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Course Criteria
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1.00 Credits
The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's and Cramer's Theorems). We then will cover large deviations for diffusion processes and the Wentsel-Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk-sensitive control; weak convergence methods; Hamilton-Jacobi-Bellman equations; Monte Carlo methods. Prerequisites: AM 263 and 264.
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1.00 Credits
Various general techniques have been developed for control and system problems. Many of the methods are indirect. For example, control problems are reduced to a problem involving a differential equation (such as the partial differential equation of Dynamic Programming) or to a system of differential equations (such as the canonical system of the Maximum Principle). Since these indirect methods are not always effective alternative approaches are necessary. In particular, direct methods are of interest.
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1.00 Credits
The course will provide presentation of the biology and mathematical models/algorithms for a variety of topics, including: (1) The analysis and interpretation of tandem mass spectrometry data for protein identification and determination of signaling pathways, (2) Identification of Phosphorylation sites and motifs and structural aspects of protein docking problems. Prerequisites: The course is recommended for graduate students. It will be self-contained; students will be able to fill in knowledge by reading material to be indicated by the instructor.
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1.00 Credits
SPDEs is an interdisciplinary area at the crossroads of stochastic processes and partial differential equations (PDEs). The topics of the course include: geneses of SPDEs in real life applications, mathematical foundations and analysis of SPDEs, numerical and computational aspects of SPDEs, applications of SPDEs to fluid dynamics, population biology, hidden Markov models, etc. Prerequisites: familiarity with stochastic calculus and PDEs (graduate level).
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1.00 Credits
Topics that will be covered include: WKB method: zeroth and first orders; turning points; Perturbation theory: regular perturbation, singular perturbation and boundary layers; Homogenization methods for ODE's, elliptic and parabolic PDE's; Homogenization for SDE's, diffusion processes in periodic and random media; Averaging principle for ODE's and SDE's. Applications will be discussed in class and in homework problems.
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1.00 Credits
This course gives an introduction to the the basic concepts of a posteriori estimates of finite element methods. After an overview of different techniques the main focus will be shed on residual based estimates where as a starting point the Laplace operator is analyzed. Effectivity and reliability of the error estimator will be proven. In a second part of the course, students will either study research articles and present them or implement the error estimates for some specific problem and present their numerical results. Recommended prerequisites: basic knowledge in finite elements, APMA 2550, 2560, 2570.
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1.00 Credits
The purpose of the course is to lay the foundation for the development and analysis of numerical methods for solving systems of ordinary differential equations. With a dual emphasis on analysis and efficient implementations, we shall develop the theory for multistage methods (Runge-Kutta type) and multi-step methods (Adams/BDF methods). We shall also discuss efficient implementation strategies using Newton-type methods and hybrid techniques such as Rosenbruck methods. The discussion includes definitions of different notions of stability, stiffness and stability regions, global/local error estimation, and error control. Time permitting, we shall also discuss more specialized topics such as symplectic integration methods and parallel-in-time methods. A key component of the course shall be the discussion of problems and methods designed with the discretization of ODE systems originating from PDE's in mind. Topics include splitting methods, methods for differential-algebraic equations (DAE),deferred correction methods. and order reduction problems for IBVP, TVD and IMEX methods. Part of the class will consist of student presentations on more advanced topics, summarizing properties and known results based on reading journal papers.
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1.00 Credits
Topics that will be covered include: the averaging principle for stochastic dynamical systems and in particular for Hamiltonian systems; metastability and stochastic resonance. We will also discuss applications in class and in homework problems. In particular we will consider metastability issues arising in chemistry and biology, e.g. in the dynamical behavior of proteins. The course will be largely self contained, but a course in graduate probability theory and/or stochastic calculus will definitely help.
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1.00 Credits
No course description available.
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1.00 Credits
Turbulence is the last mystery of classical physics. It surrounds us everywhere – in the air, in the ocean, in pipes carrying fluids and even in human body arteries. The course helps to understand what makes modeling the turbulence so difficult and challenging. The course covers the following issues: The nature of turbulence, characteristics of turbulence and classical constants of turbulence; Turbulent scales; Navier-Stokes equations, Reynolds stresses and Reynolds-Averaged Navier-Stokes (RANS) equations; RANS turbulence models: algebraic models, one-equation models, two-equation models; Low-Reynolds number turbulence models; Renormalization Group (RNG) turbulence model; Large-Eddy Simulation (LES); Students will be provided with user-friendly computer codes to run different benchmark cases. The final grade is based on two take home projects - computing or published papers analysis, optionally.
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