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

This project is concerned with the development of efficient and accurate numerical methods for the solution of the equations of radiative hydrodynamics. A successful conclusion to this project will have an impact on computations in astrophysics, chemically reacting flows, and flows with strong body forces.

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.

Professors Osher, Engquist, and others will continue their individual and joint research on the development analysis and applications of numerical methods for nonlinear partial differential equations. Specific methods to be studied include: essentially nonoscillatory shock capturing techniques, front capturing algorithms, multiresolution and wavelet based methods, numerical homogenization, effective boundary conditions, kinetic models, spectral and viscosity methods, particle methods, and stochastic difference methods. Engineering and physical applications to be studied include: aerodynamical properties of high speed vehicles, hydrodynamic device models, miscible flow in porous media, combustion, reacting gas flows, shock turbulence interactions, microwave scattering, Boltzmann equations, nuclear fusion reactors, and core-annular flows. This broad range of activity has value both in its own right and with regard to extensive applications.

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.

This three-year award supports U.S.-Italy cooperative research between Russel Caflisch, Bjorn Engquist, and Stanley Osher of the University of California at Los Angeles and Giovanni Russo at the University of Aquila, and Maurizio Falcone, University of Rome, `La Sapienza,` in Italy, on analytical and numerical methods for simulation and control of fluids. Their research will focus on five topics -- singularity formation, level set methods for capturing multivalued solutions and for computing unstable fronts, numerical homogenization, bubbly flow, and approximation schemes for Hamilton-Jacobi equations and control applications. The two research groups possess complementary areas of expertise in scientific computation, fluid mechanics, and partial differential equations. The level set and shock capturing methods developed at UCLA will be extended and applied to control problems as formulated in Rome. Particle methods developed in L'Aquila will be applied to vortical flows to look for singularities and validation of the level set method. The credentials of the L'Aquila group in the study of particle methods will contribute to the integration of the expertise of the UCLA group on conservation laws into active CFD codes. The cooperation of these two groups should lead to an expansion of the understanding and the numerical methodology in several important classes of problems in fluid dynamics.

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.

0074152
Caflisch

This project will apply mathematical modeling and computational
simulation to epitaxial growth of thin films, in particular semiconductor
materials grown by MBE (molecular beam epitaxy). The research topics
are characterized by complex geometry and a hierarchy of length scales,
requiring significantly new mathematical and numerical approaches.
Specific topics include strain in thin films, accelerated numerical methods,
coarse-graining and the dynamics of interacting defect lines.

This project will apply mathematical modeling and computational
simulation to material science, in particular to semiconductor materials
grown by MBE (molecular beam epitaxy). Although MBE is just one of
many possible growth techniques, it is the growth method for many of the
most demanding applications, such as high-performance, low-power
systems for wireless communications. Modeling and simulation are not
generally well developed for MBE growth, and current computational
methods are not sufficient to address many of the important problems of
epitaxy. The goal is to develop new mathematical models and
computational methods that will significantly advance the state of the art.
This project is multi-disciplinary, involving mathematics (both applied
and core) and materials science, as well as participation from both industry
and university. The distinguishing feature of this collaboration is that it
combines expertise in modeling, simulation and experimentation in one
cohesive group.

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.

This project seeks to develop a comprehensive research and education program in the area of computational
methods and simulations of physical systems described by high order Partial Differential Equations (PDEs). The program will unify algorithmic, visualization, theoretical, and experimental efforts as well as address applications in areas of science and technology, including computer graphics, image processing, biology, and fluids.
Intellectual merit of the proposed activity
This project advances knowledge in the area of high order PDEs, with particular emphasis on curved surface data, and produces enabling technology to address fundamental problems in biology, image processing, computer graphics, and fluids in general. The novel science is in the computational
techniques, experimental research, and diverse applications addressed by a multi-disciplinary team. This
project brings together the five fields of computer science, applied mathematics, mechanical engineering, physics, and electrical and computer engineering.
Broader impacts of the proposed activity
With the increasing interest in high order PDEs, the computational tools and experience resulting from this project impact beyond the particular applications in this proposal. Students will receive unusually broad
interdisciplinary training and the workshop planned further brings experts from different fields together. New public domain software incorporating the developed algorithms enables researchers from different fields using higher order PDEs to perform state-of-the-art numerical simulations and graphics rendering of their application of interest.
Educational initiatives of this research program include: (1) new interdisciplinary training of graduate students and postdocs through co-mentoring by PIs in different fields; (2) new interdisciplinary courses in computer graphics, numerical analysis, and modeling/simulation of physical phenomena described by higher order PDEs; (3) a workshop bringing together for the first time diverse scientific researchers using high order PDEs.

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.

AST-0442015
Green

The emergence of powerful new mathematical techniques motivates the need for training a new generation in the many challenges relevant to national security. This award supports a three-week summer school to be held in 2005 at the Institute for Pure and Applied Mathematics (IPAM) at the University of California-Los Angeles. Leading experts will lecture on the techniques of interpreting data from images, high dimensional geometric structures, and graphs, and approximately half of the program will be devoted to high dimensional data analysis. Main themes will include graph mining, relational data mining, and social networks analysis. IPAM, a national institute with an intrinsically interdisciplinary mission to connect mathematicians, scientists, and engineers, is well suited to run such a program, designed to integrate cutting-edge research with workforce development. With roughly 200 participants each week, including students, postdocs, faculty, industry, and intelligence community staff, there will be a broad range of impacts, including training in the mathematical challenges facing the intelligence community, and the acquisition of important professional contacts.

