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2018 International Graduate Summer School on "Frontiers of Applied and Computational Mathematics"

Introduction

Applied and computational mathematics plays an important role in physics, mechanics, biology, engineering, finance and other fields. In recent decades, many new research areas have emerged. We plan to organize a summer school from July 9 to July 20, 2018, which covers the basic theories, latest achievements and hot topics in some chosen areas of applied and computational mathematics. This summer school is supported by the School of Mathematical Sciences and the Institute of Natural Sciences of Shanghai Jiao Tong University.

In the latest US NEWS rankings for mathematical sciences, SJTU Math ranked 29th globally. The School of Mathematical Sciences (SMS) has been included in the project “Network of International Centers of Education in China” (three in total nationwide), which facilitates its development in internationalization.

Institute of Natural Sciences (INS), SJTU, is becoming an intellectual center for world-class interdisciplinary studies. The INS faculty consists of internationally-established scientists as well as distinguished young research fellows, and with specialties in fields ranging from applied mathematics, physics, computer sciences, to engineering, biology and life sciences.

Date

July 9 - 20, 2018

Venue

601 Pao Yue-Kong Library, Minhang Campus, Shanghai Jiao Tong University, No.800 Dongchuan Road

Application and Registration

Please register online. Apply Online
Preference is given to, but not limited to, graduate students and young researchers with a basic understanding of applied mathematics and computational mathematics. The deadline for registration is May 31, 2018 and there is no registration fee.

Steering Committee

Organizers

Lecturers

Target and Characteristics

  1. To promote academic exchange on the frontiers of applied and computational mathematics.
  2. To promote the cultivation of graduate students and young researchers.
  3. To strengthen the cooperation and exchange between SJTU and other institutions.

In view of the characteristics of applied and computational mathematics, we emphasize students’ understanding of the research background of related interdisciplinary fields, enhance students’ mathematical modeling ability and ability to use computational tools in applied mathematics research. By combining coursework, seminars, poster presentation and panel discussions, we want to help students to broaden their horizons and lay a solid foundation in order to to make progress in the frontiers of relevant research fields.

Course Contents and Practical Activities

This summer school include 7 mini-courses in the field of applied and computational mathematics and workshop about some recent advances from July 9 to July 21, 2018. We will spend 6-8 hours for teaching and discussion every day.

Schedule

Week 1

Date Time Speaker
July 9 09:30 ~ 11:00 Apala Majumdar
July 9 13:00 ~ 14:30 Itamar Procaccia
July 10 09:30 ~ 11:00 Itamar Procaccia
July 10 13:00 ~ 14:30 Apala Majumdar
July 10 15:00 ~ 16:30 Apala Majumdar
July 11 09:30 ~ 11:00 Apala Majumdar
July 11 13:00 ~ 14:30 Itamar Procaccia
July 12 09:30 ~ 11:00 Itamar Procaccia
July 12 13:00 ~ 14:30 Nicholas Zabaras
July 13 09:30 ~ 11:00 Nicholas Zabaras
July 13 13:00 ~ 14:30 Nicholas Zabaras
July 13 15:00 ~ 16:30 Francois Golse

Week 2

Date Time Speaker
July 16 09:30 ~ 11:00 Nicholas Zabaras
July 16 13:00 ~ 14:30 Francois Golse
July 16 15:00 ~ 16:30 Kazuo Aoki
July 17 09:30 ~ 11:00 Walter Schirmacher
July 17 13:00 ~ 14:30 Nicholas Zabaras
July 17 15:00 ~ 16:30 Carsten Carstensen
July 18 09:30 ~ 11:00 Walter Schirmacher
July 18 13:00 ~ 14:30 Francois Golse
July 18 15:00 ~ 16:30 Kazuo Aoki
July 19 09:30 ~ 11:00 Walter Schirmacher
July 19 13:00 ~ 14:30 Kazuo Aoki
July 19 15:00 ~ 16:30 Carsten Carstensen
July 20 09:30 ~ 11:00 Carsten Carstensen
July 20 13:00 ~ 14:30 Walter Schirmacher

Program

Theory and Numerical Approach in Kinetic Theory of Gases

Kazuo Aoki, National Cheng Kung University

The present short course consists of three part:
Part 1: Fundamentals of kinetic theory of gases
Part 2: Kinetic theory and fluid dynamics
Part 3: Numerical approach to flows caused by thermal effects

In Part 1, the fundamental equation, the Boltzmann equation, is introduced, and its basic properties, such as the conservation laws, the equilibrium solution, and the Boltzmann H theorem, are explained. The boundary condition for the Boltzmann equation is also discussed. Part 2 is concerned with the relation between kinetic theory and fluid dynamics. The classical Chapman-Enskog and Hilbert expansions, which lead to fluid-dynamic equations when the mean free path of gas molecules is small, are explained briefly. The major part of Part 2 is devoted to the derivation of the slip boundary conditions for the compressible Navier-Stokes equations by the analysis of the kinetic boundary layer (Knudsen layer). In part 3, steady flows induced by temperature fields without the help of external forces, which are peculiar to rarefied gases, are discussed, together with the numerical methods that are used to analyze such flows.

