Convex Analysis, Optimization and Variational Problems
- 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)
Ivar Ekeland Roger Teman, Convex analysis and variational problems.
- Rockafellar, Convex Analysis
- L.Evans Partial Differential Equation(chapter 3,8)
- Boyd Convex Optimization(Part 1)
- Prof. Wotao Yin's Lecture Notes:link
Partial Differential Equation
- 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
L.Evans Partial Differential Equation(chapter 4,9)
Weinan E. Principle Of multiscale methods.
- Basic Knowledge:link
- Other Lecture Notes:link
As a reward for following such hard seminar and giving presentations. (This week we will cover the course ‘An introduction
to applied math’ and give discuss about several advanced topics in NLA and Numerical PDEs)
- Numerical linear algebra: conjugate gradient, space iteration, eigenvalue problems
- Numerical analysis(except FFT)
- Numerical ODE and PDE:Fourier analysis, modified equation, Dispersion and dissipation
Also known as PDE constrained optimization
- Linear problem, Bang-bang principle
- PMP and Hamilton-Jacobi
- Dynamic programming and game theory
- Basic of optimal transport
- Jordan-Kinderlehrer-Otto scheme, Gradient flow
- Optimal Transport Geometry.
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.
- 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
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.
- Brownian Motion, Stochastic Fourier Analysis
- Gaussian measure in Hilbert space
- Ito Formula, Martingale Representation Theorem
- Stochastic Differential Equation and numerical methods
- Topics in UQ
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.