Graduate Classes
MA 574: Mathematical Modeling of Physical and Biological Processes II: – Instructor: Mohammad Farazmand – This course reviews model development in continuum mechanics using conservation laws and variational principles. In particular, this course focuses on mathematical models in fluid mechanics, acoustic and water waves, and solid mechanics.
MA 793: Special Topics In Differential Equations: Data-driven Modeling and Analysis of Dynamic Systems: – Instructor: Mohammad Farazmand – Governing equations of many natural and engineering systems are described by dynamical systems. Experiments and numerical simulations of these systems result in large data sets. How would we analyze and make sense of this data? On the other hand, many natural systems are dynamical systems but their governing equations are unknown. Can we discover the governing equations from experimentally gathered data? This course discusses a variety of data-driven methods, such as Proper Orthogonal Decomposition, Dynamic Mode Decomposition, Reduced-order Modeling, and Machine Learning, to address these questions.
BMA 790/MA 797: Machine Learning in Biology – Instructor: Kevin Flores – This course introduces students to machine learning concepts, including supervised and unsupervised approaches. The foundations of machine learning are described in the context of Bayes decision theory. Then, a wide range of machine learning techniques are covered including neural networks, SVMs, decision trees, random forests, and k-means clustering. We discuss the theory and practical application of specific neural network architectures to data types which are especially relevant to biology, i.e., imaging (convolutional neural networks for segmentation and classification) and sequential data (recurrent neural networks for natural language processing or time series forecasting). Finally, dimensionality reduction techniques are covered, including linear (PCA) and non-linear (auto-encoders, diffusion maps) methods. Students will learn to program in Scikit learn, Keras, and Pytorch. Example code is provided and covered in lectures. In-class examples and homework assignments include real data sets from biomedicine, ecology, and genomics. In addition, students will be required to complete and present a class project, applying machine learning methods to a data set of their choice. At the end of the course, guest speakers (e.g., from Microsoft Research and SAS) will discuss applications of machine learning in industry.
MAE 518: Introduction to principles of acoustic radiation from vibrating bodies and their related fields –Instructor: Marie Muller – Radiation of simple sources, propagation of sound waves in confined spaces and transmission through different media.
Course Objectives: This course will provide students with the tools necessary to calculate the acoustic pressure field generated by vibrating surfaces, and to evaluate the pressure transmitted and reflected in various media.
MA 540: Fall (Distance Education), Spring (In-Person) – Instructor: Ralph Smith- Topics: Introduction to uncertainty quantification for physical and biological models. Topics include fundamental concepts from probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, surrogate model construction, quantification of model discrepancy, and local and global sensitivity analysis. Prerequisites: MA 341 and basic knowledge of probability, linear algebra, differential equations, and introductory numerical analysis.
MA 573: Topics: Introduction to model development for physical and biological processes. – Instructor: Ralph Smith- Topics include compartmental analysis and conservation laws including advection, convection and diffusion processes, conservation of mass and the material derivative, and traffic flow models and analysis. Topics also include energy conservation and heat conduction processes and models, the development of population models, and the development of various disease models including SIR models. Supporting topics include analytical and numerical solution techniques for ordinary and partial differential equations and aspects of deterministic and frequentist model calibration. Applications include development of models for a catalytic converter, size-structured population models, traffic flow models, and neutron transport models. Prerequisites: MA 341 and knowledge of a high-level programming language; e.g., MATLAB.
MA 591-002: Applied Harmonic Analysis – Instructor: Fatma Terzioglu – Applied Harmonic Analysis is the study (and generalization) of the notions of Fourier analysis and is used in a variety of areas, ranging from engineering to medicine and finance, for data representation and analysis. Applications include signal and image processing, denoising, data compression, and image reconstruction. The course will cover the following topics: Fourier Series, Fourier Transform, Discrete (Fast) Fourier Transform, Interpolation and Sampling, Wavelet Analysis, and Discrete Wavelet Transform and its applications. Emphasis will be placed upon mathematical foundations of applicable algorithms, as well as on the ability to implement these algorithms.
MA797-Mathematics of Radar Imaging-Section 003 –Instructor: Semyon Tsynkov-Radar imaging is a mature technology with a broad range of applications. Yet in spite of its many demonstrated successes, a number of difficult issues are still outstanding, e.g., mitigation of the various distortions. Addressing these issues requires going beyond the standard engineering practices and calls for employing an array of mathematical methods. Accordingly, in the course we adopt the interpretation of radar imaging as a mathematical inverse problem. Our goal is to cover the key aspects of construction and analysis of the pertinent mathematical models using tools from differential equations, perturbation theory, and Fourier analysis. Students will learn the main quantitative concepts of radar imaging: matched filtering, synthetic aperture, imaging operator, focusing, resolution, signal compression, and others. They will also be exposed to the fundamentals of electromagnetic wave propagation and scattering, from the standpoint of both physics and mathematics. Those include Maxwell’s, d’Alembert (wave), and Helmholtz equations, geometrical optics, diffraction (Fresnel and Fraunhofer regimes), Doppler effects, first Born approximation and Neumann series, Bragg scattering, and more.
Data-Driven Methods for Biological Modeling in Industry: CRSC associate faculty members, Kevin Flores, Ryan Murray, and Hien Tran, developed a new graduate course called to be offered in the Fall of 2023. In the course, graduate students will work with CRSC associate faculty and scientists from industry and government laboratories on non-academic real-world problems. The goal is to solicit funding from the industry and government laboratory to run this as a fee-based course. This is an innovative approach to provide first and second year graduate students with the opportunity to participate in real-world research. The course has two main components: (i) Hands-on experience with real-world projects created by industry partners, and (ii) Lectures on state-of-the-art topics in modeling, parameter estimation, uncertainty quantification, optimal control, and machine learning that are tailored to the projects.