Despite much recent progress in analyzing algorithms in the linear MDPs and their variants, the understanding of more general transition models is still very restrictive. We study provably efficient RL algorithms for the MDP whose state transition is given by a multinomial logistic model. We establish the regret guarantees for the algorithms based on multinomial logistic function approximation. We also comprehensively evaluate our proposed algorithm numerically and show that it consistently outperforms the existing methods, hence achieving both provable efficiency and practical superior performance.

In this talk, we are going to consider the Fermi-Pasta-Ulam (FPU) system with innitely many oscil- lators. We particularly see that Harmonic analysis approaches allow us to observe dispersive properties of solutions to a reformulated FPU system, and with this observation, solutions to the FPU system can be approximated by counter-propagating waves governed by the Korteweg de-Vries (KdV) equation as the lattice spacing approaches zero. Additionally, we see dierent phenomena detected in the periodic FPU system.

Domain adaptation (DA) is a statistical learning problem that arises when the distribution of the source data used to train a model differs from that of the target data used to test the model. While many DA algorithms have demonstrated considerable empirical success, the unavailability of target labels in DA makes it challenging to determine their effectiveness in new datasets without a theoretical basis. Therefore, it is essential to clarify the assumptions required for successful DA algorithms and quantify the corresponding guarantees. In this work, we focus on the assumption that conditionally invariant components (CICs) useful for prediction exist across the source and target data. Under this assumption, we demonstrate that CICs found via conditional invariant penalty (CIP) play three essential roles in providing guarantees for DA algorithms. First, we introduce a new CIC-based algorithm called importance-weighted conditional invariant penalty (IW-CIP), which has target risk guarantees beyond simple settings like covariate shift and label shift. Second, we show that CICs can be used to identify large discrepancies between source and target risks of other DA algorithms. Finally, we demonstrate that incorporating CICs into the domain invariant projection (DIP) algorithm helps to address its known failure scenario caused by label-flipping features. We support our findings via numerical experiments on synthetic data, MNIST, CelebA, and Camelyon17 datasets.

The compressible Euler system (CE) is one of the oldest PDE models in fluid dynamics as a representative model that describes the flow of compressible fluids with singularities such as shock waves. But, CE is regarded as an ideal model for inviscid gas, and may be physically meaningful only as a limiting case of the corresponding Navier-Stokes system(NS) with small viscosity and heat conductivity that can be negligible. Therefore, any stable physical solutions of CE should be constructed by inviscid limit of solutions of NS. This is known as the most challenging open problem in mathematical fluid dynamics (even for incompressible case). In this talk, I will present my recent works that tackle the open problem, using new methods: the (so-called) weighted relative entropy method with shifts (for controlling shocks) and the viscous wave-front tracking method (for handling general solution with small total variation).

We present how to construct a stochastic process on a finite interval with given roughness and finite joint moments of marginal distributions. Our construction method is based on Schauder representation along a general sequence of partitions and has two ramifications. The variation index of a process (the infimum value p such that the p-th variation is finite) may not be equal to the reciprocal of Hölder exponent. Moreover, we can construct a non-Gaussian family of stochastic processes mimicking (fractional) Brownian motions. Therefore, when observing a path of process in a financial market such as a price or volatility process, we should not measure its Hölder roughness by computing p-th variation and should not conclude that a given path is sampled from Brownian motion or fractional Brownian motion even though it exhibits the same properties of those Gaussian processes. This talk is based on joint work with Erhan Bayraktar and Purba Das.

In geometric variational problems and non-linear PDEs, challenges often reduce down to questions on the asymptotic behavior near singularity and infinity. In this talk, we discuss the rate and direction of convergence for slowly converging solutions. Previously, they were constructed under so called the Adams-Simon positivity condition on the limit. We conversely prove that every slowly converging solution necessarily satisfies such a condition and the condition dictates possible dynamics. The result can be placed as a generalization of Thom's gradient conjecture. This is a joint work with Pei-Ken Hung at Minnesota

We obtain uniform in time L^\infty -bounds for the solutions to a class of thermo-diffusive systems. The nonlinearity is assumed to be at most sub-exponentially growing at infinity and have a linear behavior near zero. This is a joint work with Lenya Ryzhik and Jean-Michel Roquejoffre.

We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on the unit circle. We show that this system is locally well-posed for L^p data as well as for atomic measures, that is logarithmic spiral vortex sheets. We prove global well-posedness for almost bounded logarithmic spirals and give a complete characterization of the long time behavior of logarithmic spirals. This is due to the observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time. We are then able to show a dichotomy in the long time behavior, solutions either blow up (in finite or infinite time) or completely homogenize. In particular, bounded logarithmic spirals converge to constant steady states. For vortex logarithmic spiral sheets the dichotomy is shown to be even more drastic where only finite time blow up or complete homogenization of the fluid can and does occur.

The Harnack inequality plays a crucial role in elliptic and parabolic PDEs. In particular, one can characterize ancient positive solutions to parabolic PDEs by using the Harnack inequality. In this talk, we consider the mean curvature flow, a parabolic PDE of hypersurfaces. To study its stability, it is important to show the uniqueness of ancient flows staying in an one-side of self-similarly shrinking flows. After rescaling the ancient one-sided flow converges to the static self-similar solution, and so it is the graph of an evolving positive function defined over the self-similar solution. Then, the positive function is a solution to a parabolic PDE, and we can show the uniqueness by using the Harnack inequality.