Selected topics on analysis which may be useful in applied math.

**Detailed Plan:**pdf

**Speaker:**Yiping Lu

**Objectives:**

- Variational problems, Euler-Lagrange Equation, Eignvalue problems
- Hamilton-Jacobi Equations, Hopf-lax formula
- Pointwise supremum
- Hahn-Banach Thm and Application, Lower semi-continuous function
- Conjugate functions, biconjugate
- Subdifferentiability, Proximal Mapping, Duality of convex optimization.
- Optimal Transport: An example of optimization(Kantorovich description of optimal transportation, Brenier’s theorem)

**Text Book:**

`Ivar Ekeland Roger Teman, Convex analysis and variational problems.`

**Reference**

- Rockafellar, Convex Analysis
- L.Evans Partial Differential Equation(chapter 3,8)
- Boyd Convex Optimization(Part 1)
- Prof. Wotao Yin's Lecture Notes:link

Lecture note:link,Assignment:link,reference:link

**Speaker:**Yiping Lu

**Objectives:**

- Asymptotic Analysis, Multi-scale Analysis and Model Reduction In Dynamic System
- Transform method(Fourier, Laplace), Plane and travelling wave, separation of variable, Power series.
- *Weak solution, Lax-Milgram Thm, Regularity
- *Systems of conservation Law.

**class note will be available soon**

**Text Book:**

```
L.Evans Partial Differential Equation(chapter 4,9)
Weinan E. Principle Of multiscale methods.
```

**Reference:**

**Objectives:**

- Numerical linear algebra: conjugate gradient, space iteration, eigenvalue problems
- Numerical analysis(except FFT)
- Numerical ODE and PDE:Fourier analysis, modified equation, Dispersion and dissipation

**Objectives:**

- Linear problem, Bang-bang principle
- PMP and Hamilton-Jacobi
- Dynamic programming and game theory

**Discussion Weak**

**Objectives:**

- Basic of optimal transport
- Jordan-Kinderlehrer-Otto scheme, Gradient flow
- Optimal Transport Geometry.
- Application.

**Text Book:**

```
Villani,2008, Optimal transport, old and new
Santambrogio, 2015, Optimal Transport for Applied Mathematicians.
Ambrosio and Gigli, 2011, A User’s Guide to Optimal Transport.
```

**Reference:**

- CMU CNA Semina:link
- Richard Jordan, David Kinderlehrer, Felix Otto
**The variational formulation of the Fokker-planck equation** - Liero and Mielke.
**Systems of equations as gradient flows: Reaction diffusion systems** - Martin Burger, Marzena Franek, Carola-Bibiane Schonlieb
**Regularised Regreesion and Density Estimation based on Optimal Transport** -
Adrian Tudorascu
**On the Jordan-Kinderlehrer-Otto variational scheme and constrained optimization in the Wasserstein metric** -
Liero, Mielke, Savare preprints on
**geometry of optimal transport with mass loss and creation. Global distance is the Hellinger-Kantorovich distance.paper1,paper2** -
Otto, 2001,
**The geometry of dissipative evolution equations: the porous medium equation** -
Li W, Yin P, Osher S.
**Computations of optimal transport distance with Fisher information regularization[J]. 2017.**

**Speaker:**Yiping Lu

**Outline:**

- Brownian Motion, Stochastic Fourier Analysis
- Gaussian measure in Hilbert space
- Ito Formula, Martingale Representation Theorem
- Stochastic Differential Equation and numerical methods
- Topics in UQ

**Text Book:**

```
L.Evans Introduction to Stochastic Differential Equation
Bernt Oksendal. Stochastic Differential Equation.
Xiaoliang Wan Introduction to uncertain certainty quantification.
```

**Presentation Next Week**

- David Balduzzi, Marcus Frean et al.The shattered gradient problems: If resnet are the answer, then what is the question?
- Pratik Chaudhari, Adam Oberman et al.Deep relaxation: Partial Differential Equations for optimizing deep neural networks.(Mainly used the knowledge in week2)
- Qianxiao Li, Cheng Tai, Weinan E. Stochastic modified equations and adaptive stochastic gradient algorithms.

- Unsupervised/semi-supervised deep learning:slides
- Information Geometry
- Computational Fluid Dynamics.