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.

Treatment of supersonic flows with embedded subsonic regions

Abstract: A nonlinear method based on the full potential equation in conservation form, cast in an arbitrary coordinate system, has been developed to treat predominantly supersonic flows with embedded subsonic regions. This type of flow field occurs frequently near the fuselage/canopy junction area and wing leading-edge regions for a moderately swept fighter configuration. The method uses the theory of characteristics to accurately monitor the type-dependent flowfield. A conservative switching scheme is developed to handle the transition from the supersonic marching algorithm to a subsonic relaxation procedure, and vice versa. An implicit approximate factorization scheme is employed to solve the finite differenced equation. Results are shown for a few configurations, including a wing/body/wake realistic fighter model having embedded subsonic regions.

FourierFormer: Transformer Meets Generalized Fourier Integral Theorem

Abstract: Multi-head attention empowers the recent success of transformers, the state-of-the- 1 art models that have achieved remarkable success in sequence modeling and beyond. 2 These attention mechanisms compute the pairwise dot products between the queries 3 and keys, which results from the use of unnormalized Gaussian kernels with the 4 assumption that the queries follow a mixture of Gaussian distribution. There is no 5 guarantee that this assumption is valid in practice. In response, we first interpret 6 attention in transformers as a nonparametric kernel regression. We then propose 7 the FourierFormer, a new class of transformers in which the dot-product kernels 8 are replaced by the novel generalized Fourier integral kernels. Different from the 9 dot-product kernels, where we need to choose a good covariance matrix to capture 10 the dependency of the features of data, the generalized Fourier integral kernels can 11 automatically capture such dependency and remove the need to tune the covariance 12 matrix. We theoretically prove that our proposed Fourier integral kernels can effi- 13 ciently approximate any key and query distributions. Compared to the conventional 14 transformers with dot-product attention, FourierFormers attain better accuracy 15 and reduce the redundancy between attention heads. We empirically corroborate 16 the advantages of FourierFormers over the baseline transformers in a variety of 17 practical applications including language modeling and image classification. 18

Computational Methods for First-Order Nonlocal Mean Field Games with Applications

Abstract: We introduce a novel framework to model and solve first-order mean field game systems with nonlocal interactions, extending the results in [L. Nurbekyan and J. Saude, Port. Math., 75 (2018), pp. 36...

An L1 Penalty Method for General Obstacle Problems

Abstract: We construct an efficient numerical scheme for solving obstacle problems in divergence form. The numerical method is based on a reformulation of the obstacle in terms of an L1-like penalty on the variational problem. The reformulation is an exact regularizer in the sense that for large (but finite) penalty parameter, we recover the exact solution. Our formulation is applied to classical elliptic obstacle problems as well as some related free boundary problems, for example the two-phase membrane problem and the Hele-Shaw model. One advantage of the proposed method is that the free boundary inherent in the obstacle problem arises naturally in our energy minimization without any need for problem specific or complicated discretization. In addition, our scheme also works for nonlinear variational inequalities arising from convex minimization problems.

An ill-posed problem for a strictly hyperbolic equation in two unknowns near a corner

Abstract: In an earlier note [4] we gave a simple example of an ill-posed problem for a system of hyperbolic equations in a region whose boundary has a corner. The system was diagonal with coupling only at the boundary. Earlier we derived necessary and sufficient conditions for well-posedness [2] for a wide class of constant coefficient hyperbolic systems in such regions. In [3] we examined in some detail the phenomena which occur when these conditions are violated. The fundamental work for hyperbolic problems in regions with smooth boundaries was done by Kreiss [1]. It was pointed out by Sarason and Smoller [5] that the work of Strang [6] for the half-space problem implies that the corner problem is well posed for a strictly hyperbolic system in two unknowns iff the corresponding half-space problems are well posed. They constructed, using geometrical optics, a four dependent variable ill-posed example, where the half-space extensions were well posed. In all the above-mentioned work, the boundary conditions imposed were local, i.e., of the form Bu—f at ^ = 0 , where B is a matrix and u is the unknown vector on the boundary. We have noticed that much of the theory can be extended to nonlocal pseudo-differential boundary conditions. In particular, conditions of the form

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.