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立命館大学数理工学セミナー（2022年5月26日（木））

2022.05.19 Thu up
<<立命館大学幾何学セミナー>>

日時：2022年5月26日（木） 16:30～18:00

タイトル：On the role and the design of loss functions in machine learning

講演者： Nguyen Tien Zung (Institut de Mathématiques de Toulouse)

アブストラクト：
Most people who are doing machine learning use some “standard” loss functions, such as the cross entropy and the mean square loss. However, for the same machine learning problem, one may use many different loss functions, and some of them may turn out to work much better than the “standard” ones. In this talk I want to discuss some simple general ideas about the loss functions, how do they affect the machine learning programs, and how to design them. I’ll try to use some examples from what we’re doing at Torus AI (https://torus.ai) for illustration.

開催方法：Zoom配信での開催です．

問い合わせ先：立命館大学理工学部数理科学科 多羅間 大輔

数理科学科談話会 （2022/5/19）

2022.05.16 Mon up
日時：5月19日(木) 16:30-17:30
開催方法：対面とオンラインのハイブリッド開催
場所：ウエストウイング 6階 談話会室（対面）および Zoom
講師：福泉麗華（東北大学）
講演題目：A nonlinear Kronig-Penney model
講演概要：
We consider the 1D nonlinear Schroedinger equation with focusing nonlinearities concentrated at some points.
In the linear model, there are many studies on the spectrum properties of the Schrodinger operator depending on the position of those points (quasi-periodic, random, etc).
In this talk we address the asymptotic behavior of the global solution for the nonlinear model, and its applications. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.


問い合せ先：立命館大学理工学部数理科学科　平良晃一　
 
オンライン参加の方は平良 ktaira@fc.ritsumei.ac.jp までご連絡いただけたら，談話会当日にzoomのアドレスをお送りいたします．

Math-Fi seminar on 28 Apr.

2022.04.27 Wed up
  • Date: 28 Apr. (Thu.)
  • Place: On the Web
  • Time: 17:00-18:30
  • Speaker:  Arturo Kohatsu-Higa (Ritsumeikan University)
  • Title:   Simulation of Reflected Brownian motion on two dimensional wedges
  • Abstract: 
We study Brownian motion in two dimensions, which is reflected, stopped or killed in a wedge represented as the intersection of two half-spaces. First, we provide explicit density formulas, hinted by the method of images. These explicit expressions rely on infinite oscillating sums of Bessel functions and may demand computationally costly procedures. We propose suitable recursive algorithms for the simulation of the laws of reflected and stopped Brownian motion which are based on generalizations of the reflection principle in two dimensions. We study and give bounds for the complexity of the proposed algorithms. (Joint with P. Bras.)

Math-Fi seminar on 21 Apr.

2022.04.20 Wed up
  • Date: 21 Apr. (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
  • Time: 17:00-18:00
  • Speaker:  Jiro Akahori (Ritsumeikan University)
  • Title:  Variational approach to optimal stopping problems revisited
  • Abstract: 
After reviewing besoussan-lions’s variational approach, I will discuss its applications to numerical problems; discretization error, deep solver, and so on.  The talk will be in English.

Math-Fi seminar on 14 Apr.

2022.04.14 Thu up
  • Date: 14 Apr. (Thu.)
  • Place: On the Web
  • Time: 17:00-18:30
  • Speaker:  Umut Cetin (London School of Economics)
  • Title:  Speeding up the Euler scheme for killed diffusions
  • Abstract:
Let X be a linear diffusion taking values in  (l,r) and consider the standard Euler discretisation to compute the fair price of a Barrier option written on X that becomes worthless if X hits one of the barriers before the maturity date T. It is well-known since Gobet’s work that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to N^{-1/2} with N being the number of discretisatons. We introduce a drift-implicit Euler method to bring the convergence rate back to 1/N, i.e. the optimal rate in the absence of killing, using the theory of recurrent transformations. Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions.
 

Math-Fi seminar on 7 Apr.

