Abstract: The method described here for recovering the shape of a surface from a shaded image can deal with complex, wrinkled surfaces. Integrability can be enforced easily because both surface height and gradient are represented (A gradient field is integrable if it is the gradient of some surface height function). The robustness of the method stems in part from linearization of the reflectance map about the current estimate of the surface orientation at each picture cell (The reflectance map gives the dependence of scene radiance on surface orientation). The new scheme can find an exact solution of a given shape-from-shading problem even though a regularizing term is included. The reason is that the penalty term is needed only to stabilize the iterative scheme when it is far from the correct solution; it can be turned off as the solution is approached. This is a reflection of the fact that shape-from-shading problems are not ill-posed when boundary conditions are available, or when the image contains singular points. This paper includes a review of previous work on shape from shading and photoclinometry. Novel features of the new scheme are introduced one at a time to make it easier to see what each contributes. Included is a discussion of implementation details that are important if exact algebraic solutions of synthetic shape-from-shading problems are to be obtained. The hope is that better performance on synthetic data will lead to better performance on real data.

Standard multigrid algorithms have proven ineffective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's effectiveness through theoretical analysis of a model problem and experimental results. In particular, we show that the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as two hundred. For fixed wave numbers, it displays grid-independent convergence rates and has costs proportional to the number of unknowns.

This article gives a tutorial overview of essential components of scale-space theory --- a framework for multi-scale signal representation, which has been developed by the computer vision community to analyse and interpret real-world images by automatic methods. 1 The need for multi-scale representation of image data An inherent property of real-world objects is that they only exist as meaningful entities over In: Proc. CERN School of Computing, Egmond aan Zee, The Netherlands, 8--21 September, 1996. certain ranges of scale. A simple example is the concept of a branch of a tree, which makes sense only at a scale from, say, a few centimeters to at most a few meters, It is meaningless to discuss the tree concept at the nanometer or kilometer level. At those scales, it is more relevant to talk about the molecules that form the leaves of the tree, and the forest in which the tree grows, respectively. This fact, that objects in the world appear in different ways depending on the scale of ...

An overview of current multigrid techniques for unstructured meshes is given. The basic principles of the multigrid approach are first outlined. Application of these principles to unstructured mesh problems is then described, illustrating various different approaches, and giving examples of practical applications. Advanced multigrid topics, such as the use of algebraic multigrid methods, and the combination of multigrid techniques with adaptive meshing strategies are dealt with in subsequent sections. These represent current areas of research, and the unresolved issues are discussed. The presentation is organized in an educational manner, for readers familiar with computational fluid dynamics, wishing to

Abstract. In this paper we study geometrical properties of the iterative 4-triangles longest-side partition of triangles (and of a 3-triangles partition), as well as practical algorithms based on these partitions, used both directly for the triangulation refinement problem, and as a basis for point insertion strategies in Delaunay refinement algorithms. The 4-triangles partition is obtained by joining the midpoint of the longest side with the opposite vertex and the midpoints of the two remaining sides. By means of simple geometrical arguments we show that the iterative partition of obtuse triangles systematically improves the triangles (while they remain obtuse) in the following sense: the sequence of smallest angles monotonically increases while the sequence of largest angles monotonically decreases in an amount (at least) equal to the smallest angle of each iteration. This allows us to improve the known bound on the smallest angle (without making use of previous results), and to obtain a better a priori bound on the number of similarly distinct triangles, as a function of the geometry of the initial triangle. Numerical evidence, showing that the practical behavior of the 4-triangles partition is in complete agreement with this theory, is included. A 4-triangles refinement algorithm is also discussed and illustrated. Furthermore, we show that the time cost of the algorithm is linear independently of the size of the triangulation. 1. Introduction: The

Abstract. The relative performance of a non-linear FAS multigrid algorithm and an equivalent linear multigrid algorithm for solving two di erent non-linear problems is investigated. The rst case consists of a transient radiation-di usion problem for which an exact linearization is available, while the second problem involves the solution of the steady-state Navier-Stokes equations, where a rst-order discrete Jacobian is employed as an approximation to the Jacobian of a second-order accurate discretization. When an exact linearization is employed, the linear and non-linear multigrid methods converge at identical rates, asymptotically, and the linear method is found to be more e cient due to its lower cost per cycle. When an approximate linearization is employed, as in the Navier-Stokes cases, the relative e ciency of the linear approach versus the non-linear approach depends both on the degree to which the linear system approximates the full Jacobian as well as the relative cost of linear versus non-linear multigrid cycles. For cases where convergence is limited by a poor Jacobian approximation, substantial speedup can be obtained using either multigrid method as a preconditioner to a Newton-Krylov method.

