
Introduction to Scientific Computing
For students in any discipline that may involve numerical computations. Includes instruction for programming in MATLAB. Applications discussed include solution of linear equations (with vectors and matrices) and nonlinear equations (by bisection, iteration, and Newton's method), interpolation, and curvefitting, difference equations, iterated maps, numerical differentiation and integration, and differential equations. Prerequisite: MATH 0100 or its equivalent.

Methods of Applied Mathematics I, II
This course will cover mathematical techniques involving ordinary differential equations used in the analysis of physical, biological, and economic phenomena. The course emphasizes established methods and their applications rather than rigorous foundation. Topics include: first and second order differential equations, an introduction to numerical methods, series solutions, and Laplace transformations.
 Primary Instructor
 Akopian

Methods of Applied Mathematics I, II
Mathematical techniques involving differential equations used in the analysis of physical, biological and economic phenomena. Emphasis on the use of established methods, rather than rigorous foundations. I: First and second order differential equations. II: Applications of linear algebra to systems of equations; numerical methods; nonlinear problems and stability; introduction to partial differential equations; introduction to statistics. Prerequisite: MATH 0100, 0170, 0180, 0190, 0200, or 0350, or advanced placement.
 Primary Instructor
 Dobrushkin

Applied Ordinary Differential Equations
This course provides a comprehensive introduction to ordinary differential equations and their applications. During the course, we will see how applied mathematicians use ordinary differential equations to solve practical applications, from understanding the underlying problem, creating a differentialequations model, solving the model using analytical, numerical, or qualitative methods, and interpreting the findings in terms of the original problem. We will also learn about the underlying rigorous theoretical foundations of differential equations. Format: lectures and problemsolving workshops.
Prerequisites: MATH 0100, MATH 0170, MATH 0180, MATH 0190, MATH 0200, MATH 0350 or advanced placement. MATH 0520 (can be taken concurrently).
 Primary Instructor
 Dafermos

Applied Partial Differential Equations I
Course builds on APMA 0350 which covers ordinary differential equations and systems involving a single independent variable. We will look at processes with two or more independent variables formulated as partial differential equations (PDE) using concepts from multivariable calculus. We will see how problems are described quantitatively as PDEs, how seemingly unrelated contexts can result in similar equations; and develop methods for solution using analytical, numerical or qualitative methods. Contexts include first order equations; the second order wave equation and problems involving diffusion processes; steady state balances for systems in two or three dimensions; together with insights from theory.

Essential Statistics
A first course in probability and statistics emphasizing statistical
reasoning and basic concepts. Topics include visual and numerical
summaries of data, representative and nonrepresentative samples,
elementary discrete probability theory, the normal distribution,
sampling variability, elementary statistical inference, measures of
association. Examples and applications from the popular press and the
life, social and physical sciences. No prerequisites.
 Primary Instructor
 Lawrence

Quantitative Models of Biological Systems
Quantitative dynamic models help understand problems in biology and there has been rapid progress in recent years. The course provides an introduction to the concepts and techniques, with applications to population dynamics, infectious diseases, enzyme kinetics, aspects of cellular biology. Additional topics covered will vary. Mathematical techniques will be discussed as they arise in the context of biological problems. Prerequisites: APMA 0330, 0340 or 0350, 0360, or written permission.
 Primary Instructor
 Bienenstock

An Introduction to Numerical Optimization
This course provides a thorough introduction to numerical methods and algorithms for solving nonlinear continuous optimization problems. A particular attention will be given to the mathematical underpinnings to understand the theoretical properties of the optimization problems and the algorithms designed to solve them. Topics will include: line search methods, trustregion methods, nonlinear conjugate gradient methods, an introduction to constrained optimization (KarushKuhnTucker conditions, minimaximization, saddlepoints of Lagrangians). Some applications in signal and image processing will be explored. Basic programming skills at the level of APMA 16 or CSCI 40 are assumed.

