986436
Convergence of gauge method for incompressible flow
A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field a and a gauge variable Φ, u = a + ⊇Φ, was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation and verifies the computational advantages. We discuss the implicit gauge method, which uses backward Euler or Crank-Nicolson in time discretization. However, the boundary conditions for the auxiliary field a are implemented explicitly (vertical extrapolation). The resulting momentum equation is decoupled from the kinematic equation, and the computational cost is reduced to solving a standard heat and Poisson equation. Moreover, such explicit boundary conditions for the auxiliary field a will be shown to be unconditionally stable for Stokes equations. For the full nonlinear Navier-Stokes equations the time stepping constraint is reduced to the standard CFL constraint Δt/Δx < C. We also prove first order convergence of the gauge method when we use MAC grids as our spatial discretization. The optimal error estimate for the velocity field is also obtained.
2139488
Projection method III: Spatial discretization on the staggered grid
In E & Liu (SIAM J Numer. Anal., 1995), we studied convergence and the structure of the error for several projection methods when the spatial variable was kept continuous (we call this the semi-discrete case). In this paper, we address similar questions for the fully discrete case when the spatial variables are discretized using a staggered grid. We prove that the numerical solution in velocity has full accuracy up to the boundary, despite the fact that there are numerical boundary layers present in the semi-discrete solutions.
18986222
Concepts and Application of Time-Limiters to High Resolution Schemes
A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.
828752
Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions
Solutions of conservation laws satisfy the monotonicity property: the number of local extrema is a non-increasing function of time, and local maximum/minimum values decrease/increase monotonically in time. This paper investigates this property from a numerical standpoint. We introduce a class of fully discrete in space and time, high order accurate, difference schemes, called generalized monotone schemes. Convergence toward the entropy solution is proven via a new technique of proof, assuming that the initial data has a finite number of extremum values only, and the flux-function is strictly convex. We define discrete paths of extrema by tracking local extremum values in the approximate solution. In the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum. As a corollary, we obtain a proof of convergence for a MUSCL-type scheme that is second order accurate away from sonic points and extrema.
14714835
A Method of Spatial Reasoning Based on Qualitative Trigonometry
Abstract Due to the lack of exact quantitative information or the difficulty associated with obtaining or processing such information, qualitative spatial knowledge representation and reasoning often become an essential means for solving spatial constraint problems as found in science and engineering. This paper presents a computational approach to representing and reasoning about spatial constraints in two-dimensional Euclidean space, where the a priori spatial information is not precisely expressed in quantitative terms. The spatial quantities considered in this work are qualitative distances and qualitative orientation angles. Here, we explicitly define the semantics of these quantities and thereafter formulate a representation of qualitative trigonometry (QTRIG). The resulting QTRIG formalism provides the necessary inference rules for qualitative spatial reasoning. In the paper, we illustrate how the QTRIG relationships can be employed in generating qualitative spatial descriptions in two-dimensional Euclidean geometric problems, and furthermore, how the derived qualitative spatial descriptions can be used to guide a simulated-annealing-based exact quantitative value assignment. Finally, we discuss an application of the proposed spatial reasoning method to the kinematic constraint analysis in computer-aided pre-parametric mechanism design.
5682640
A dual-space approach to tracking and sensor management in wireless sensor networks
Wireless ad hoc sensor networks have the advantage of spanning a large geographical region and being able to collaboratively detect and track non-local spatio-temporal events. This paper presents a dual-space approach to event tracking and sensor resource management in sensor networks. The dual-space transformation maps a non-local phenomenon, e.g., the edge of a half-plane shadow, to a single point in the dual space, and maps locations of distributed sensor nodes to a set of lines that partitions the dual space. The detection problem becomes finding and tracking the cell that contains the point in the arrangement defined by these lines. This mechanism can be effectively used for power management of the sensor network - nodes that will not be immediately visited by an event can be turned off to save energy required for sensing, processing, and communication. The approach has been successfully demonstrated on a laboratory testbed built using the UC Berkeley motes sensors. An implemented application of detecting and tracking light shadow edges moving over a sensor field is described.
14343926
Qualitative Discovery in Medical Databases
Implication rules have been used in uncertainty reasoning systems to confirm and draw hypotheses or conclusions. However a major bottleneck in developing such systems lies in the elicitation of these rules. This paper empirically examines the performance of evidential inferencing with implication networks generated using a rule induction tool called KAT. KAT utilizes an algorithm for the statistical analysis of empirical case data, and hence reduces the knowledge engineering efforts and biases in subjective implication certainty assignment. The paper describes several experiments in which real-world diagnostic problems were investigated; namely, medical diagnostics. In particular, it attempts to show that (1) with a limited number of case samples, KAT is capable of inducing implication networks useful for making evidential inferences based on partial observations, and (2) observation driven by a network entropy optimization mechanism is effective in reducing the uncertainty of predicted events.
2027823
Evolutionary Search for Faces from Line Drawings
Single 2D line drawing is a straightforward method to illustrate 3D objects. The faces of an object depicted by a line drawing give very useful information for the reconstruction of its 3D geometry. Two recently proposed methods for face identification from line drawings are based on two steps: finding a set of circuits that may be faces and searching for real faces from the set according to some criteria. The two steps, however, involve two combinatorial problems. The number of the circuits generated in the first step grows exponentially with the number of edges of a line drawing. These circuits are then used as the input to the second combinatorial search step. When dealing with objects having more faces, the combinatorial explosion prevents these methods from finding solutions within feasible time. This paper proposes a new method to tackle the face identification problem by a variable-length genetic algorithm with a novel heuristic and geometric constraints incorporated for local search. The hybrid GA solves the two combinatorial problems simultaneously. Experimental results show that our algorithm can find the faces of a line drawing having more than 30 faces much more efficiently. In addition, simulated annealing for solving the face identification problem is also implemented for comparison.
8458381
Dynamic Resource Selection For Service Composition in The Grid
While numerous efforts have focused on service composition in the Grid environment, service selection among similar services from multiple providers has not been addressed. In particular, all service composition work done so far are based on a given selection of services under a well set environment. As a result, uncertainty (e.g., server load, network traffic, computation time of the services due to changing memory and other unexpected conditions) under a real, dynamic environment has never been considered. This paper prototypes the service selection under a Grid environment and proposes an uncertainty framework to address the issue. Experimental results show that our considerations are valid and our preliminary solution works well in our Globus Grid network.