18182246
Total variation blind deconvolution
In this paper, we present a blind deconvolution algorithm based on the total variational (TV) minimization method proposed. The motivation for regularizing with the TV norm is that it is extremely effective for recovering edges of images as well as some blurring functions, e.g., motion blur and out-of-focus blur. An alternating minimization (AM)implicit iterative scheme is devised to recover the image and simultaneously identify the point spread function (psf). Numerical results indicate that the iterative scheme is quite robust, converges very fast (especially for discontinuous blur), and both the image and the psf can be recovered under the presence of high noise level. Finally, we remark that psf's without sharp edges, e.g., Gaussian blur, can also be identified through the TV approach.
20538253
Parallel Complexity of Domain Decomposition Methods and Optimal Coarse Grid Size
Abstract In this paper, we analyze the parallel complexity of domain decomposition method through estimating the arithmetic time as well as the communication time. In most domain decomposition (DD) methods, a coarse grid solve is employed to provide global coupling and obtain an optimal convergence rate. Generally, a small coarse grid size H improves the convergence rate at the cost of a larger coarse grid problem and a large number P of processors results in more costly global communication, whereas a large H and a small P have the opposite effect. Therefore, there often exists an optimal H such that the execution time on the parallel machine reaches a minimum. Our main goal here is to find this optimal coarse grid size H in the parallel environment. We assume the same solver is used for the subdomains as well as the coarse problem. By expressing the parallel complexity as a function of H and h, we can derive the analytic optimal coarse grid size Hopt. When the number of processors is equal to the number of subdomains, we can prove that the choice of Hopt = √h asymptotically minimises the execution time as h tends to zero. This optimal coarse grid size is independent of the choice of subproblem solvers and the spatial dimension. However, when the number of processors is much less than the number of subdomains, the optimal coarse grid size depends on the number of processors, the subdomain solver and dimension.
2980605
A Simplified Approach to the Order Conditions of Integration Methods
We present an approach to the numerical integration of ordinary differential equations based on the algebraic theory of Butcher (Math. Comp. 26, 79–106, 1972) and the -series theory of Hairer and Wanner (Computing 13, 1–15, 1974). We clarify the differences of these two approaches by equating the elementary weight functions and showing the differences of the composition rules. By interpreting the elementary weight function as a mapping from input values to output values and introducing some special mappings, we are able to derive the order conditions of several types of integration methods in a straight-forward way. The simplicity of the derivation is illustrated by linear multistep methods that use the second derivative as an input value, Runge-Kutta type methods that use the second as well as first derivatives, and general two-step Runge-Kutta methods. We derive new high stage-order methods in each example. In particular, we found a symmetric and stiffly-accurate method of order eight in the second example.
26929194
Balanced k-colorings
Abstract While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k -coloring, for k ⩾2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest difference between the size of any two color classes in that subset. The discrepancy is the minimum possible imbalance. We show that the discrepancy is always at most 4 d −3, where d (the “dimension”) is the maximum number of subsets containing a common vertex. For 2-colorings, the bound on the discrepancy is atmost max{2 d −3,2}. Finally, we prove that several restricted versions of computing the discrepancy are NP-complete.
15785799
Semi-online maintenance of geometric optima and measures
We give the first nontrivial worst-case results for dynamic versions of various basic geometric optimization and measure problems under the semi-online model, where during the insertion of an object we are told when the object is to be deleted. Problems that we can solve with sublinear update time include the Hausdorff distance of two point sets, discrete 1-center, largest empty circle, convex hull volume in three dimensions, volume of the union of axis-parallel cubes, and minimum enclosing rectangle. The decision versions of the Hausdorff distance and discrete 1-center problems can be solved fully dynamically. Some applications are mentioned.
9903009
A note on maximum independent sets in rectangle intersection graphs
Finding the maximum independent set in the intersection graph of n axis-parallel rectangles is NP-hard. We re-examine two known approximation results for this problem. For the case of rectangles of unit height, Agarwal, van Kreveld and Suri [Comput. Geom. Theory Appl. 11 (1998) 209–218] gave a (1+1/k)-factor algorithm with an O(nlogn+n2k−1) time bound for any integer constant k⩾1; we describe a similar algorithm running in only O(nlogn+nΔk−1) time, where Δ⩽n denotes the maximum number of rectangles a point can be in. For the general case, Berman, DasGupta, Muthukrishnan and Ramaswami [J. Algorithms 41 (2001) 443–470] gave a ⌈logkn⌉-factor algorithm with an O(nk+1) time bound for any integer constant k⩾2; we describe similar algorithms running in O(nlogn+nΔk−2) and nO(k/logk) time.
3002414
More algorithms for all-pairs shortest paths in weighted graphs
In the first part of the paper, we reexamine the all-pairsshortest paths (APSP) problem and present a newalgorithm with running time approaching O(n3/log2n), which improves all known algorithms for general real-weighted dense graphs andis perhaps close to the best result possible without using fast matrix multiplication, modulo a few log log n factors.
 In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of "geometrically weighted" graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n3-(3-Ω)/(2d+4)), where Ω < 2.376 ; in two dimensions, this is O(n2.922). Our framework greatly extends the previously considered case of small-integer-weighted graphs, and incidentally also yields the first truly subcubic result (near O(n3-(3-Ω)/4)=O(n2.844) time) forAPSP in real-vertex-weighted graphs, as well as an improved result (near O(n(3+Ω)/2)=O(n2.688) time) for the all-pairs lightest shortest path problem for small-integer-weighted graphs.