Stanley Osher

Bio: Stanley Osher is an academic researcher from University of California, Los Angeles. The author has contributed to research in topics: Level set method & Hyperbolic partial differential equation. The author has an hindex of 114, co-authored 510 publications receiving 104028 citations. Previous affiliations of Stanley Osher include University of Minnesota & University of Innsbruck.

Total Variation Based Image Cartoon-Texture Decomposition

Abstract: : This paper studies algorithms for decomposing a real image into the sum of cartoon and texture based on total variation minimization and second-order cone programming (SOCP). The cartoon is represented as a function of bounded variation while texture (and noise) is represented by elements in the space of oscillating functions, as proposed by Yves Meyer. Our approach gives more accurate results than those obtained previously by Vese-Osher's approximation to Meyer's model, which we also formulate and solve as an SOCP. The model of minimizing total variation with an L1-norm fidelity term is also considered and empirically shown to achieve even better results when there is no noise. This model is analyzed and shown to be able to select features of an image according to their scales.

Regularization of ill-posed problems via the level set approach

Abstract: We introduce a new formulation for the motion of curves in R2 (easily extendable to the motion of surfaces in R3), when the original motion generally corresponds to an ill-posed problem such as the Cauchy--Riemann equations. This is, in part, a generalization of our earlier work in [6], where we applied similar ideas to compute flows with highly concentrated vorticity, such as vortex sheets or dipoles, for incompressible Euler equations. Our new formulation involves extending the level set method of [12] to problems in which the normal velocity is not intrinsic. We obtain a coupled system of two equations, one of which is a level surface equation. This yields a fixed-grid, Eulerian method which regularizes the ill-posed problem in a topological fashion. We also present an analysis of curvature regularizations and some other theoretical justification. Finally, we present numerical results showing the stability properties of our approach and the novel nature ofthe regularization, including the development o...

Level set methods in image science

Abstract: In this article, we discuss the question "what level set methods can do for image science". We examine the presence of these and related methods in image science and introduce some relevant level set techniques that are potentially useful for this class of applications. We will show that image science demands multidisciplinary knowledge and flexible but still robust methods, and that is why the level set method has become a thriving technique in this field.

Improved finite-difference schemes for transonic potential flow calculations

Abstract: Engquist and Osher (1980) have introduced a finite difference scheme for solving the transonic small disturbance equation, taking into account cases in which only compression shocks are admitted. Osher et al. (1983) studied a class of schemes for the full potential equation. It is proved that these schemes satisfy a new discrete 'entropy inequality' which rules out expansion shocks. However, the conducted analysis is restricted to steady two-dimensional flows. The present investigation is concerned with the adoption of a heuristic approach. The full potential equation in conservation form is solved with the aid of a modified artificial density method, based on flux biasing. It is shown that, with the current scheme, expansion shocks are not possible.

Going deeper with convolutions

Abstract: We propose a deep convolutional neural network architecture codenamed Inception that achieves the new state of the art for classification and detection in the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC14). The main hallmark of this architecture is the improved utilization of the computing resources inside the network. By a carefully crafted design, we increased the depth and width of the network while keeping the computational budget constant. To optimize quality, the architectural decisions were based on the Hebbian principle and the intuition of multi-scale processing. One particular incarnation used in our submission for ILSVRC14 is called GoogLeNet, a 22 layers deep network, the quality of which is assessed in the context of classification and detection.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Fast parallel algorithms for short-range molecular dynamics

Abstract: Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Distributed Optimization and Statistical Learning Via the Alternating Direction Method of Multipliers

Abstract: Many problems of recent interest in statistics and machine learning can be posed in the framework of convex optimization. Due to the explosion in size and complexity of modern datasets, it is increasingly important to be able to solve problems with a very large number of features or training examples. As a result, both the decentralized collection or storage of these datasets as well as accompanying distributed solution methods are either necessary or at least highly desirable. In this review, we argue that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas. The method was developed in the 1970s, with roots in the 1950s, and is equivalent or closely related to many other algorithms, such as dual decomposition, the method of multipliers, Douglas–Rachford splitting, Spingarn's method of partial inverses, Dykstra's alternating projections, Bregman iterative algorithms for l1 problems, proximal methods, and others. After briefly surveying the theory and history of the algorithm, we discuss applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others. We also discuss general distributed optimization, extensions to the nonconvex setting, and efficient implementation, including some details on distributed MPI and Hadoop MapReduce implementations.