Operations Research: Probabilistic Models
Basic probabilistic problems and methods in operations research and management science. Methods of problem formulation and solution. Markov chains, birthdeath processes, stochastic service and queueing systems, the theory of sequential decisions under uncertainty, dynamic programming. Applications. Prerequisite: APMA 1650, 1655 or MATH 1610, or equivalent.
 Primary Instructor
 Matzavinos

Applied Dynamical Systems
This course gives an overview of the theory and applications of dynamical systems modeled by differential equations and maps. We will discuss changes of the dynamics when parameters are varied, investigate periodic and homoclinic solutions that arise in applications, and study the impact of additional structures such as time reversibility and conserved quantities on the dynamics. We will also study systems with complicated "chaotic" dynamics that possess attracting sets which do not have an integer dimension. Applications to chemical reactions, climate, epidemiology, and phase transitions will be discussed.
 Primary Instructor
 Bramburger

Statistical Inference I
APMA 1650 is an integrated first course in mathematical statistics. The first half of APMA 1650 covers probability and the last half is statistics, integrated with its probabilistic foundation. Specific topics include probability spaces, discrete and continuous random variables, methods for parameter estimation, confidence intervals, and hypothesis testing.
Prerequisite: One year of universitylevel calculus. At Brown, this corresponds to MATH 0100, MATH 0170, MATH 0180, MATH 0190, MATH 0200, or MATH 0350. A score of 4 or 5 on the AP Calculus BC exam is also sufficient.

Statistical Inference II
APMA 1660 is designed as a sequel to APMA 1650 to form one of the alternative tracks for an integrated year's course in mathematical statistics. The main topic is linear models in statistics. Specific topics include likelihoodratio tests, nonparametric tests, introduction to statistical computing, matrix approach to simplelinear and multiple regression, analysis of variance, and design of experiments. Prerequisite: APMA 1650, 1655 or equivalent, basic linear algebra.

Monte Carlo Simulation with Applications to Finance
The course will cover the basics of Monte Carlo and its applications to financial engineering: generating random variables and simulating stochastic processes; analysis of simulated data; variance reduction techniques; binomial trees and option pricing; BlackScholes formula; portfolio optimization; interest rate models. The course will use MATLAB as the standard simulation tool. Prerequisites: APMA 1650 or MATH 1610
 Primary Instructor
 Mukherjee

Recent Applications of Probability and Statistics
This course develops the mathematical foundations of
modern applications of statistics to the computational,
cognitive, engineering, and neural sciences. The
course is rigorous, but the emphasis is on application.
Topics include: Gibbs ensembles and their relation to
maximum entropy, large deviations, exponential
models, and information theory; statistical estimation
and the generative, discriminative and algorithmic
approaches to classification; graphical models,
dynamic programming, MCMC computing, parameter
estimation, and the EM algorithm. For 2,000level
credit enroll in 2610; for 1,000level credit enroll in
1740. Rigorous calculusbased statistics,
programming experience, and strong mathematical
background are essential. For 2610, some graduate level
analysis is strongly suggested.

Race and Gender in the Scientific Community
This course examines the (1) disparities in representation in the scientific community, (2) issues facing different groups in the sciences, and (3) paths towards a more inclusive scientific environment. We will delve into the current statistics on racial and gender demographics in the sciences and explore their background through texts dealing with the history, philosophy, and sociology of science. We will also explore the specific problems faced by underrepresented and wellrepresented racial minorities, women, and LGBTQ community members. The course is reading intensive and discussion based. To be added to the waitlist for this course, please go to https://goo.gl/forms/foK0fyGxm5Eu2irA2
 Primary Instructor
 Sandstede

Wavelets and Applications
The aim of the course is to introduce you to: the relatively new and
interdisciplinary area of wavelets; the efficient and elegants algorithms to
which they give rise including the wavelet transform; and the mathematical
tools that can be used to gain a rigorous understanding of wavelets. We will
also cover some of the applications of these tools including the compression of
video streams, approximation of solution of partial differential equations, and
signal analysis.
 Primary Instructor
 Ainsworth

