The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); (ii) to conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining "°°-order" approximations and low n, even when singularities are present. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problemsconfirm theoretical predictions. Similar techniques for initial-value problems are briefly discussed. 1. Introduction. In most numerical procedures for solving partial differential equations, the analyst first discretizes the problem, choosing approximating algebraic equations on a finite-dimensional approximation space, and then devises a numerical process to (nearly) solve this huge system of discrete equations. Usually, no real interplay is allowed between discretization and solution processes. This results in enormous waste: The discretization process, being unable to predict the proper resolution and the proper order of approximation at each location, produces a mesh which is too fine. The algebraic system thus becomes unnecessarily large in size, while accuracy usually remains rather low, since local smoothness of the solution is not being properly exploited. On the other hand, the solution process fails to take advantage of the fact that the algebraic system to be solved does not stand by itself, but is actually an approximation to continuous equations, and therefore can itself be similarly approximated by other (much simpler) algebraic systems. The purpose of the work reported here is to study how to intermix discretization and solution processes, thereby making both of them orders-of-magnitude more effective. The method to be proposed is not "saturated", that is, accuracy grows indefinitely as computations proceed. The rate of convergence (overall error E as function of computational work IV) is in principle of "infinite order", e.g., E ~ expi-ß^W) for a d-dimensional problem which has a solution with scale-ratios > ß > 0; or E ~ expi-W^2), for problems with arbitrary thin layers (see Section 9). The basic idea of the Multi-Level Adaptive Techniques (MLAT) is to work not

Abstract.The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); (ii) to conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining "°°-order" approximations and low n, even when singularities are present. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problemsconfirm theoretical predictions. Similar techniques for initial-value problems are briefly discussed.

We present a bottom-up aggregation approach to image segmentation. Beginning with an image, we execute a sequence of steps in which pixels are gradually merged to produce larger and larger regions. In each step, we consider pairs of adjacent regions and provide a probability measure to assess whether or not they should be included in the same segment. Our probabilistic formulation takes into account intensity and texture distributions in a local area around each region. It further incorporates priors based on the geometry of the regions. Finally, posteriors based on intensity and texture cues are combined using “a mixture of experts” formulation. This probabilistic approach is integrated into a graph coarsening scheme, providing a complete hierarchical segmentation of the image. The algorithm complexity is linear in the number of the image pixels and it requires almost no user-tuned parameters. In addition, we provide a novel evaluation scheme for image segmentation algorithms, attempting to avoid human semantic considerations that are out of scope for segmentation algorithms. Using this novel evaluation scheme, we test our method and provide a comparison to several existing segmentation algorithms.

The subject of this paper is the application of the multi-grid method to the solution of -V. (D(x, y)VU(x, y))+(x, y)U(x, y)=f (x, y) in a bounded region f of R where D is positive and D, tr, and f are allowed to be discontinuous across internal boundaries F of fL The emphasis is on discontinuities of orders of magnitude in D, when special techniques must be applied to restore the multi-grid method to good efficiency. These techniques are based on the continuity of D VU across F. Two basic methods are derived, one in which the approximating finite difference operators on coarser grids are five point operators (assuming the finite difference operator on the finest grid is a five point one) and one in which they are nine point operators.

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