Computational Science and Engineering PhD. Doctoral program offered jointly with eight participating departments, focusing on the development of new computational methods relevant to science and engineering disciplines. Students specialize in a computation-related field of their choice through coursework and a doctoral thesis.
The specialization in computational science and engineering is highlighted by specially crafted thesis fields. Course P builds on the Bachelor of Science in Computation and Cognition to provide additional depth in the subject areas through advanced coursework and a substantial thesis. Master of Engineering program Course 6-P provides the depth of knowledge and the skills needed for advanced graduate study and for professional work, as well as the breadth and perspective essential for engineering leadership.
Emphasizes expressing all hardware designs in a high-level hardware language and synthesizing the designs. Introduction to mathematical modeling of computational problems, as well as common algorithms, algorithmic paradigms, and data structures used to solve these problems. Emphasizes the relationship between algorithms and programming, and introduces basic performance measures and analysis techniques for these problems.
Enrollment may be limited. Institute LAB. Introduces probabilistic modeling for problems of inference and machine learning from data, emphasizing analytical and computational aspects. Distributions, marginalization, conditioning, and structure, including graphical and neural network representations. Belief propagation, decision-making, classification, estimation, and prediction.
Sampling methods and analysis. Introduces asymptotic analysis and information measures. Computational laboratory component explores the concepts introduced in class in the context of contemporary applications. Students design inference algorithms, investigate their behavior on real data, and discuss experimental results. Introduces fundamental concepts of programming. Designed to develop skills in applying basic methods from programming languages to abstract problems.
Topics include programming and Python basics, computational concepts, software engineering, algorithmic techniques, data types, and recursion.
Lab component consists of software design, construction, and implementation of design. Boning, A. Chlipala, S. Devadas, A. An integrated introduction to electrical engineering and computer science, taught using substantial laboratory experiments with mobile robots. Key issues in the design of engineered artifacts operating in the natural world: measuring and modeling system behaviors; assessing errors in sensors and effectors; specifying tasks; designing solutions based on analytical and computational models; planning, executing, and evaluating experimental tests of performance; refining models and designs.
Issues addressed in the context of computer programs, control systems, probabilistic inference problems, circuits and transducers, which all play important roles in achieving robust operation of a large variety of engineered systems.
Freeman, A. Hartz, L. Kaelbling, T. Covers signals, systems and inference in communication, control and signal processing. Topics include input-output and state-space models of linear systems driven by deterministic and random signals; time- and transform-domain representations in discrete and continuous time; and group delay. State feedback and observers. Probabilistic models; stochastic processes, correlation functions, power spectra, spectral factorization.
Least-mean square error estimation; Wiener filtering. Hypothesis testing; detection; matched filters. Studies interaction between materials, semiconductor physics, electronic devices, and computing systems. Develops intuition of how transistors operate. Topics range from introductory semiconductor physics to modern state-of-the-art nano-scale devices. Considers how innovations in devices have driven historical progress in computing, and explores ideas for further improvements in devices and computing.
Students apply material to understand how building improved computing systems requires knowledge of devices, and how making the correct device requires knowledge of computing systems. Includes a design project for practical application of concepts, and labs for experience building silicon transistors and devices.
Akinwande, J. Kong, T. Palacios, M. Analysis and design of modern applications that employ electromagnetic phenomena for signals and power transmission in RF, microwaves, optical and wireless communication systems. Fundamentals include dynamic solutions for Maxwell's equations; electromagnetic power and energy, waves in media, metallic and dielectric waveguides, radiation, and diffraction; resonance; filters; and acoustic analogs.
Lab activities range from building to testing of devices and systems e. Students work in teams on self-proposed maker-style design projects with a focus on fostering creativity, teamwork, and debugging skills. Subject meets with 6. Study of electromagnetics and electromagnetic energy conversion leading to an understanding of devices, including electromagnetic sensors, actuators, motors and generators.
Quasistatic Maxwell's equations and the Lorentz force law. Studies of the quasistatic fields and their sources through solutions of Poisson's and Laplace's equations. Boundary conditions and multi-region boundary-value problems. Steady-state conduction, polarization, and magnetization.
Charge conservation and relaxation, and magnetic induction and diffusion. Extension to moving materials. Electric and magnetic forces and force densities derived from energy, and stress tensors.
