Stanley J. Osher, Ph.D. - US grants

The objectives of this project are the following: (1) To investigate transient phenomena which lead to incomplete combustion of hazardous wastes. (2) To assess the relationship between incinerator size (cost) and reliability, and to identify the chemical and fluid mechanical phenomena which may limit the ultimate degree of waste destruction in small systems. (3) To evaluate the use of high volumetric heat release rate aerospace technology in the development of small incinerators, including how combustion acoustics might be used to improve and/or monitor incinerator performance. The objectives above are to be addressed by means of an integrated program of experiments and numerical simulations. The experiments are designed to follow the instantaneous velocity, temperature and hydroxyl radical concentration fields in the combusting fluid, from which the three T's of incineration (residence time, temperature, and turbulence intensity or mixing) can be calculated. Experimental results are to be used in the development of a 2-D transient combustion model based on the numerical solution of the conservative form of the conservation equations. Realistic chemistry and transport, capable of accurately representing the interaction of the flame with large scale vortical structures found in the combustor, are to be used. The results are expected to contribute to the development of small, reliable incinerators which can be used effectively on the site of small waste generators.

The principal investigators and their collaborators study the development, analysis, and applications of numerical methods for nonlinear partial differential equations. The three main topics addressed are: (1) high resolution methods, including shock capturing, front capturing, numerical homogenization, and particle methods, (2) multiscale analysis applied to these and other related problems, and (3) fast methods for linear evolution equations based on multiscale analysis and high frequency asymptotics. Engineering and physical applications studied include combustion, fluid dynamics, crystal growth, Stefan problems, microwave scattering, and reacting gases. These innovative numerical methods are used to simulate real world problems in the areas of aeronautics, oil recovery, materials science, and environmental science. Transition to industrial and military application is made in the above areas, as well as in low observables, semiconductor device modelling, and shape recognition. These methods are applicable wherever the phenomena to be studied have sharp changes of state variables or are beyond the resolving capabilities of standard methods.

9706827 Stanley Osher This research is concerned with the accurate and efficient computation of "irregular" solutions of partial differential equations (PDEs). This includes solutions with discontinuities, singularities, fine scale structure, or persistent oscillations. The main topics include (1) kinetic formulations of nonlinear PDE's and applications; (2) interface capturing through the level set method; (3) discontinuity capturing based on ideas developed for the numerical solution of conservation laws; and (4) numerical and analytical study of oscillations and critical threshold phenomena. The proposed research will impact numerous areas of science and technology. With the advent of modern computers, formerly intractable problems can be solved accurately. This, of course, requires accurate and convergent algorithms for these difficult nonlinear and computationally intense problems. The algorithms formerly developed by this group are already in wide use throughout the country in national laboratories and industry. The proposed methods to be developed here will be useful in a host of applications including combustion, oil recovery, crystal growth, electromagnetic and acoustic scattering, thin film semiconductor growth, aircraft design, to name just a few.

NSF Proposal: DMS-0074735

"Advances in Level Set and Related Methods: New Technologies and Applications"

Principal Investigator: Stanley J. Osher


ABSTRACT

The level set method devised by Osher and Sethian in 1988 has proven to be phenomenally successful as a numerical and theoretical device for representing and analyzing the motion of curves in R^2 and surfaces in R^3. A level set calculus has been developed, and recent extensions include the ghost fluid method, convolution generated motion, dynamic surface extension, the variational level set approach, and the motion of higher codimensional objects. Recent applications include multiphase fluid dynamics, the island dynamics model for epitaxial growth, level set based interpolation of unorganized points, and fast methods in image restoration. This work was partially supported by our previous NSF grants. The goal of this proposed research is to extend the technology and the range of applications through the following two projects:
(1) Convolution generated motion for filaments.
(2) Fast algorithms for steady state geometric Hamilton-Jacobi equations and
the induced motion of fronts.

The level set method is rapidly becoming the method of choice to simulate on the computer a host of important physical, biological, materials science, image processing, computer vision, electromagnetic and other real world problems. In particular areas of nanotechnology will also be impacted. Improvements of the numerical methods used to simulate these phenomena will ultimately be crucial in the design of computer chips, analysis of explosions, recognition of objects and many other areas of modern technology. This proposal addresses further improvements of the level set and related methods.

