The Mathematics Of Nonlinear Programming (Undergraduate Texts In Mathematics) [CRACKED]
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This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, nonlinear optimization, and Fourier transforms. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.
The course is intended for Courant Institute graduate students who are either mathematics students on the applied half of the spectrum or computer science students with an interest in numerical computation. ODEs, the starting point of analysis and dynamics, are one of the two foundation stones of applied mathematics (the other is linear algebra). This course will deepen your appreciation of them fundamentally.
This course is intended to provide a practical introduction to problem solving. Topics covered include: the notion of well-conditioned and poorly conditioned problems, with examples drawn from linear algebra; the concepts of forward and backward stability of an algorithm, with examples drawn from floating point arithmetic and linear-algebra; basic techniques for the numerical solution of linear and nonlinear equations, and for numerical optimization, with examples taken from linear algebra and linear programming; principles of numerical interpolation, differentiation and integration, with examples such as splines and quadrature schemes; an introduction to numerical methods for solving ordinary differential equations, with examples such as multistep, Runge Kutta and collocation methods, along with a basic introduction of concepts such as convergence and linear stability; An introduction to basic matrix factorizations, such as the SVD; techniques for computing matrix factorizations, with examples such as the QR method for finding eigenvectors; Basic principles of the discrete/fast Fourier transform, with applications to signal processing, data compression and the solution of differential equations.
Linear algebra is two things in one: a general methodology for solving linear systems, and a beautiful abstract structure underlying much of mathematics and the sciences. This course will try to strike a balance between both. We will follow the book of our own Peter Lax, which does a superb job in describing the mathematical structure of linear algebra, and complement it with applications and computing. The most advanced topics include spectral theory, convexity, duality, and various matrix decompositions.
Prerequisites: A familiarity with rigorous mathematics, proof writing, and the epsilon-delta approach to analysis, preferably at the level of MATH-GA 1410, 1420 Introduction to Mathematical Analysis I, II.
This is a first-year course for all incoming PhD and Master students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations
The course should be interesting for graduate students, and postdocs in pure and applied mathematics, physics, engineering, and climate, atmosphere, ocean science interested in turbulent dynamical systems as well as other complex systems.
Turbulent dynamical systems are ubiquitous complex systems in geoscience and engineering and are characterized by a large dimensional phase space and a large dimension of strong instabilities which transfer energy throughout the system. They also occur in neural and material sciences. Key mathematical issues are their basic mathematical structural properties and qualitative features, their statistical prediction and uncertainty quantification (UQ), their data assimilation, and coping with the inevitable model errors that arise in approximating such complex systems. These model errors arise through both the curse of small ensemble size for large systems and the lack of physical understanding. This is a research expository course on the applied mathematics of turbulent dynamical systems through the paradigm of modern applied mathematics involving the blending of rigorous mathematical theory, qualitative and quantitative modelling, and novel numerical procedures driven by the goal of understanding physical phenomena which are of central importance. The contents include the general mathematical framework and theory, instructive qualitative models, and concrete models from climate atmosphere ocean science. New statistical energy principles for general turbulent dynamical systems are discussed with applications, linear statistical response theory combined with information theory to cope with model errors, reduced low order models, and recent mathematical strategies for UQ in turbulent dynamical systems. Also recent mathematical strategies for online data assimilation of turbulent dynamical systems as well as rigorous results are briefly surveyed. Accessible open problems are often mentioned. This research expository book is the first of its kind to discuss these important issues from a modern applied mathematics perspective.
This will be an introduction to entropy and its many roles in different branches of mathematics, especially information theory, probability, ergodic theory and statistical mechanics. The aim is to give a quick overview of many topics, emphasizing a few basic combinatorial problems that they have in common and which are responsible for the ubiquity of entropy.
