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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.


Papers
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Journal ArticleDOI
TL;DR: An explicit time differencing technique is introduced to approximate nonlinear conservation laws and convergence to the correct physical solution is proven given only a local CFL condition.
Abstract: Numerical approximations to the initial value problem for nonlinear systems of conservation laws are considered The considered system is said to be hyperbolic when all eigenvalues of every real linear combination of the Jacobian matrices are real Solutions may develop discontinuities in finite time, even when the initial data are smooth In the investigation, explicit finite difference methods which use locally varying time grids are considered The global CFL restriction is replaced by a local restriction The numerical flux function is studied from a finite volume viewpoint, and a differencing technique is developed at interface points between regions of distinct time increments

190 citations

Journal ArticleDOI
TL;DR: This paper applies a recently developed second order accurate symmetric discretization of the Poisson equation to the simulation of the dendritic crystallization of a pure melt and finds that the d endrite tip velocity and tip shapes are in excellent agreement with solvability theory.
Abstract: In this paper, we present a level set approach for the modeling of dendritic solidification. These simulations exploit a recently developed second order accurate symmetric discretization of the Poisson equation, see l12r. Numerical results indicate that this method can be used successfully on complex interfacial shapes and can simulate many of the physical features of dendritic solidification. We apply this algorithm to the simulation of the dendritic crystallization of a pure melt and find that the dendrite tip velocity and tip shapes are in excellent agreement with solvability theory. Numerical results are presented in both two and three spatial dimensions.

190 citations

Journal ArticleDOI
TL;DR: This work investigates the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis and finds that this method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms.
Abstract: We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms.

189 citations

Journal ArticleDOI
TL;DR: The procedure is used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced, which retain the desirable properties of simplicity and high resolution and yield Riemann-solver-free recipes which avoid dimensional splitting.
Abstract: We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408--463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892--1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.

186 citations

Journal ArticleDOI
TL;DR: This work proposes a fast algorithm for solving the Basis Pursuit problem, min u, and claims that in combination with a Bregman iterative method, this algorithm will achieve a solution with speed and accuracy competitive with some of the leading methods for the basis pursuit problem.
Abstract: We propose a fast algorithm for solving the Basis Pursuit problem, minu $\{|u|_1\: \Au=f\}$, which has application to compressed sensing We design an efficient method for solving the related unconstrained problem minu $E(u) = |u|_1 + \lambda \||Au-f\||^2_2$ based on a greedy coordinate descent method We claim that in combination with a Bregman iterative method, our algorithm will achieve a solution with speed and accuracy competitive with some of the leading methods for the basis pursuit problem

185 citations


Cited by
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Proceedings ArticleDOI
07 Jun 2015
TL;DR: Inception as mentioned in this paper is a deep convolutional neural network architecture that achieves the new state of the art for classification and detection in the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC14).
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.

40,257 citations

Journal ArticleDOI

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08 Dec 2001-BMJ
TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.
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 …

33,785 citations

01 May 1993
TL;DR: 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.
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.

29,323 citations

Book
23 May 2011
TL;DR: It is argued 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.
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.

17,433 citations