## Scientific Activities

### Graduate Courses

#### Advanced Topics in Analysis: Finite Element Methods for Evolution Equations - Fall 2014

**Speaker:** N. Walkington (PIRE senior personnel)

**Abstract:** This course considers the construction and analysis of numerical schemes for evolution
equations. Construction of time stepping schemes which inherit the stability properties will be
emphasized. Schemes for parabolic equations, rst order evolution equations, and the wave
equation will be analyzed. If time permits, the Crandall Liggett theorem and Kato Trotter
theorem will be covered.

#### Advanced Topics in Analysis: Collective Dynamics and Structure in Infinite-Dimensional Systems - Fall 2014

**Speaker:** R. Pego (PIRE senior personnel)

**Abstract:** This course will survey recent progress regarding the emergence of coherent scaling
behavior in important PDE and kinetic models, focusing on phase transitions, aggregation, and
random shock clustering. We deal with models at various levels of microscopic detail: diffuse-interface,
sharp-interface, particles, kinetic equations, stochastic models. Dynamical concepts
motivated by the theory of stable laws and infinite divisibility in probability play a significant
role.

#### Advanced Topics in Analysis: Finite Element Methods - Spring 2014

**Speaker:** N. Walkington (PIRE senior personnel)

**Abstract:** Finite element methods for elliptic boundary value problems. Analysis of errors, approximation
by finite element spaces. Efficient implementation of finite element algorithms, finite
element methods for parabolic problems, effects of curved boundaries. Numerical quadrature,
non-conforming methods.

#### Advanced Topics in Analysis: Techniques of Applied Analysis in Image Processing and Data Analysis - Spring 2014

**Speaker:** D. Slepčev (PIRE senior personnel)

**Abstract:** The course focused on applications of variational and PDE techniques in data analysis
and image processing. This included learning about tasks of data analysis such as clustering and
classification and image processing tasks such as image comparison, registration and warping.
The mathematical techniques of calculus of variations and optimal transportation were presented
and used in modeling and analysis.

#### Advanced Topics in Analysis - Fall 2013

**Speaker:** D. Kinderlehrer (PIRE senior personnel)

**Abstract:** In the past ten years, Mass Transport methods have burgeoned to the extent that
every educated analyst should be familiar with their range of application and the basic methods.
For the first half of this course, our intention is to present the basic features of Mass Transport
methods for the study of PDE, in particular evolution equations and systems. If possible we
shall cover modeling issues too. This part of the course is introductory. In the second half,
we shall read papers from the recent literature, expanding on the Spring semester repertoire.
The course begins with a reprise of basic mass transport theory in order to introduce the new
students to the methods.

- This course was videotaped and recorded with open access. Instructions for accessing the live stream and course materials were posted on http://mm.math.cmu.edu/

#### Advanced Topics in Analysis - Spring 2013

**Speaker:** D. Kinderlehrer (PIRE senior personnel)

**Abstract:** This course consisted of mass transport methods and gradient
ows in the metric space of probability measures. The main lectures followed the Ambrosio and Gigli notes 'A
users guide to mass transport.' The books of Ambrosio, Gigli, and Savare' and Villani were
also major references. We then analyzed in detail several papers about gradient
ows. These included especially work by Blanchet, Calvez, and Carillo and by Blanchet and Laurencot, the
latter on the ow interchange method. Other issues were the analysis of liquid crystal systems
and transport systems. A feature of this course was that most of the lectures were by the
postdocs and the students.

- This course was videotaped and recorded with open access. Instructions for accessing the live stream and course materials were posted on http://mm.math.cmu.edu/

#### Calculus of Variations - Caltech, Spring 2013

**Speaker:** Kaushik Bhattacharya (PIRE Co-PI, Caltech)

**Abstract: **First and second variations; Euler-Lagrange equation; Hamiltonian formalism; action principle; Hamilton-Jacobi theory; stability; local and global minima; direct methods and relaxation; isoperimetric inequality; asymptotic methods and gamma convergence; selected applications to mechanics, materials science, control theory and numerical methods

- More information can be found at http://www.its.caltech.edu/~am127.

