A Multigrid Method of the Second Kind for Solving Linear Systems of Odes Discretized by Continuous Runge-Kutta Methods

A Waveform Relaxation method as applied to a linear system of ODEs is the Picard iteration for a linear Volterra integral equation of the second kind ({cal I} - {cal K})y = b eqno (1) called Waveform Relaxation second kind equation. A corresponding Waveform Relaxation Runge-Kutta method is the Picard iteration for a discretized version ({cal I} - {cal K}_l )y_l = b_l eqno (2) of the integral equation (1), where y l is the continuous solution of the original linear system of ODE provided by the so called limit method. We consider a W-cycle multigrid method, with Picard iteration as smoothing step, for iteratively computing y l . This multigrid method belongs to the class of multigrid methods of the second kind as described in Hackbusch [3, chapter 16]. In the paper we prove that the truncation error after one iteration is of the same order of the discretization error y l @ y of the limit method and the truncation error after two iterations has order larger than the discretization error. Thus we can see the multigrid method as a new numerical method for solving the original linear system of ODE which provides, after one iteration, a continuous solution of the same order of the solution of the limit method, and after two iterations, a solution with asymptotically the same error of the solution of the limit method. On the other hand the computational cost of the multigrid method is considerably smaller than the limit method.     

A Mathematical Morphology Based Scale Space Method for the Mining of Linear Features in Geographic Data

This paper presents a spatial data mining method MCAMMO and its extension L_MCAMMO designed for discovering linear and near linear features in spatial databases. L_MCAMMO can be divided into two basic steps: first, the most suitable re-segmenting scale is found by MCAMMO, which is a scale space method with mathematical morphology operators; second, the segmented result at this scale is re-segmented to obtain the final linear belts. These steps are essentially a multi-scale binary image segmentation process, and can also be treated as hierarchical clustering if we view the points under each connected component as one cluster. The final number of clusters is the one which survives (relatively, not absolutely) the longest scale range, and the clustering which first realizes this number of clusters is the most suitable segmentation. The advantages of MCAMMO in general and L_MCAMMO in particular, are: no need to pre-specify the number of clusters, a small number of simple inputs, capable of extracting clusters with arbitrary shapes, and robust to noise. The effectiveness of the proposed method is substantiated by the real-life experiments in the mining of seismic belts in China.