Extensive use of engineering examples. Students taking graduate version complete additional assignments. Studies key concepts, systems, and algorithms to reliably communicate data in settings ranging from the cellular phone network and the Internet to deep space. Weekly laboratory experiments explore these areas in depth. Topics presented in three modules - bits, signals, and packets - spanning the multiple layers of a communication system.
Bits module includes information, entropy, data compression algorithms, and error correction with block and convolutional codes. Signals module includes modeling physical channels and noise, signal design, filtering and detection, modulation, and frequency-division multiplexing. Packets module includes switching and queuing principles, media access control, routing protocols, and data transport protocols. Same subject as 2. Integrated overview of the biophysics of cells from prokaryotes to neurons, with a focus on mass transport and electrical signal generation across cell membrane.
First third of course focuses on mass transport through membranes: diffusion, osmosis, chemically mediated, and active transport. Second third focuses on electrical properties of cells: ion transport to action potential generation and propagation in electrically excitable cells. Synaptic transmission. Electrical properties interpreted via kinetic and molecular properties of single voltage-gated ion channels.
Final third focuses on biophysics of synaptic transmission and introduction to neural computing. Laboratory and computer exercises illustrate the concepts. Students taking graduate version complete different assignments. Preference to juniors and seniors. Application of the principles of energy and mass flow to major human organ systems.
Anatomical, physiological and clinical features of the cardiovascular, respiratory and renal systems. Mechanisms of regulation and homeostasis. Systems, features and devices that are most illuminated by the methods of physical sciences and engineering models.
Required laboratory work includes animal studies. See description under subject See description under subject 2. Enrollment limited. Slocum, G. Hom, E. Roche, N. Same subject as HST. Fundamentals of digital signal processing with emphasis on problems in biomedical research and clinical medicine. Basic principles and algorithms for processing both deterministic and random signals. Topics include data acquisition, imaging, filtering, coding, feature extraction, and modeling.
Lab projects, performed in MATLAB, provide practical experience in processing physiological data, with examples from cardiology, speech processing, and medical imaging. Lectures cover signal processing topics relevant to the lab exercises, as well as background on the biological signals processed in the labs. Greenberg, E. Adalsteinsson, W. Explores biomedical signals generated from electrocardiograms, glucose detectors or ultrasound images, and magnetic resonance images.
Topics include physical characterization and modeling of systems in the time and frequency domains; analog and digital signals and noise; basic machine learning including decision trees, clustering, and classification; and introductory machine vision.
Labs designed to strengthen background in signal processing and machine learning. Students design and run structured experiments, and develop and test procedures through further experimentation. Introduces fundamental principles and techniques of software development: how to write software that is safe from bugs, easy to understand, and ready for change.
Topics include specifications and invariants; testing, test-case generation, and coverage; abstract data types and representation independence; design patterns for object-oriented programming; concurrent programming, including message passing and shared memory concurrency, and defending against races and deadlock; and functional programming with immutable data and higher-order functions.
Includes weekly programming exercises and larger group programming projects. Topics on the engineering of computer software and hardware systems: techniques for controlling complexity; strong modularity using client-server design, operating systems; performance, networks; naming; security and privacy; fault-tolerant systems, atomicity and coordination of concurrent activities, and recovery; impact of computer systems on society. Case studies of working systems and readings from the current literature provide comparisons and contrasts.
Includes a single, semester-long design project. Students engage in extensive written communication exercises. Introduces representations, methods, and architectures used to build applications and to account for human intelligence from a computational point of view. Covers applications of rule chaining, constraint propagation, constrained search, inheritance, statistical inference, and other problem-solving paradigms. Also addresses applications of identification trees, neural nets, genetic algorithms, support-vector machines, boosting, and other learning paradigms.
Considers what separates human intelligence from that of other animals. Analyzes issues associated with the implementation of higher-level programming languages. Fundamental concepts, functions, and structures of compilers. The interaction of theory and practice. Using tools in building software. Includes a multi-person project on compiler design and implementation. Introduces principles, algorithms, and applications of machine learning from the point of view of modeling and prediction; formulation of learning problems; representation, over-fitting, generalization; clustering, classification, probabilistic modeling; and methods such as support vector machines, hidden Markov models, and neural networks.