2022.04.11 Mon up
  • Date: 7 Apr. (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker:  Andrey Pilipenko (Ukraine National Academy of Sciences)
  • Title: Limit behavior of perturbed random walks
  • Abstract:
A particle moves randomly over the integer points of the real line. Jumps of the particle outside the membrane (a fixed “locally perturbating set”) are i.i.d., have zero mean and finite variance, whereas jumps of the particle from the membrane have other distributions with finite means which may be different for different points of the membrane; furthermore, these jumps are mutually independent and independent of the jumps outside the membrane. We prove that the weak scaling limit of the particle position is a skew Brownian motion.

立命館大学幾何学セミナー（2022年1月31日（月））

2022.01.25 Tue up
<<立命館大学幾何学セミナー>>

日時： 2022年1月31日（月） 16:30～18:00

タイトル： A geometric approach to stochastic extensions of nonholonomic constraints

講演者： François Gay-Balmaz（CNRS – LMD – Ecole Normale Supérieure）

アブストラクト：
We propose several stochastic extensions of nonholonomic constraints for mechanical systems and study the effects on the dynamics and on the conservation laws. Our approach relies on a stochastic extension of the Lagrange-d’Alembert framework. The mechanical system we focus on is the example of a Routh sphere, i.e., a rolling unbalanced ball on the plane. We interpret the noise in the constraint as either a stochastic motion of the plane, random slip or roughness of the surface. Without the noise, this system possesses three integrals of motion: energy, Jellet and Routh. Depending on the nature of noise in the constraint, we show that either energy, or Jellet, or both integrals can be conserved, with probability 1. We also present some exact solutions for particular types of motion in terms of stochastic integrals.
 
Inspired by this example, we then consider two different ways of including stochasticity in nonholonomic systems. We show that when the noise preserves the linearity of the constraints, then energy is.  preserved. For other types of noise in the constraint, e.g., in the case of an affine noise, the energy is not conserved. This approach is illustrated with a class of Lagrangian mechanical systems on Lie groups, with constraints of “rolling ball type”. We conclude with numerical simulations illustrating our theories, and some pedagogical examples of noise in constraints for other nonholonomic systems popular in the literature, such as the nonholonomic particle, the rolling disk and the Chaplygin sleigh.

開催方法：Zoom配信での開催です．

問い合わせ先：立命館大学理工学部数理科学科 多羅間 大輔

立命館大学幾何学セミナー（2022年1月24日（月））

2022.01.20 Thu up
<<立命館大学幾何学セミナー>>

日時：2022年1月24日（月） 15:00～16:00

タイトル：微分可能同相写像群の基本群

講演者： 前田 吉昭（東北大学）

アブストラクト：
微分可能同相写像群の基本群を調べるための不変量としてWodzicki-Chern-Simons クラスを定義する。この応用として、基本群が無限となる例として、シンプレクティック多様体上のサークル束について調べる。

開催方法：Zoom配信での開催です．

問い合わせ先：立命館大学理工学部数理科学科 多羅間 大輔

Math-Fi seminar on 20 Jan.

2022.01.19 Wed up
  • Date: 20 Jan. (Thu.)
  • Place: On the Web
  • Time: 16:30 – 18:00
  • Speaker: Stefano Pagliarani (University of Bologna)
  • Title: A Yosida’s parametrix approach to Varadhan’s estimates for a degenerate diffusion under the weak Hörmander condition
  • Abstract: 
We adapt and extend Yosida’s parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak H\”ormander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain a Varadhan-type formula for the asymptotic behavior of the logarithm of the transition density, in terms of the optimal cost function of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.

Math-Fi seminar on 13 Jan.

2022.01.12 Wed up
Date: 13 Jan. (Thu.)
Place: On the Web
Time: 16:30 – 18:00
Speaker: Benjamin Jourdain (CERMICS)
Title: Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q — L ρDrift Coefficient and Additive Noise
Abstract: 
We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q — L ρ drift coefficient when the condition d ρ + 2 q < 1, under which Krylov and R{ö}ckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 — (d ρ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove that both the diffusion and this Euler scheme admit transition densities and that the difference between these densities is bounded from above by the time-step to this order multiplied by some centered Gaussian density.

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