Lecture Notes
Lecture 1
Lecture 2
Lecture 3


Axioms of Adaptivity

Carsten Carstensen, Humboldt University

The lecture series on the optimal rates of adaptive mesh-refining algorithms in computational PDEs provides an introduction to the mathematics of optimal rates based on the standard D"orfler marking in a collective refinement strategy. The focus is on the thorough insight into the mathematics for the simplest meaningful setting with elementary tools like the trace inequality, inverse estimate, plus several forms of triangle and Cauchy inequalities.

Part 1 Solely four axioms guarantee the optimality in terms of the error estimators outlined in Part 1 of the lectures. This general framework covers a huge part of the existing literature on optimal rates of adaptive schemes and is exemplified for the 2D Poisson model problem on polygonal domains.

Part 2 gives the outline of the proof of optimal rates with linear convergence and a comparison lemma as Stevenson’s key argument for the optimality. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. The local discrete efficiency of the error estimator is neither needed to prove convergence nor utilised for the quasi-optimal convergence behaviour of the error estimator. Efficiency exclusively characterises the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on any efficiency constant. Some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the R-linear convergence of the error estimator.

Part 3 discusses the lowest-order conforming finite element method based on triangles and provides proofs of the stability, reduction, and discrete localised reliability of the explicit residual-based error estimators.

The power of the axiomatic framework towards separate marking will be mentioned only shortly with applications to mixed or least squares finite element methods and so leads towards the current research of non-standard adaptive finite element schemes.

References

  1. Open-Access Reference: C-Feischl-Page-Praetorius: Axioms of Adaptivity. Computer & Mathematics with Applications 67 (2014) 1195-1253.

  2. C. Carstensen and H. Rabus. Axioms of adaptivity for separate marking. arXiv:1606.02165v1. SIAM J. Numer. Anal. 56 (2018) 2644–2665.

Lecture Notes
Lecture 1
Lecture 2
Lecture 3


Mean-Field Limits for the Dynamics of Large Particle Systems

Francois Golse, Ecole Polytechnique

The purpose of this course is to describe the evolution of large systems of identical particles in classical or quantum mechanics. The first major result in this direction is the derivation of the Vlasov equation from Newton’s second law of motion written for each particle, assuming that the particle interaction is defined in terms of a C1,1 pairwise potential. Dobrushin’s estimate of the convergence rate for this limit in terms of Wasserstein distances will be presented in detail. The analogous result in quantum mechanics, due to Spohn, is the derivation of the time-dependent Hartree equation from the Schrödinger N-body problem, which wil be discussed in terms of the notion of BBGKY hierarchy, and by another, more recent method due to Pickl. Finally, the course will focus on the joint mean-field (large N) and semiclassical (small Planck constant) limits, providing a derivation of the Vlasov equation from the quantum N-body problem.

References

  1. R.L. Dobrushin : Funct. Anal. Appl. 13, 115–123 (1979) 

  2. H. Spohn : Rev. Mod. Phys. 52, 600–640 (1980) 

  3. P. Pickl : Lett. Math. Phys. 97, 151– 
164 (2011) 

  4. F. Golse, C. Mouhot, T. Paul : Commun. Math. Phys. 343, 165–205 (2016) 

  5. F. Golse, T. Paul : Arch. Rational Mech. Anal. 223, 57–94 (2017)

Lecture Notes
Lecture 1
Lecture 2
Lecture 3


The Mathematics of Liquid Crystals - Analysis, Simulations and Applications

Apala Majumdar, University of Bath

Nematic liquid crystals are classical examples of mesogenic materials that combine the fluidity of liquids with the orientational order of solids. Nematics are, informally speaking, anisotropic materials with distinguished directions of preferred averaged molecular alignment, referred to as “directors”. Consequently, nematics have an anisotropic response and strong coupling to external electric and magnetic fields, and incident light and these unique electromagnetic and optical responses make nematics the working material of choice for several electro-optic applications notably the multi-billion dollar liquid crystal display industry.