Independent Study
Section numbers vary by instructor. Please check Banner for the correct section number and CRN to use when registering for this course.
 Primary Instructor
 Bienenstock
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 I: Independent Study/Research
 Primary Instructor
 Dafermos
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 I: Independent Study/Research
 Primary Instructor
 Dupuis
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 I: Independent Study/Research
 Primary Instructor
 Rozovsky
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 I: Independent Study/Research
 Primary Instructor
 Ramanan
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 I: Independent Study/Research
 Primary Instructor
 Morrow
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 I: Independent Study/Research
 Primary Instructor
 Geman
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 I: Independent Study/Research
 Primary Instructor
 Gidas
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 I: Independent Study/Research
 Primary Instructor
 Darbon
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 I: Independent Study/Research
 Primary Instructor
 Matzavinos
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 I: Independent Study/Research
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 Primary Instructor
 Karniadakis
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 I: Independent Study/Research
 Primary Instructor
 Lawrence
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 I: Independent Study/Research
 Primary Instructor
 Maxey
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 I: Independent Study/Research
 Primary Instructor
 McClure
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 I: Independent Study/Research
 Primary Instructor
 Shu
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 I: Independent Study/Research
 Primary Instructor
 Klivans
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 I: Independent Study/Research
 Primary Instructor
 Dobrushkin
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 I: Independent Study/Research
 Primary Instructor
 Harrison
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 I: Independent Study/Research
 Primary Instructor
 Kunsberg
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 I: Independent Study/Research
 Primary Instructor
 Wang
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 I: Independent Study/Research
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 I: Independent Study/Research
 Primary Instructor
 Dong
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 I: Independent Study/Research
 Primary Instructor
 Guo
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 I: Independent Study/Research
 Primary Instructor
 Guzman
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 I: Independent Study/Research
 Primary Instructor
 MalletParet
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 I: Independent Study/Research
 Primary Instructor
 Menon
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 I: Independent Study/Research
 Primary Instructor
 Sandstede
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 I: Independent Study/Research
 Primary Instructor
 Su
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 Primary Instructor
 Crawford
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 I: Independent Study/Research

Hilbert Spaces and Their Applications
A continuation of APMA 2110: metric spaces, Banach spaces, Hilbert spaces, the spectrum of bounded operators on Banach and Hilbert spaces, compact operators, applications to integral and differential equations.

Nonlinear Dynamical Systems: Theory and Applications
Basic theory of ordinary differential equations, flows, and maps. Twodimensional systems. Linear systems. Hamiltonian and integrable systems. Lyapunov functions and stability. Invariant manifolds, including stable, unstable, and center manifolds. Bifurcation theory and normal forms. Nonlinear oscillations and the method of averaging. Chaotic motion, including horseshoe maps and the Melnikov method. Applications in the physical and biological sciences.
 Primary Instructor
 MalletParet

Numerical Solution of Partial Differential Equations II
An introduction to weighted residual methods, specifically spectral, finite element and spectral element methods. Topics include a review of variational calculus, the RayleighRitz method, approximation properties of spectral end finite element methods, and solution techniques. Homework will include both theoretical and computational problems.

Computational Fluid Dynamics
The course will focus primarily on finite difference methods for viscous incompressible flows.
Other topics will include multiscale methods, e.g. molecular dynamics, dissipative particle dynamics and lattice Boltzmann methods. We will start with the mathematical nature of the NavierStokes equations and their simplified models, learn about highorder explicit and implicit methods, time stepping, and fast solvers. We will then cover advectiondiffusion equations and various forms of the NavierStokes equations in primitive variables and in vorticity/streamfunction formulations. In addition to the homeworks the students are required to develop a NavierStokes solver as a final project.
 Primary Instructor
 Karniadakis

Recent Applications of Probability and Statistics
This course develops the mathematical foundations of
modern applications of statistics to the computational,
cognitive, engineering, and neural sciences. The
course is rigorous, but the emphasis is on application.
Topics include: Gibbs ensembles and their relation to
maximum entropy, large deviations, exponential
models, and information theory; statistical estimation
and the generative, discriminative and algorithmic
approaches to classification; graphical models,
dynamic programming, MCMC computing, parameter
estimation, and the EM algorithm. For 2,000level
credit enroll in 2610; for 1,000level credit enroll in
1740. Rigorous calculusbased statistics,
programming experience, and strong mathematical
background are essential. For 2610, some graduate level
analysis is strongly suggested.