Mathematical methods for solving decision problems encountered inbusiness situations. Emphasis on problem formulation and application ofspreadsheet-based algorithms for solution. Linear models and linearprogramming. Sensitivity analysis. Network models. Integer and nonlinearprogramming. Decision analysis and value of information. Dynamicanalysis and the principle of optimality. Prerequisite: OMIS 41 or ECON41 and 42. (5 units)
Above: Herb Keller and Richard Tapia (second and third from left) of the CRPCdiscuss career opportunities with participants in the program, "Computers:The Machines, Science, People, and Jobs!" held at the California Instituteof Technology. Graduate and Postgraduate ProgramsComputational Science CurriculaAt Caltech, Syracuse University, and Rice University, the CRPC has helpedto establish ground-breaking computational science programs that emphasizeparallel computing. Syracuse University's School of Computer andInformation Science offers a graduate program of courses in computationalscience. These courses include the study of computational techniques inphysics, biology, geology, mathematics, and engineering. Other coursesfocus on new algorithms, languages, and models in computer science andapplied mathematics. At Rice University, the Computational Science andEngineering graduate degree program offers degrees in computational scienceat both the master's and Ph.D. levels. Students learn methods inparallel-vector processing, scientific visualization, networking, compilertechnology, programming environments, parallel algorithms, numericalmethods, and modeling with an emphasis on a particular area in science orengineering.Danny Sorensen's current research activities involve iterative techniquesfor the solution of very large linear systems of equations and forlarge-scale algebraic eigenvalue problems. The techniques he is mostinterested in are based upon projection methods (such as Lanczos, Arnoldi,and GMRES). He has also worked extensively in the area of nonlinearnumerical optimization. Sorensen was an Assistant Professor of Mathematicsat the University of Kentucky from 1977 to 1980, was a Senior ComputerScientist in the Mathematics and Computer Science Division of ArgonneNational Laboratory from 1980 to 1989, and has been a professor in theComputational And Applied Mathematics Department at Rice University since1989. He was one of the founders of the Advanced Computing ResearchFacility at Argonne National Laboratory, one of the first facilities toprovide public access to a variety of parallel computers.CRPC researchers have also implemented graduate and undergraduate coursesin parallel computing at Argonne, Caltech, Rice University, SyracuseUniversity, and the University of Tennessee. Topics of study includeparallel processing, scientific visualization, networking, compilertechnology, programming environments and templates, numerical methods,structured parallel programming, and modeling with an emphasis onapplications in a particular science or engineering field. The CRPC atArgonne offers semester-long courses using various parallel machinessupported in part by CRPC funding. PCN and Compositional C++ are being usedat Caltech to teach undergraduate courses in programming and algorithms.Short courses have been taught in conjunction with the annual CRPC researchreview, with several SIAM and SUPERCOMPUTING conferences, and at many othersymposia.Courses for Supercomputer Center StaffIn a joint effort with the National Center for Supercomputer Applications,which is an NSF supercomputer center at the University of Illinois, and theUniversity of Illinois at Urbana-Champaign Computer Science Department, theCRPC has designed a two-week course for supercomputer staff members whowill be teaching short courses on parallel computation to users. Thiscourse includes material on parallel computer architectures, parallelprogramming paradigms, languages for parallel programming (including HighPerformance Fortran and HPC++), performance tuning and debugging tools,algorithms and mathematical software, parallel input/output, andvisualization. The course incorporates hands-on experience working withusers to parallelize scientific applications. A key goal of the course isto provide course materials and software technologies that can be used inteaching parallel computation to users. This course, introduced in fall1993, is offered annually and is open to staff members from any state ornational center with a commitment to teaching parallel computing courses toits users.Undergraduate ProgramsUndergraduate ResearchThe CRPC has been very active in its efforts to provide researchopportunities for undergraduates, especially in the summer. Several ofthese programs have emphasized participation by minority and femalestudents. The overall goal of these programs is to attract more students toresearch careers in computational science and engineering. Research Experiences for Undergraduates (REU) - Syracuse University. The REU program introduces promising students to opportunities inhigh-performance computing and provides them with formative researchexperience similar to graduate study. Students work closely with staffresearchers and faculty at the CRPC and Syracuse University, learningproject formulation, methodology, solution, and interpretation of results.The program is wide in its scope, allowing research in areas such asscientific visualization, multimedia technology, image processing,computational fluid