The mathematics of nematic liquid crystals is undergoing a modern-day renaissance as researchers from different walks of mathematics - calculus of variations and partial differential equations, topology, geometric analysis, numerical analysis and scientific computation, work with physicists, chemists and engineers to work on the theory, numerical algorithms and applications of the next generation of liquid crystal-based materials. This mini-course is a review of two popular continuum theories for nematic liquid crystals - the Oseen-Frank theory and the Landau-de Gennes theories, the relationship between the two theories and how they are connected to other variational theories in materials science such as nonlinear elasticity and theory of superconductivity, how these theories are used to model and make predictions about liquid crystal experiments and devices and new strategic research directions.

The lectures will be organized as follows: Lecture 1: Introduction to the Mathematics and Physics of Nematic Liquid Crystals
Lecture 2: The Oseen-Frank Theory for Nematic Liquid Crystals
Lecture 3: The Landau-de Gennes Theory for Nematic Liquid Crystals
Lecture 4: Asymptotic Analysis of the Landau-de Gennes Theory - Part I
Lecture 5: Asymptotic Analysis of the Landau-de Gennes Theory - Part II
Lecture 6: Case Study I: Nematic Equilibria in Square Wells - Part I
Lecture 7: Case Study I: Nematic Equilibria in Square Wells - Part II
Lecture 8: Case Study II: Nematic Equilibria in Annular Wells

Lecture Notes
china2018lec1.pdf
china2018lec2.pdf
china2018lec3.pdf
china2018lec4.pdf


Statistical Physics of Glasses and Amorphous Solids

Itamar Procaccia, The Weizmann Institute of Science

In this course I will present the recent advances in the statistical physics of amorphous solids. I will introduce the material physics of glasses, and discuss their mechanical and magnetic properties. I will then develop that theory of plasticity, instabilities, and material failure. The lectures will draw information from experiments, numerical simulations and theory.


Mean-Field Theories for the Description of Disordered Materials

Walter Schirmacher,University of Mainz

The mathematical description of the physical properties of disordered materials is hampered by the absence of the crystalline order. While in materials, which form a crystalline lattice Bloch’s theorem is a powerful tool for solving Schrödinger and other dynamical equations, amorphous and liquid materials must be dealt with taking the disorder into account.

Most mathematical descriptions of such materials are based on numerical simulations. While such simulations are very useful for designing new materials, they are of limited help for understanding the salient disorder-induced features of disordered materials. For this understanding one needs analytical mean-field theories.

In this lecture series it will be shown, how one can incorporate the statistical properties of the disorder for constructing mathematically coarse-grained schemes to calculate analytically the salient physical features. These features are those which are described by partial differential equations with spatially fluctuating coefficients, namely diffusion and wave propagation. If transformed into the frequency domain all these phenomena (including the propagation of electrons in disordered media) can be treated by the same formalism, which is the Green’s function formalism. Further tools are multiple-scattering theory and functional integrals, which are being introduced in a pedagogical way.

The mean-field theories to be constructed arise as saddle points of effective field theories. For weak disorder one can use the self-consistent Born approximation (SCBA), for strong disorder the coherent potential approximation (CPA).

It will be shown, how the following (very different) disorder-induced phenomena can be treated and explained using these mathematical tools: Anomalous diffusion in disordered systems; Rayleigh scattering; boson peak and associated vibrational properties; Anderson localization of electrons, acoustic and electromagnetic waves, anomalous Maxwell relaxation near the glass-to liquid transition.

Lecture Notes
Lecture 1
Lecture 2
Lecture 3
Lecture 4


Monte Carlo Methods in Scientific Computing

Nicholas Zabaras, University of Notre Dame

The course provides an introduction to Monte Carlo methods and their applications to the Bayesian data-driven simulation of high-dimensional engineering and scientific applications. A review of basic probabilistic concepts will be provided together with the fundamentals of Monte Carlo and sampling methods. An introduction to Bayesian inference will also be given including conjugate and non-informative priors, likelihood and posterior evaluation. Topics such as model selection and validation will be highlighted. Markov Chain Monte Carlo (MCMC) methods will then be introduced including Metropolis-Hastings, Gibbs sampler, Simulated Tempering/Annealing, Trans-Dimensional samplers and other. The course will conclude with introduction to Sequential Monte Carlo methods as applied to Hidden Markov Models, State Space Models, Static and Dynamic problems. Applications will be discussed throughout the course including the solution of stochastic PDEs, optimization and rare event modeling, system identification and other. The course aims to acquaint students with the necessary background to pursue research in various scientific fields in the era of Big Data and Deep Learning.

Please refer to the attached website for more details:
https://www.zabaras.com/monte-carlo-scientific-comp-2018

Lecture Notes
http://ins.sjtu.edu.cn/people/jinglaili/files/Summer_School_2018.zip


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