Theory of Probability
Part two of a two semester course that provides an introduction to probability theory based on measure theory. Standard topics covered in the secondsemester (APMA 2640) include the following: discrete time martingale theory, weak convergence (also called convergence in distribution) and the central limit theorem, and a study of Brownian motion. Optional topics include the ergodic theorem and large deviation theory. Prerequisites—undergraduate probability and analysis, corequisite—graduate real analysis.
 Primary Instructor
 Harrison

Mathematical Statistics II
The course covers modern nonparametric statistical methods. Topics include: density estimation, multiple regression, adaptive smoothing, crossvalidation, bootstrap, classification and regression trees, nonlinear discriminant analysis, projection pursuit, the ACE algorithm for time series prediction, support vector machines, and neural networks. The course will provide the mathematical underpinnings, but it will also touch upon some applications in computer vision/speech recognition, and biological, neural, and cognitive sciences. Prerequisite: APMA 2670.

Introduction to Parallel Computing on Heterogeneous (CPU+GPU) Systems
This course we will learn fundamental aspects of parallel computing on heterogeneous systems composed of multicore CPUs and GPUs. This course will contain lectures and handson parts. We will cover the following topics: shared memory and distributed memory programming models, parallelization strategies and nested parallelism. We will also learn techniques for managing memory and data on systems with heterogeneous memories (DDR for CPUs and HBM for GPUs); parallelization strategies using OpenMP and MPI; programming GPUs using OpenMP4.5 directives and CUDA. We will focus on programming strategies and application performance. Grading will be based on homework assignments, and final project.

Research in Applied Mathematics
Section numbers vary by instructor. Please check Banner for the correct section number and CRN to use when registering for this course.
 Primary Instructor
 Bienenstock
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dafermos
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 I: Independent Study/Research
 Primary Instructor
 Dupuis
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 I: Independent Study/Research
 Primary Instructor
 Rozovsky
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 I: Independent Study/Research
 Primary Instructor
 Ramanan
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 I: Independent Study/Research
 Primary Instructor
 Matzavinos
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 I: Independent Study/Research
 Primary Instructor
 Geman
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 I: Independent Study/Research
 Primary Instructor
 Gidas
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 I: Independent Study/Research
 Primary Instructor
 Harrison
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 I: Independent Study/Research
 Primary Instructor
 Wang
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 I: Independent Study/Research
 Primary Instructor
 Crawford
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 I: Independent Study/Research
 Primary Instructor
 Karniadakis
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 I: Independent Study/Research
 Primary Instructor
 Lawrence
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 I: Independent Study/Research
 Primary Instructor
 Maxey
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 I: Independent Study/Research
 Primary Instructor
 McClure
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 I: Independent Study/Research
 Primary Instructor
 Shu
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 I: Independent Study/Research
 Primary Instructor
 Darbon
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 I: Independent Study/Research
 Primary Instructor
 Menon
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 I: Independent Study/Research
 Primary Instructor
 Dobrushkin
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 I: Independent Study/Research
 Primary Instructor
 Guo
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 I: Independent Study/Research
 Primary Instructor
 Klivans
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 I: Independent Study/Research
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dong
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 I: Independent Study/Research
 Primary Instructor
 Guzman
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 I: Independent Study/Research
 Primary Instructor
 MalletParet
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 I: Independent Study/Research
 Primary Instructor
 Sandstede
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 I: Independent Study/Research
 Primary Instructor
 Su
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 I: Independent Study/Research
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 I: Independent Study/Research
 Primary Instructor
 Ainsworth
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 I: Independent Study/Research

Thesis Preparation
For graduate students who have met the residency requirement and are continuing research on a full time basis.
 Schedule Code
 E: Grad Enrollment Fee/Dist Prep