Meets with 6. Recommended prerequisites: 6. Boning, P. Jaillet, L. Studies the structure and interpretation of computer programs which transcend specific programming languages. Demonstrates thought patterns for computer science using Scheme.
Includes weekly programming projects. REST Credit cannot also be received for An introduction to probability theory, the modeling and analysis of probabilistic systems, and elements of statistical inference. Probabilistic models, conditional probability. Discrete and continuous random variables. Expectation and conditional expectation, and further topics about random variables. Limit Theorems. Bayesian estimation and hypothesis testing. Elements of classical statistical inference.
Bernoulli and Poisson processes. Markov chains. Bresler, P. Jaillet, J. Same subject as Elementary discrete mathematics for science and engineering, with a focus on mathematical tools and proof techniques useful in computer science. Topics include logical notation, sets, relations, elementary graph theory, state machines and invariants, induction and proofs by contradiction, recurrences, asymptotic notation, elementary analysis of algorithms, elementary number theory and cryptography, permutations and combinations, counting tools, and discrete probability.
Mathematical introduction to the theory of computing. Rigorously explores what kinds of tasks can be efficiently solved with computers by way of finite automata, circuits, Turing machines, and communication complexity, introducing students to some major open problems in mathematics. Builds skills in classifying computational tasks in terms of their difficulty. Discusses other fundamental issues in computing, including the Halting Problem, the Church-Turing Thesis, the P versus NP problem, and the power of randomness.
Techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics include sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; greedy algorithms; amortized analysis; graph algorithms; and shortest paths. Advanced topics may include network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing. Covers the algorithmic and machine learning foundations of computational biology, combining theory with practice.
Principles of algorithm design, influential problems and techniques, and analysis of large-scale biological datasets. Topics include a genomes: sequence analysis, gene finding, RNA folding, genome alignment and assembly, database search; b networks: gene expression analysis, regulatory motifs, biological network analysis; c evolution: comparative genomics, phylogenetics, genome duplication, genome rearrangements, evolutionary theory. These are coupled with fundamental algorithmic techniques including: dynamic programming, hashing, Gibbs sampling, expectation maximization, hidden Markov models, stochastic context-free grammars, graph clustering, dimensionality reduction, Bayesian networks.
Same subject as 7. See description under subject 7. Students must provide their own laptop and software. Introduces fundamental concepts for 6. Students engage in problem solving, using Mathematica and MATLAB software extensively to help visualize processing in the time frequency domains. Electric circuit theory with application to power handling electric circuits.
Modeling and behavior of electromechanical devices, including magnetic circuits, motors, and generators. Operational fundamentals of synchronous. Interconnection of generators and motors with electric power transmission and distribution circuits. Power generation, including alternative and sustainable sources.
Incorporation of energy storage in power systems. Same subject as EC. Intuition-based introduction to electronics, electronic components and test equipment such as oscilloscopes, meters voltage, resistance inductance, capacitance, etc. Emphasizes individual instruction and development of skills, such as soldering, assembly, and troubleshooting. Students design, build, and keep a small electronics project to put their new knowledge into practice.
Intended for students with little or no previous background in electronics. Same subject as CMS. See description under subject CMS. Limited to Tan, S. Verrilli, R. Eberhardt, A. Introduction to embedded systems in the context of connected devices, wearables, and the "Internet of Things" IoT. Topics include microcontrollers, energy utilization, algorithmic efficiency, interfacing with sensors, networking, cryptography, and local versus distributed computation.
Students design, make, and program an Internet-connected wearable or handheld device. In the final project, student teams design and demo their own server-connected IoT system. Enrollment limited; preference to first- and second-year students. Mueller, J. Steinmeyer, J.
Prereq: None U Fall Units arranged. Prereq: Permission of instructor U Fall Not offered regularly; consult department Units arranged Can be repeated for credit. Prereq: Permission of instructor U Fall; second half of term Not offered regularly; consult department Units arranged Can be repeated for credit. Individual experimental work related to electrical engineering and computer science. Student must make arrangements with a project supervisor and file a proposal endorsed by the supervisor.
Departmental approval required. Written report to be submitted upon completion of work. Consult Department Undergraduate Office.
