Multiplicative Operator Splittings in Nonlinear Diffusion: From Spatial Splitting to Multiple Timesteps

Operator splitting is a powerful concept used in many diversed fields of applied mathematics for the design of effective numerical schemes. Following the success of the additive operator splitting (AOS) in performing an efficient nonlinear diffusion filtering on digital images, we analyze the possibility of using multiplicative operator splittings to process images from different perspectives.We start by examining the potential of using fractional step methods to design a multiplicative operator splitting as an alternative to AOS schemes. By means of a Strang splitting, we attempt to use numerical schemes that are known to be more accurate in linear diffusion processes and apply them on images. Initially we implement the Crank-Nicolson and DuFort-Frankel schemes to diffuse noisy signals in one dimension and devise a simple extrapolation that enables the Crank-Nicolson to be used with high accuracy on these signals. We then combine the Crank-Nicolson in 1D with various multiplicative operator splittings to process images. Based on these ideas we obtain some interesting results. However, from the practical standpoint, due to the computational expenses associated with these schemes and the questionable benefits in applying them to perform nonlinear diffusion filtering when using long timesteps, we conclude that AOS schemes are simple and efficient compared to these alternatives.We then examine the potential utility of using multiple timestep methods combined with AOS schemes, as means to expedite the diffusion process. These methods were developed for molecular dynamics applications and are used efficiently in biomolecular simulations. The idea is to split the forces exerted on atoms into different classes according to their behavior in time, and assign longer timesteps to nonlocal, slowly-varying forces such as the Coulomb and van der Waals interactions, whereas the local forces like bond and angle are treated with smaller timesteps. Multiple timestep integrators can be derived from the Trotter factorization, a decomposition that bears a strong resemblance to a Strang splitting. Both formulations decompose the time propagator into trilateral products to construct multiplicative operator splittings which are second order in time, with the possibility of extending the factorization to higher order expansions. While a Strang splitting is a decomposition across spatial dimensions, where each dimension is subsequently treated with a fractional step, the multiple timestep method is a decomposition across scales. Thus, multiple timestep methods are a realization of the multiplicative operator splitting idea. For certain nonlinear diffusion coefficients with favorable properties, we show that a simple multiple timestep method can improve the diffusion process.     

Efficient removal of boundary-divergence errors in time-splitting methods

A normal mode analysis is presented and numerical tests are performed to assess the effectiveness of a new time-splitting algorithm proposed recently in Karniadakiset al. (1990) for solving the incompressible Navier-Stokes equations. This new algorithm employs high-order explicit pressure boundary conditions and mixed explicit/implicit stiffly stable time-integration schemes, which can lead to arbitrarily high-order accuracy in time. In the current article we investigate both the time accuracy of the new scheme as well as the corresponding reduction in boundary-divergence errors for two model flow problems involving solid boundaries. The main finding is that time discretization errors, induced by the nondivergent splitting mode, scale with the order of the accuracy of the integration rule employed if a proper rotational form of the pressure boundary condition is used; otherwise a first-order accuracy in time similar to the classical splitting methods is achieved. In the former case the corresponding errors in divergence can be completely eliminated, while in the latter case they scale asO(vt)_1/2.     

Linear-time snapshot implementations in unbalanced systems

Anatomic snapshot memory object in shared memory systems enables a set of processes, calledscanners, to obtain a consistent picture of the shared memory while other processes, calledupdaters, keep updating memory locations concurrently. In this paper we present two conversion methods of snapshot implementations. Using the first conversion method we obtain a new snapshot implementation in which the scan operation has linear time complexity and the time complexity of the update operation becomes the sum of the time complexities of the original implementation. Applying the second conversion method yields similar results, where in this case the time complexity of the update protocol becomes linear. Although our conversion methods use unbounded space, their space complexity can be bounded using known techniques. One of the most intriguing open problems in distributed wait-free computing is the existence of a linear-time implementation of this object. Using our conversion methods and known constructions we obtain the following results:

?Consider a system ofn processes, each an updater and a scanner. We present an implementation in which the time complexity of either the update or the scan operation is linear, while the time complexity of the second operation isO(n logn).

?We present an implementation with linear time complexity when the number of either updaters or scanners isO(n/logn), wheren is the total number of processes.

?We present an implementation with amortized linear time complexity when one of the protocols (either upate or scan) is executed significantly more often than the other protocol.

