Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

羟氨苄青霉素引起大疱性类天疱疮样疹

Sphere Packings, Lattices and Groups

Network Optimization: Continuous and Discrete Models

Subdivision is a powerful paradigm for the generation of surfaces of arbitrary topology. Given an initial triangular mesh the goal is to produce a smooth and visually pleasing surface whose shape is controlled by the initial mesh. Of particular interest are interpolating
schemes since they match the original data exactly, and play an important role in fast multiresolution and wavelet techniques. Dyn, Gregory, and Levin introduced the Butterfly scheme, which yields C^1 surfaces in the topologically regular setting. Unfortunately it
exhibits undesirable artifacts in the case of an irregular topology. We examine these failures and derive an improved scheme, which retains the simplicity of the Butterfly scheme, is interpolating, and results in smoother surfaces.

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Interpolating Subdivision for meshes with arbitrary topology

To improve the visual quality of existing triangle-based art in realtime entertainment, such as computer games, we propose replacing flat triangles with curved patches and higher-order normal variation. At the hardware level, based only on the three vertices and three vertex normals of a given flat triangle, we substitute the geometry of a three-sided cubic Bezier patch for the triangle’s flat geometry, and a quadratically varying normal for Gouraud shading. These curved point-normal triangles, or PN triangles, require minimal or no change to existing authoring tools and hardware designs while providing a smoother, though not necessarily everywhere tangent continuous, silhouette and more organic shapes. CR Categories: I.3.5 [surface representation, splines]: I.3.6— graphics data structures

Curved PN triangles

Given a polyhedron, construct a new polyhedron by connecting every edge-midpoint to its four neighboring edge-midpoints. This refinement rule yields a C1 surface and the surface has a piecewise quadratic parametrozation except at a finite number of isolated points. We analyze and improve the construction.

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The simplest subdivision scheme for smoothing polyhedra

A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented The main challenge here is the verification of injectivity of the characteristic map The tools are sufficiently versatile and easy to wield to allow, as an application, a full analysis of algorithms generalizing biquadratic and bicubic B-spline subdivision In the case of generalized biquadratic subdivision the analysis yields a hitherto unknown sharp bound strictly less than 1 on the second largest eigenvalue of any smoothly converging subdivision

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Analysis of Algorithms Generalizing B-Spline Subdivision

The interpolation of a mesh of curves by a smooth regularly parametrized surface with one polynomial piece per facet is studied. Not every mesh with a well-defined tangent plane at the mesh points has such an interpolant: the curvature of mesh curves emanating from mesh points with an even number of neighbors must satisfy an additional “vertex enclosure constraint.” The constraint is weaker than previous analyses in the literature suggest and thus leads to more efficient constructions. This is illustrated by an implemented algorithm for the local interpolation of a cubic curve mesh by a piecewise [bi]quarticC
1 surface. The scheme is based on an alternative sufficient constraint that forces the mesh curves to interpolate second-order data at the mesh points. Rational patches, singular parametrizations, and the splitting of patches are interpreted as techniques to enforce the vertex enclosure constraint.

Smooth interpolation of a mesh of curves

By organizing the control mesh of subdivision in texture memory so that irregularities occur strictly inside independently refinable fragment meshes, all major features of subdivision algorithms can be realized in the framework of highly parallel stream processing. Our implementation of Catmull-Clark subdivision as a GPU kernel in programmable graphics hardware can model features like semi-smooth creases and global boundaries; and a simplified version achieves near-realtime depth-five re-evaluation of moderate-sized subdivision meshes. The approach is easily adapted to other refinement patterns, such as Loop, Doo-Sabin or √3 and it allows for postprocessing with additional shaders.

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Jörg Peters

Papers

Curved PN triangles

The simplest subdivision scheme for smoothing polyhedra

Analysis of Algorithms Generalizing B-Spline Subdivision

Smooth interpolation of a mesh of curves

A realtime GPU subdivision kernel