dynamics, signal processing, optimization algorithms,computational geometry, computational physics, digital system design, andfinancial modeling. The program accepts majors from all disciplines,including computer science, physics, engineering, mathematics, and finance. Summer Program in Parallel Computing for Minority Undergraduates -California Institute of Technology. Through this program, minorityundergraduates in computer science and mathematics spend two months duringthe summer working with Caltech scientists on research projects in areassuch as parallel programming methods, algorithms, and scientific computing.Students are intimately involved in the day-to-day activities ofleading-edge computational research, using high-performance parallelcomputers. Working within this stimulating research environment, theparticipants are encouraged to continue their education through graduateschool. Spend a Summer with a Scientist - Rice University. The Spend aSummer with a Scientist program at Rice University provides opportunitiesfor talented minority undergraduate students to participate in universityresearch, motivating them to attend graduate school in science,mathematics, or engineering. Participants work with center researchers,faculty, and graduate students from six different departments at Rice, andwith researchers from the Keck Center for Computational Biology, acollaboration between Rice and the Baylor College of Medicine. The majorityof past participants in this program are either currently enrolled ingraduate school or planning to apply. Below: Daniel Bauman, a participant in the "Research Experiences forUndergraduates" (REU) program at Syracuse University, funded by theNational Science Foundation. CRPC summer programs give students valuableresearch experience and motivate them to pursue careers in science andengineering.Undergraduate Courses in Computational ScienceSeveral courses on parallel computing and computational science have beenintroduced at institutions participating in the CRPC. For instance, at RiceUniversity, the "Introduction to Computational Science" course introducesundergraduates to the basic principles of computational science, focusingon vector and parallel computer architectures, parallel numericalalgorithms, scientific visualization, analysis and enhancement ofperformance, and use of programming tools and environments. Studentsreceive hands-on experience with high-performance computers. At Caltech, a concurrent scientific computing course has been offered since1988. Although it is a graduate-level course, it is heavily attended byadvanced undergraduates. The course introduces basic numerical methods forlinear algebra and partial differential equation problems. Through theapplication of stepwise refinement, concurrent implementations areobtained. A consistent methodology is applied to problems ranging fromLU-decomposition fast Fourier transform to multigrid methods and fastparticle methods. Students gain practical experience on concurrentcomputers through an extensive set of homework problems. The lecture notesfor this course are being published by Springer-Verlag. Another novel course at Caltech introduces students to parallel andsequential programming together. The course also integrates algorithms andprograms in Fortran, C, C++, and Ada with performance tuning for parallelmachines and software engineering issues. Course lectures are based onmultimedia text that allows students to study on their own. The text isorganized around programming "archetypes" or "templates" that help studentstake a systematic approach to programming. Each archetype has an abstractcode that can be specialized to get code for a specific application,documentation explaining program correctness, performance analysis onparallel and sequential machines, or recommendations for test suites.Rick Stevens is pursuing the following research activities: evaluatingarchitecture and performance of high-performance computing systems,developing scientific algorithms for use of MPP systems, combining symbolicand numerical methods in scientific applications, improving modeling andCAD systems for molecular nanotechnology, and using virtual reality in thevisualization of scientific data and processes. Most recently he has ledthe joint ANL/IBM activity in parallel computing, which includes theinstallation of a large SP1 system at ANL.Programs for Undergraduate Teaching FacultyThe CRPC has been developing workshops to educate undergraduate teachingfaculty in the fundamentals of computational science, so that these facultycan take these materials back to their home institutions and incorporatethem into courses. The goal is to attract more students to careers incomputational science and engineering. If this effort is successful, itwill ultimately help in providing industry with a well-educated workforcethat has experience with emerging technologies in high-performancecomputing. Argonne National Laboratory Outreach Programs. Argonne NationalLaboratory has two summer research programs for graduate students and aprogram for faculty from underrepresented groups. Argonne has also offeredone-week "immersion workshops" that were general introductions to parallelcomputing, using the computers and parallel programming tools at Argonne.Topics included review of different architectures, performance evaluation,porting of codes, parallel program writing and visualization. Computational Science Undergraduate Programs for