the classical orders apply, for general nonlinear problems, however, two types of order reductions occur. There a distinction has to be made between stiff problems where only $\| {f_y } \|$ is “large” and stiff problems where also some of the other derivatives $f_t ,f_{tt} ,f_{yy} , \cdots $, are “large.” For these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h. The order levels for both classes of problems are considerably lower than the respective conventional orders. These order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems.

Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems

Iterated Defect Correction (IDeC)-methods based on the implicit Euler scheme are shown to have a fixed point. This fixed point coincides with the solution of certain implicit multi-stage Runge-Kutta methods (equivalent to polynomial collocation). Sufficient conditions for the convergence of the iterates to the fixed point are given for linear problems. These results indicate that for a large variety of general non-linear stiff problems, fixed-point-convergence can be expected, and moreover they indicate that the rate of convergence to the fixed point is very high for very stiff problems. Thus the proposed methods combine the high orders and the high accuracy of multistage-methods with the low computational effort of single-stage methods.

Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations

New stability concepts—$BS$-stability and $BSI$-stability (internal $BS$-stability)—are introduced. $BS$-stability is a modification and extension of B-stability and enables the derivation of order results for Runge–Kutta methods applied to general nonlinear stiff initial value problems. These order results (B-consistency, B-convergence) are not affected by stiffness. Several classes of implicit Runge–Kutta methods are shown to be $BS$-stable: Gauss, Radau IA and Radau IIA schemes.

Stability Properties of Implicit Runge–Kutta Methods

The structure of the global discretization error is studied for the implicit midpoint and trapezoidal rules applied to nonlinearstiff initial value problems The point is that, in general, the global error contains nonsmooth (oscillating) terms at the dominanth 2-level However, it is shown in the present paper that for special classes of stiff problems these nonsmooth terms contain an additional factor ? (where-1/? is the magnitude of the stiff eigenvalues) In these cases a "full" asymptotic error expansion exists in thestrongly stiff case (? sufficiently small compared to the stepsizeh) The general case (where the oscillating error components areO(h 2) and notO(?h 2)) and applications of our results (extrapolation and defect correction algorithims) will be studied in separate papers

Reinhard Frank

Papers

The Concept of B-Convergence

Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems

Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations

Stability Properties of Implicit Runge–Kutta Methods

Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case