Mon, 18 Mar 2024 14:15 -
Tue, 19 Mar 2024 15:00
L2

Euler Equations and Mixed-Type Problems in Gas Dynamics and Geometry

Professor Dehua Wang
(University of Pittsburgh)
Further Information

This course is running as part of the National PDE Network Meeting being held in Oxford 18-21 March 2024, and jointly with the 13th Oxbridge PDE conference.

The course is broken into 3 sessions over two days, with all sessions taking place in L2:

14:15-14:55:    Short Course I-1 Monday 18 March

9:45-10:25:    Short Course I-2 Tuesday 19 March

14:15-14:55:    Short Course I-3 Tuesday 19 March

Euler Equations and Mixed-Type Problems in Gas Dynamics and Geometry WANG_Oxford2024.pdf

Abstract

 In this short course, we will discuss the Euler equations and applications in gas dynamics and geometry. First, the basic theory of Euler equations and mixed-type problems will be reviewed. Then we will present the results on the transonic flows past obstacles, transonic flows in the fluid dynamic formulation of isometric embeddings, and the transonic flows in nozzles. We will discuss global solutions and stability obtained through various techniques and approaches. The short course consists of three parts and is accessible to PhD students and young researchers.

Thu, 23 Nov 2023
14:00
Lecture Room 3

Making SGD parameter-free

Oliver Hinder
(University of Pittsburgh)
Abstract

We develop an algorithm for parameter-free stochastic convex optimization (SCO) whose rate of convergence is only a double-logarithmic factor larger than the optimal rate for the corresponding known-parameter setting. In contrast, the best previously known rates for parameter-free SCO are based on online parameter-free regret bounds, which contain unavoidable excess logarithmic terms compared to their known-parameter counterparts. Our algorithm is conceptually simple, has high-probability guarantees, and is also partially adaptive to unknown gradient norms, smoothness, and strong convexity. At the heart of our results is a novel parameter-free certificate for the step size of stochastic gradient descent (SGD), and a time-uniform concentration result that assumes no a-priori bounds on SGD iterates.

Additionally, we present theoretical and numerical results for a dynamic step size schedule for SGD based on a variant of this idea. On a broad range of vision and language transfer learning tasks our methods performance is close to that of SGD with tuned learning rate. Also, a per-layer variant of our algorithm approaches the performance of tuned ADAM.

This talk is based on papers with Yair Carmon and Maor Ivgi.

Mon, 05 Nov 2018

16:00 - 17:00
L4

On the Monge-Ampere equation via prestrained elasticity

Marta Lewicka
(University of Pittsburgh)
Abstract

In this talk, we will present results regarding the regularity and

rigidity of solutions to the Monge-Ampere equation, inspired by the role

played by this equation in the context of prestrained elasticity. We will

show how the Nash-Kuiper convex integration can be applied here to achieve

flexibility of Holder solutions, and how other techniques from fluid

dynamics (the commutator estimate, yielding the degree formula in the

present context) find their parallels in proving the rigidity. We will indicate

possible avenues for the future related research.

Mon, 08 Mar 2010

12:00 - 13:00
L3

New approaches to problems posed by Sir Roger Penrose

George Sparling
(University of Pittsburgh)
Abstract

I will outline two areas currently under study by myself and my co-workers, particularly Jonathan Holland: one concerns the relation between the exceptional Lie group G_2 and Einstein's gravity; the second will introduce and apply the concept of a causal geometry.

Mon, 09 Nov 2009

17:00 - 18:00
Gibson 1st Floor SR

Elastic models for growing tissues: scaling laws and derivation by Gamma convergence

Reza Pakzad
(University of Pittsburgh)
Abstract

Certain elastic structures and growing tissues (leaves, flowers or marine invertebrates) exhibit residual strain at free equilibria. We intend to study this phenomena through an elastic growth variational model. We will first discuss this model from a differential geometric point of view: the growth seems to change the intrinsic metric of the tissue to a new target non-flat metric. The non-vanishing curvature is the cause of the non-zero stress at equilibria.

We further discuss the scaling laws and $\Gamma$-limits of the introduced 3d functional on thin plates in the limit of vanishing thickness. Among others, given special forms of growth tensors, we rigorously derive the non-Euclidean versions of Kirchhoff and von Karman models for elastic non-Euclidean plates. Sobolev spaces of isometries and infinitesimal isometries of 2d Riemannian manifolds appear as the natural space of admissible mappings in this context. In particular, as a side result, we obtain an equivalent condition for existence of a $W^{2,2}$ isometric immersion of a given $2$d metric on a bounded domain into $\mathbb R3$.

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