The topic of this paper is motivated by the Navier-Stokes equations in rotation form. Linearization and application of an implicit time stepping scheme results in a linear stationary problem of Oseen type. In well-known solution techniques for this problem such as the Uzawa (or Schur complement) method, a subproblem consisting of a coupled non-symmetric system of linear equations of diusion-reaction type must be solved to update the velocity vector eld. In this paper we analyse a standard nite element method for the discretization of this coupled system and we introduce and analyse a multigrid solver for the discrete problem. Both for the discretization method and the multigrid solver the question of robustness with respect to the amount of diusion and variation in the convection eld is addressed. We prove stability results and discretization error bounds for the Galerkin nite element method. We present a convergence analysis of the multigrid method which shows the robustness of the solver. Results of numerical experiments are presented which illustrate the stability of the discretization method and the robustness of the multigrid solver. AMS subject classications. 65N30, 65N55, 76D17, 35J55 Key words. nite elements, multigrid, convection-diusion, Navier-Stokes equations, rotation form, vorticity 1.

Over the years, multigrid has been demonstrated as an efficient technique for solving inviscid flow problems. However, for viscous flows, convergence rates often degrade. This is generally due to the required use of stretched meshes (i.e. the aspect-ratio AR = \Deltay=\Deltax !! 1) in order to capture the boundary layer near the body. Usual techniques for generating a sequence of grids that produce proper convergence rates on isotropic meshes are not adequate for stretched meshes. This work focuses on the solution of Laplace's equation, discretized through a Galerkin finite-element formulation on unstructured stretched triangular meshes. A coarsening strategy is proposed and results are discussed. Introduction Multigrid method has been shown to be successful for solving elliptic problems. This is mainly due to its good damping properties which result from two very simple principles. A usual Fourier analysis demonstrates that most of the commonly used solvers effectively damp the high...

Abstract. We consider saddle point problems that result from the finite element discretization of stationary and instationary Stokes equations. Three efficient iterative solvers for these problems are treated, namely the preconditioned CG method introduced by Bramble and Pasciak, the preconditioned MINRES method and a method due to Bank et al. We give a detailed overview of algorithmic aspects and theoretical convergence results. For the method of Bank et al a new convergence analysis is presented. A comparative study of the three methods for a 3D Stokes problem discretized by the Hood-Taylor P2 − P1 finite element pair is given. AMS subject classifications. 65N30, 65F10 Key words. Stokes equations, inexact Uzawa methods, preconditioned MINRES, multigrid

We study the role of preconditioning strategies recently developed for coercive problems in connection with a two-step iterative method (HSS) proposed by Bai, Golub and Ng for the solution of nonsymmetric linear systems whose real part is coercive. As a model problem we consider Finite Dierences (FD) matrix sequences fAn(a; p)gn discretizing the elliptic (convection-diusion) problem Aa;pu r[a(x)ru(x)] +r[p(x)u(x)] = f(x); x Dirichlet BC; (1) with with a(x) being a uniformly positive function and p(x) denoting the Reynolds function. More precisely, in connection with preconditioned HSS/GMRES like methods, we consider the preconditioning sequence fPn (a)gn , Pn(a) := D n (a)An(1; 0)D n (a) where Dn (a) is the suitable scaled main diagonal of An (a; 0). If a(x) is positive and regular enough, then the preconditioned sequence shows a strong clustering at the unity so that the sequence fPn (a)gn turns out to be a superlinear preconditioning sequence for fAn(a; 0)gn where An(a; 0) represents a good approximation of Re(An(a; p)) namely the real part of An(a; p).