     

Automated Meta-Control for Adaptable Real-Time Software

The software meta-controller is an online agent responsible for dynamically adapting an application's software configuration, e.g. altering operational modes and migrating tasks, to best accommodate varying runtime circumstances. In distributed real-time applications such adaptations must be carried out in a manner which maintains the schedulability of all critical tasks while maximizing some notion of system value for all other tasks. For large-scale real-time applications, considering all possible adaptations at the task-level is computationally intractable. This paper presents an automated aggregate approach to software meta-control, appropriate for large-scale distributed real-time systems. The aggregate automated meta-control problem is still NP-hard, but it has very practical approximate solutions. Introduced, here, are two very-effective approximation algorithms, QDP and GG, with very reasonable polynomial time complexity. Both algorithms also provide us with upper bounds for optimum system values, useful for deriving absolute, albeit somewhat pessimistic, measures of actual performance. Extensive Monte Carlo analysis is used to illustrate that expected performance for both algorithms is generally suboptimal by no more than a few percent. Our flexible software meta-control model is also shown to be readily applied to a wide range of time-sensitive applications.     

Analyzing expected time by scheduler-luck games

We introduce a novel technique, the scheduler luck game (in short sl-game) for analyzing the performance of randomized distributed protocols. We apply it in studying uniform self-stabilizing protocols for leader election under read/write atomicity. We present two protocols for the case where each processor in the system can communicate with all other processors and analyze their performance using the sl-game technique     

Effect of prostate biopsy on the results of the PSA RT-PCR test

In this investigation, the presence of NKA-immunoreactive substances was determined in pineal glands from intact, castrated and castrated, testosterone-treated male rats. The effect of environmental light, melatonin treatment and superior cervical ganglionectomy on pineal NKA-immunoreactive substances was also investigated. The results obtained show that NKA is present in measurable amounts in the rat pineal, and NPK is probably also present, Orchidectomy was followed by an increase in the content of NKA-immunoreactive substances in the pineal gland. The replacement treatment with testosterone propionate in castrated rats blocked this effect. NKA-immunoreactive substances were not significantly different quantitatively in pineals from rats killed under light or under darkness. The removal of the superior cervical ganglia was followed by a significant increase in the NKA-immunoreactive substance content in the pineal gland of male rats. These results indicate that NKA and other tachykinins are present in the pineal gland of the male rat, and they seem to be regulated by gonadal hormones and the innervation originated from the superior cervical ganglia.     

Laminar compressible flow between close rotating disks: An asymptotic and numerical study

Laminar steady compressible flow between close rotating thermally conducting axisymmetric disks with inflow was investigated by means of a numerical solution of the Navier-Stokes equation and an asymptotic analysis. The approximate solution, obtained for small ?, E and H (Rossby and Ekman numbers, and height/radius, respectively) is valid for “merged”, “close” and “separate” boundary layers on the disks, corresponding to β? 1, $\text{β ? 1}$ and β? 1, respectively (where $\text{β =}\text{H}\text{2}\text{√}\text{E}\text{ρ}$, and ρ is the non-dimensional density). These three cases may appear simultaneously in different regions of the same system due to the large variation of ρ in the radial direction. The small ? (i.e. negligible convection terms) does not necessarily imply small perturbations of the pressure, and a special treatment of the pressure term was used in order to account for this feature, which sometimes culminates in inversion of the radial pressure gradient. Thenumerical solution was obtained by a finite-difference, modified Cheng-Allen method, using a non-uniform mesh. The numerical and the approximate solution are in good agreement.     

Optimal time byzantine agreement for t <n/8 with linear-messages

Summary.  This paper presents a Byzantine Agreement protocol with n=8t+1, optimal number of rounds (namely min{ f+2, t+1} where f is number of actual faults), and messages of linear size (namely m≦n+O(log n), where m stands for message size). All previous protocols that stop in optimal time and tolerate t=O(n) faults require messages of size at least O(n ²). The new protocol uses a novel technique called Reconstructed Traversal which is based on the Reconstruction Principle and on the Coordinated Traversal protocol. Received: August 1992/Accepted: January 1995l     

Residual stress determination by continuous electroetching

The production of precision parts requires manufacturing processes which procedure low residual stresses. A technique has been developed to continuously electroetch a specimen and simultaneously measure the change in its dimension, and then to compute the residual stress profile in that specimen. A ‘Specific Instability Potential’ derived from the strain energy of the residual stresses was found to relate directly to the machining parameters.