#### Calculus of Variations (MATH-GA 2660.001, Advanced Topics in Analysis), Spring 2013

**Speaker:** Robert V. Kohn

**Abstract:**A modern introduction to the Calculus of Variations, with equal emphasis on theory and applications. Topics will include: existence of solutions and convergence of numerical schemes; convex duality; one-dimensional variational problems; multidimensional nonconvex problems; relaxation; Gamma convergence; and length scale selection via singular perturbation. Along the way, we'll discuss many applications including minimal surfaces, optimal control, nonlinear elasticity, optimal design, martensitic phase transformations, and the wrinkling of thin sheets.

#### 21-820 Modern Methods in the Calculus of Variations in Sobolev Spaces - Fall 2012

**Speaker:** Irene Fonseca (PIRE PI, CMU): Videotaped , T Th 9:30am - 11am

**Abstract:** The objective of this course is to organize and unify contemporary developments in the Calculus of Variations and PDE, and to present applications spanning from fracture mechanics and thin films to micromagnetism. The material discussed in this course will be included in a book, with the same title, that Irene Fonseca and Giovanni Leoni are currently writing.

- Instructions for accessing the live stream and course materials are posted on http://mm.math.cmu.edu/irene2012.html

#### Atomistic-to-Continuum Coupling - July 15, 2012 - August 11, 2012

**Speaker:** Mitchell Luskin (PI, Minnesota), University of Heidelberg Internationales Wissenschaftsforums (IWF) Interdisciplinary Center for Scientific Computing

**Abstract:** Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long-range elastic fields with a much larger region that cannot be computed atomistically. Materials scientists have proposed many methods to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform on the atomistic scale.

During the past several years, a mathematical structure has been given to the description and formulation of atomistic-to-continuum coupling methods, and corresponding numerical analysis has clarified the relation between the various methods and their sources of error. This course will begin by introducing the physical and mathematical background and then present the current state of numerical analysis for atomistic-to-continuum coupling methods.

#### Minicourse on "A Mathematical Perspective on the Structure of Matter" - May 14 - 18, 2012

**Speaker:** Richard D. James (Co-PI, University of Minnesota)

**Abstract:** Beginning with some observations about the periodic and nonperiodic structures commonly adopted by elements in the periodic table, I will introduce a definition ("objective structures") of a mathematically small but physically well represented class of molecular structures. This definition will be seen to have an intimate relation to the invariance of the equations of quantum mechanics, statistical mechanics and continuum mechanics. The resulting framework can be used to design various multiscale methods, and offers an unusual perspective on experimental science. Open mathematical problems will be highlighted.

#### MATH 8450 Topics in Numerical Analysis: Multiscale Numerical Analysis for Materials - Spring 2012

**Speaker: **Mitchell Luskin (PI, Minnesota)

**Abstract:** Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long-range elastic fields with a much larger region that cannot be computed atomistically. Materials scientists have proposed many methods to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform on the atomistic scale.

During the past several years, a mathematical structure has been given to the description and formulation of atomistic-to-continuum coupling methods, and corresponding numerical analysis has clarified the relation between the various methods and their sources of error. This course will begin by introducing the physical and mathematical background and then presenting the current state of numerical analysis for atomistic-to-continuum coupling methods.

#### Micromechanics ME/MS 260 - Spring 2011-2012

**Speaker:** Prof. Kaushik Bhattacharya

**Abstract: **The course concerned contemporary experimental and theoretical issues concerning the effective properties of heterogeneous materials. It attracted graduate students from multiple disciplines including mechanical engineering, materials science, chemical engineering, applied mathematics and aeronautics. It introduced rigorous framework of homogenization in the linear setting, and illustrated it by considering overall elastic, dielectric, transport and magnetic moduli. The course then turned to nonlinear problems, and demonstrated key concepts through selected examples in finite elasticity, plasticity, discrete to continuum limits. The course concluded with time-dependent problems including viscoelasticity, dispersion, scattering of waves and pinning. The students were assigned homework, readings from the contemporary literature and a term project.

- Website: http://www.its.caltech.edu/~me260/

#### 21-820 Perspectives on Microstructure - Spring 2012

**David Kinderlehrer:** (PIRE senior personnel, CMU). Videotaped

**Abstract:** Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries, and thus comprise cellular networks. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales.

A central problem in materials is to develop technologies capable of producing an arrangement of grains that provides for a desired set of material properties. In this course we will investigate properties of these microstructures and the theories that are employed to characterize them. There will be opportunities for student participation and further research. We anticipate a number of guest lectures.

- Instructions for accessing the live stream and course materials are posted on http://mm.math.cmu.edu/