Stanley J. Osher and Luminita A. Vese
The investigators together with junior faculty, postdoctoral
fellows, and students develop novel and efficient computational
techniques for partial differential equations of fourth order
arising in imaging science, including medical imaging, image
processing, computer vision, and graphics, as well as material
science. They begin with analysis done on their previous TV model
by Yves Meyer. This gives a unique blend of nonlinear partial
differential equation and functional analysis as well as new and
very useful models for image decomposition. This leads the
investigators to the analysis of more general fourth order
equations used in imaging science, with geometric applications
and interpretations, as well as new efficient computational and
numerical methods to approximate these and other related partial
differential equations. Problems considered include image
decomposition into cartoon plus texture, Wulff evolution of
shapes in crystal growth, image disocclusion by the Euler
elastica model, image restoration via geometry of level sets, as
well as reconstruction of surfaces from unorganized data points
(such as LIDAR, LADAR, 3D cloud or geoscience data) in the form
of patches belonging to the same unknown surface.
This investigation dramatically advances the
state-of-the-art in material science (e.g., microchip design),
image analysis, computer vision, and graphics. These fields are
of great strategic value in the US information technology
industry, in nanoscience, in homeland security, and in medical
imaging. The level set method and applications, developed by the
investigators and collaborators, has impacted numerous areas of
technology -- see, e.g., the Google website for close to 5,000
hits on "Level Set Methods" with a spectacular range of
applications from Hollywood graphics to underwater explosions to
control of passenger aircraft to developing new microchip designs
and beyond. The investigators' new models reduce the burden of
laborious human operations for the processing of large-scale data
sets, enhance the efficiency of domain scientists as well as
ordinary users, facilitate modeling and rendering tasks, and
streamline the pipeline of information and materials processing.

Abstract

Award: DMS-0439872
Principal Investigator: Mark L. Green, Stanley Osher

The Institute for Pure and Applied Mathematics (IPAM) is a national
research institute that focuses on fostering interactions that bring
together researchers in the mathematical sciences with scientists and
engineers. The Institute operates without permanent faculty or
research staff. IPAM conducts semester-length programs, short
workshops, a summer research program for undergraduates, graduate
summer schools, outreach programs, and workshops to develop and
enhance the careers of researchers from groups underrepresented in the
mathematical sciences.

The IPAM web site at http://www.ipam.ucla.edu/ describes upcoming and
past programs, solicits ideas for future programs, and offers
application forms for Institute activities.

The investigator and his colleagues propose a new paradigm-shifting approach towards high resolution and high contrast imaging, which combines revolutions in magnetic resonance imaging (MRI) and optical imaging with equally cutting edge mathematical developments.
The approach uses non-linear feedback between the detector and the sample, so that the measured field is fed back to the MRI magnet. The unstable feedback increases the dynamical contrast between normal and cancerous cells. This highly nontraditional approach will be complemented by incorporating compressed sensing, a data analysis technique where mathematical algorithms are used to extract specific features and images from a relatively small number of measurements.
Finally, multiple sources and detectors, which coupled with compressed sensing and feedback imaging can collect data in parallel will be implemented together, and modern filter-diagonalization techniques will be used to synthesize the data leading to faster images.

Overall, the research and development that the principal investigator and his colleagues propose will revolutionize MRI. By applying nontraditional measurement and imaging technique, the contrast between tumors and normal areas will be increased many fold. The increase will be based on the same physical phenomena, chaos, that is used to by birds and jet fighters to quickly switch their direction. The revolutionary paradigm will will eventually make it much cheaper and faster to do an MRI scan, thereby having enormously broad impacts. In 2008, an estimated 1,680,000 people in the U.S. will be diagnosed with cancer, and approximately 670,000 people will die. Between 10%-35% could have been saved with earlier detection.
This highlights the need for improved early detection methods, which could have saved many patients. A large (2-5 or more) reduction MRI acquisition time, which is not feasible with conventional methods, coupled with the enhanced feature resolution native to the proposed approach, will allow for faster and cheaper cancer screening, which is crucial to improved early detection and thus reducing deaths due to cancer. Besides cancer detection, there are numerous other imaging applications that stand to benefit from significantly decreased scan time and cost, such as industrial sensing (for example uniformity of fruits in agriculture) and homeland security applications, including highly sensitive detection of concealed materials.

Abstract
Award DMS 1440415, Principal Investigator Russell E. Caflisch


The mission of the Institute for Pure and Applied Mathematics (IPAM) is to foster the interaction of mathematics with a broad range of science and technology, to build new interdisciplinary research communities, to promote mathematical innovation, and to engage and transform the world through mathematics. Mathematics is an essential ingredient in much of current technology, including internet search engines, medical imaging such as magnetic resonance imaging and computed tomography, voice recognition systems, DNA sequencing methods, and many others. Future developments -- such as smart electrical grids, personalized medicine, and new forms of social networking -- will require further mathematical innovation. IPAM's overall goal is to foster the interaction of mathematicians with doctors, engineers, physical scientists, social scientists, and humanists to enable such future technological and social progress.