MinorityInstitutions. CRPC scientists at Rice University and Argonne NationalLaboratory have been working with the University of Houston, Downtown, todevelop a program aimed at stimulating computational science instruction atminority schools in the south central United States (Arkansas, Louisiana,New Mexico, Oklahoma, and Texas). This program will consist of a program ofinstruction and research experience for undergraduate faculty at minorityinstitutions, coupled with computational support for courses. Students whotake the resulting courses will be strongly encouraged to participate inCRPC summer undergraduate research programs.Pre-college ProgramsAwareness WorkshopsThe CRPC has pioneered a program of awareness workshops for pre-collegeteachers. The first series of such programs was offered by Richard Tapia atRice University during the summers from 1989 to 1992. These workshopsexposed high school, middle school, and elementary school teachers tocareer opportunities for students in computational science. Since theprogram was aimed at teachers from schools with large minority populations,a significant portion of these workshops was devoted to minority issues.These workshops have been so successful that they are now emulated by otherresearch centers and universities. The following workshops at CRPC siteshave similar goals of encouraging minority involvement in science. Minorities Teachers Computational Sciences and Graphics AwarenessProgram - California Institute of Technology. This program brings highschool teachers from Los Angeles and Pasadena-area schools with largeminority enrollments to Caltech for a five-day session that introduces themto the most recent developments and opportunities in the areas ofconcurrent computing and graphics. The information provided from thesesessions has enabled teachers to motivate their students to consideropportunities in science and engineering fields. This program has beensuccessful because it puts information in the hands of those people whohave some of the greatest potential to inspire students: their teachers. Computers: The Machines, Science, People, and Jobs! - CaliforniaInstitute of Technology. This two-day program at Caltech provided 100minority high school students with a stimulating first-hand exposure toproblems and issues in computer science. The program served to demystifyand humanize scientific research professionals through face-to-faceinteraction of the participants with successful minority scientists. Inaddition, the students gained new insights into what career opportunitiesare available to them in science, engineering, and mathematics.Richard Tapia's research interests are primarily in the area ofcomputational optimization theory. Under his leadership, the Rice/CRPCeffort in the area of interior-point methods for linear and nonlinearprogramming has received international visibility. Rice/CRPC educationactivities, also under his leadership, include the Spend a Summer with aScientist program for underrepresented undergraduate and graduateminorities, the Mathematical and Computational Sciences Awareness workshopfor K-12 teachers, and the recruitment and retention of underrepresentedminority students into Rice Ph.D. programs in the computational sciences.All of these efforts have received excellent national recognition andvisibility. He has been named Noah Harding Professor of Computational andApplied Mathematics at Rice, elected to the National Academy of Engineering(the first Mexican-American to receive this honor), given the HispanicEngineering Magazine National Achievement Award for Education, given theGeorge R. Brown Award for Superior Teaching, and named one of the 20 mostinfluential leaders in minority math education by the National ResearchCouncil.Curriculum DevelopmentThe CRPC is collaborating with John Muir High School of Pasadena,California, the Art Center College of Design, Jet Propulsion Laboratory,and Pasadena City College to develop a curriculum for ninth graders indanger of dropping out of school. This curriculum will use design problemsas tools to introduce these students to basic mathematics and to developtheir social and verbal skills. The project team is assisting in thedevelopment of specific design problems and contests and arranging campusvisits for selected teams of students. They are also helping the highschool in setting up a computer laboratory. This effort is led by Eric Vande Velde of Caltech. Future ProgramsIn the future, the Center for Research on Parallel Computation plans toinitiate two other programs aimed at K-12 education: Geoffrey Fox at Syracuse plans to implement a program to providescientific visualizations to schools at all levels. These visualizationscould be run on workstations to demonstrate complex scientific phenomena orused for experimentation by students. In the near future, the CRPC will expand its educational workshopsand summer research programs to include high school science faculty whohave an interest in teaching aspects of computational science to theirstudents.If successful, these programs will have a significant impact on thepre-college teaching of science in America.Sites & Affiliations | Leadership | Research & Applications | Major Accomplishments | FAQ | Search | Knowledge & Technology Transfer | Calendar of Events | Education & Outreach | Media Resources | Technical Reports & Publications | Parallel Computing Research Quarterly Newsletter | News Archives | Contact Information 2b1af7f3a8