Experimental laboratory explores the design, construction, and debugging of analog electronic circuits. Lectures and laboratory projects in the first half of the course investigate the performance characteristics of semiconductor devices diodes, BJTs, and MOSFETs and functional analog building blocks, including single-stage amplifiers, op amps, small audio amplifier, filters, converters, sensor circuits, and medical electronics ECG, pulse-oximetry.
Projects involve design, implementation, and presentation in an environment similar to that of industry engineering design teams. Instruction and practice in written and oral communication provided. Opportunity to simulate real-world problems and solutions that involve tradeoffs and the use of engineering judgment. Investigates digital systems with a focus on FPGAs. Lectures and labs cover logic, flip flops, counters, timing, synchronization, finite-state machines, digital signal processing, as well as more advanced topics such as communication protocols and modern sensors.
Prepares students for the design and implementation of a final project of their choice: games, music, digital filters, wireless communications, video, or graphics. Steinmeyer, G. Hom, A. Introduces analysis and design of embedded systems. Microcontrollers provide adaptation, flexibility, and real-time control. Emphasizes construction of complete systems, including a five-axis robot arm, a fluorescent lamp ballast, a tomographic imaging station e.
Includes a sequence of assigned projects, followed by a final project of the student's choice, emphasizing creativity and uniqueness. Provides instruction in written and oral communication. To satisfy the independent inquiry component of this subject, students expand the scope of their laboratory project.
Students taking independent inquiry version 6. Introduces basic electrical engineering concepts, components, and laboratory techniques. Covers analog integrated circuits, power supplies, and digital circuits.
Lab exercises provide practical experience in constructing projects using multi-meters, oscilloscopes, logic analyzers, and other tools. Includes a project in which students build a circuit to display their own EKG. Enrollment limited; preference to Course 20 majors and minors. Boyden, M. Jonas, S. Nagle, P. So, S. Wasserman, M. Introduces the design and construction of power electronic circuits and motor drives.
Laboratory exercises include the construction of drive circuitry for an electric go-cart, flash strobes, computer power supplies, three-phase inverters for AC motors, and resonant drives for lamp ballasts and induction heating.
Basic electric machines introduced include DC, induction, and permanent magnet motors, with drive considerations. Presents concepts, principles, and algorithmic foundations for robots and autonomous vehicles operating in the physical world. Topics include sensing, kinematics and dynamics, state estimation, computer vision, perception, learning, control, motion planning, and embedded system development.
Students design and implement advanced algorithms on complex robotic platforms capable of agile autonomous navigation and real-time interaction with the physical word. Students engage in extensive written and oral communication exercises. Autonomous robotics contest emphasizing technical AI, vision, mapping and navigation from a robot-mounted camera. Teams should have members with varying engineering, programming and mechanical backgrounds.
Culminates with a robot competition at the end of IAP. Artificial Intelligence programming contest in Java. Student teams program virtual robots to play Battlecode, a real-time strategy game. Competition culminates in a live BattleCode tournament. Assumes basic knowledge of programming. Student teams learn to design and build functional and user-friendly web applications. All teams are eligible to enter a competition where sites are judged by industry experts.
Beginners and experienced web programmers welcome, but some previous programming experience is recommended. School of Engineering Toggle School of Engineering. Aeronautics and Astronautics Toggle Aeronautics and Astronautics. Biological Engineering Toggle Biological Engineering. Chemical Engineering Toggle Chemical Engineering. Mechanical Engineering Toggle Mechanical Engineering. Anthropology Toggle Anthropology. Economics Toggle Economics. Global Languages Toggle Global Languages. History Toggle History.
Linguistics and Philosophy Toggle Linguistics and Philosophy. Literature Toggle Literature. Political Science Toggle Political Science. Management Toggle Management. School of Science Toggle School of Science. Biology Toggle Biology. Chemistry Toggle Chemistry. Schwarzman College of Computing opened in September Applicants interested in graduate education should apply to the department or graduate program conducting research in the area of interest.
Applicants interested in research being conducted by faculty in the Department of Electrical Engineering and Computer Science should apply online. Below is an alphabetical list of all the available departments and programs that offer a graduate-level degree.
Skip to main content. Aeronautics and Astronautics. Application Deadline: December Application Deadline: January 7. Biological Engineering. Application Deadline: December 1. Brain and Cognitive Sciences. Center for Real Estate. Application Deadline: January Chemical Engineering. Application Deadline: November
0コメント