IPAM fulfills its mission through workshops and long programs that connect mathematics and other disciplines or multiple areas of mathematics. These activities bring in thousands of visitors annually from academia, government, and industry. IPAM also has programs that encourage the inclusion of women and members of minorities underrepresented in the mathematics community, that serve specific needs of government agencies, and that inform the public about the excitement of modern mathematics and the important contributions that have come to society through mathematics.

Through these activities, IPAM serves the national interest. IPAM promotes the progress of science by stimulating the mathematical developments that are needed for this progress; advances the national health, prosperity, and welfare through programs that address challenges such as disease modeling, systemic financial risk assessment, and traffic control; and helps to secure the national defense through programs that address defense requirements such as advanced radar, computer-aided decision making, and new communication modalities.

The investigators intend to generate new and effective mathematical algorithms and methodologies in sensor systems for the detection of chemical and biological materials. Next, they intend to transfer this technology directly to those working towards reducing the threat to the homeland of biological and chemical attack. The new techniques they will use come primarily from information science, image science and physics, involving harmonic analysis, machine learning, optimization and partial differential equations. In particular they intend to provide useful algorithms for multi-component aerosol unmixing for active sensing using LiDAR and for mixtures of vapors in passive sensing. They will use ideas and algorithms recently developed, broadly speaking, from compressive sensing and L1 related optimization which were applied to hyperspectral imaging (recently used by Navy SEALS in the Bin Laden take down), unmixing, template matching, anomaly detection, clustering, change detection and endmember computation. They will improve relevant classical learning techniques, such as support vector machine, using their optimization techniques. They will also use ideas from machine learning with nonlocal means with prior information, in order to segment and identify objects in data collected from all sorts of sensors. Finally, they will factor in physics, such as plume dissipation, as part of the prior information needed to do spatial segmentation and identification.

The US government has been developing laser-based sensors for locating and classifying aerosols in the atmosphere at safe standoff ranges for more than a decade. There is a need to distinguish aerosols of biological origin from indifferent materials such as smoke and dust. Often, mixtures of aerosols are present and it is important to decide whether a threat exists. This project is intended to resolve data containing such a mixture into their separate components. Some success has already been obtained here by the investigators. This is an example of what this work concerns. A chemical and/or biological contamination might occur on the ground or in the air. The problem is to determine the presence of and concentration of chemical and biological threats and to track the dynamics of the cloud. The research done here is relevant to all the sensor modalities used in this type of threat detection. These include state-of-the-art LiDAR sensors, infrared radiometry and hyperpectral spensors. Plume tracking through the atmosphere is particularly important in a potential threat situation. The type of work proposed here is basic to our nation's security, given the threat posed by chemical and biological WMD's.

Graph-structured data is ubiquitous in scientific and artificial intelligence applications, for instance, particle physics, computational chemistry, drug discovery, neural science, recommender systems, robotics, social networks, and knowledge graphs. Graph neural networks (GNNs) have achieved tremendous success in a broad class of graph learning tasks, including graph node classification, graph edge prediction, and graph generation. Nevertheless, there are several bottlenecks of GNNs: 1) In contrast to many deep networks such as convolutional neural networks, it has been noticed that increasing the depth of GNNs results in a severe accuracy degradation, which has been interpreted as over-smoothing in the machine learning community. 2) The performance of GNNs relies heavily on a sufficient number of labeled graph nodes; the prediction of GNNs will become significantly less reliable when less labeled data is available. This research aims to address these challenges by developing new mathematical understanding of GNNs and theoretically-principled algorithms for graph deep learning with less training data. The project will train graduate students and postdoctoral associates through involvement in the research. The project will also integrate the research into teaching to advance data science education.<br/><br/>This project aims to develop next-generation continuous-depth GNNs leveraging computational mathematics tools and insights and to advance data-driven scientific simulation using the new GNNs. This project has three interconnected thrusts that revolve around pushing the envelope of theory and practice in graph deep learning with limited supervision using PDE and harmonic analysis tools: 1) developing a new generation of diffusion-based GNNs that are certifiable to learning with deep architectures and less training data; 2) developing a new efficient attention-based approach for learning graph structures from the underlying data accompanied by uncertainty quantification; and 3) application validation in learning-assisted scientific simulation and multi-modal learning and software development.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.