This award is supported jointly by the NSF and the Intelligence Community. The Approaches to Combat Terrorism Program in the Directorate for Mathematical and Physical Sciences supports new concepts in basic research and workforce development with the potential to contribute to national security.

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 proposed activity is targeted to present a very general regularization framework. It enhances the processing of sets of signals and images using variational techniques. The key idea is to exploit information from the entire set for the regularization of each image. This is done via a non-parametric variational approach.

This framework can improve image and signal processing techniques for many types of data-sets.
Sets which contain many repetitions of similar objects or patterns are expected to gain considerable performance. Such data-sets are common in many diverse fields. The new method is anticipated to have a significant contribution in the analysis of medical images, the enhancement of aerial and satellite imagery, and in processing biological signals and genetic data.

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.

This proposal introduces a class of novel inverse problems with applications to, and motivated by anti-terrorism efforts, such as surveillance and discovery of harmful comtamination sources in unknown battle fields as well as urban regions. Unlike the typical settings of a large class of inverse problems, the research involves inverting Radon transforms from very sparse samples and constraints involving parttial differential equations. These considerations present interesting challenges in both mathematical analysis and modeling as well as in the design and implementation of appropriate computational methods. In addition, this proposal introduces novel strategies which greatly reduce the complexity for the inversion.
State-of-the-art numerical techniques that have been in development by the PI and his collaborators, such as the use of Bregman iteration in imaging and compressed sensing and inverse problem applications will be central in meeting these challenges.


This research has immediate and direct implications for anti-
terrorism efforts, such as surveillance and discovery of harmful
contamination sources in unknown battlefields as well as urban regions.
A desired capability is to reconstruct and predict the whereabouts and the
extent of an offending chemical and/or biological cloud from passive,
remote measurements from an array of sensors. A very limited number
of stationary or moving sensors receive and record infrared radiation
from the scene containing the cloud (plume) in addition to the radiation
from other elements in the scene, such as the background and intervening
atmosphere. The sensors are assumed to be able to resolve the spectrum of
the receivedtotal radiation and the spectral signatures of chemicals of
interest may be known. This research will help to move the sensors to
optimal locations, to detect the locations and contents of thesources, to
predict the plumes' behavior and ultimately to minimize the damage caused
by such events.

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 thrust of this Computational and Applied Mathematics program is engaging students starting and finishing the critical transition point from undergraduate to PhD student in high quality university level research. Students experience both the development of independent research projects and the milestones needed to get admitted to and succeed in a top PhD program. They participate in summer research modules on topics such as crime modeling, fluid dynamics experiments and modeling, robotics and control, medical imaging, cancer stem cells, bone growth, remote sensing applications, alcohol biosensors, photovoltaic cells, and algorithm design for microscopy. The program involves faculty from Mathematics in collaboration with faculty in Medicine, Anthropology, Engineering, Chemistry, and other disciplines. The project includes a training program for postdocs and junior faculty to learn how to involve pre-PhD students in publication-level research. The training program is based at UCLA and includes undergraduate and masters student participation from nearby colleges and universities.

The program goal is to directly address diversity and access to top level PhD programs in computational and applied mathematics. The proposed program provides
(a) summer fellowships for undergraduates to participate in research in computational and applied mathematics; (b) summer research fellowships for masters students from non-PhD granting institutions; (b) a summer traineeships for faculty from non-PhD granting institutions to gain experience supervising undergraduate research and to collaborate with research faculty based at UCLA; (c) first year PhD fellowships to recruit and train a more diverse group of PhD students (d) postdoctoral traineeships for recent PhDs with an interesting in mentoring undergraduates on research problems;
(e) summer research mentorships for postdocs to work with younger students on research modules.

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.

The US continued achievement at the forefront of science and technology requires a significant investment in new research in information technology to tackle the most challenging problems created by the vast data footprint created by digital recording of human activity. This project develops novel models and methods for forecasting human activity in time and space using sparse, heterogeneous data. The goals are very general and are focused on predicting and filling in missing data. An example of the type of data this project addresses would be a year's worth of geotagged Twitter data from a major city along with other informative geospatial information from that region. This project combines expertise of senior scientists in both Mathematics and Anthropology. The project develops analytical tools for understanding a diverse array of cyber-geospatial-temporal datasets. While focused on basic research, the project has tremendous potential to impact national security. This three-year project trains postdocs, graduate students, and undergraduate researchers. The mentees will be trained in research, in presentation of their work in written and spoken formats, with an emphasis on refereed journal publications and conference presentations. They will also be connected to future employers and will be given career advice throughout the length of their training.

The project focuses on information technology at the interface between large-scale cultural, social and behavioral processes and the situational conditions that lead to the expression of specific behaviors. This work extends a general conceptualization of text-based topic modeling to handle diverse collections of data types. The project develops methods to detect situational probabilistic effects through spatially-explicit topic modeling. One goal is to organize situational effects into different categories: (a) relatively stationary (e.g., the spatially discrete, but temporally stable role that the physical airport plays in driving airport related topics), (b) intermittent (e.g., discrete holidays) and (c) ephemeral (e.g., Foursquare). Another goal is temporal forecasting while a third goal is filling in missing information from a latent space. The research approach focuses on algorithms that are flexible enough to extend to a variety of datasets. The work interweaves several very useful models and algorithms for large data including self-exciting point process models for temporal information, soft topic modeling such as nonnegative matrix factorization and latent Dirichlet allocation for linear mixture models of data, hard clustering methods built around total variation minimization on graphs and graph Laplacians, and data fusion methods to combine these ideas in which latent space information is studied for forecasting and filling in missing information.

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.