2月演講

Applied Mathematics Seminar, Professor Tamaki Tanaka, Niigata University Tuesday, February 20, 15:10—16:00 數學系3174 Title: Generalized Cone-continuity of Set-Valued Maps with Scalarization Abstract:A composite function is a function which is the nesting of two or more functions to form a single new function. Such operation frequently preserves several mathematical properties of each nested function. For instance, a composition of continuous maps is continuous on topological spaces. From the view point of vector optimization and set optimization, this kind of inheritance by composite operations is important and useful to prove extended results and to get characterizations of optimal solutions through scalarization. This is a typical approach by which optimization problems with vector-valued or set-valued maps can be easily handled by converting vectors or sets into real numbers. Recently, Ike, Liu, Ogata and Tanaka show certain results on the inheritance property of some kinds of continuity of set-valued maps via scalarization functions for sets: if a set-valued map has a kind of continuity (lower continuity or upper continuity), then the composition of its set-valued map and a certain scalarization function assures a similar semicontinuity to its scalarization function defined on the family of nonempty subsets of a real topological vector space. Their results are generalizations of results in earlier study by Kuwano, Tanaka and Yamada. However, the statements of inheritance are confined to four types out of the six set-relations proposed by Kuroiwa, Tanaka and Ha. On the other hand, Sonda, Kuwano, and Tanaka introduce two kinds of continuity with respect to cone, called ``cone continuity,'' for set-valued maps by analogy with semicontinuity for real-valued functions, and they investigate the inheritance properties on cone continuity of parent set-valued maps via scalarization. Therefore, it is interesting to investigate the inheritance of cone continuity for set-valued maps via general scalarization functions for sets in the same manner. The aim of this talk is to introduce the mechanism by which composite functions of a set-valued map and a scalarization function transmit semicontinuity of parent set-valued maps through several scalarization for sets.

Colloquium, Professor Hayato Miyazaki, Kagawa University Thursday, February 22, 14:10—15:00 數學系3174 Title: Modified Scattering Operator for Nonlinear Schrodinger Equations with a Time-Decaying Harmonic Potential Abstract:We consider nonlinear Schr\"odinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. Kawamoto and Muramatsu show that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in the frameworks of both the final state problem and the initial value problem. Also, in the equation without the potential, Hayashi and Naumkin establish the existence of a modified scattering operator. In this talk, we construct a modified scattering operator for our equation by utilizing a generator of the Galilean transformation.

Colloquium, Professor Kota Uriya, Okayama University of Science Thursday, February 22, 15:10—16:00 數學系3174 Title: Asymptotic Behavior of the Solution to a System of Cubic Nonlinear Schrodinger Equations in One Dimension Abstract: We consider a system of cubic nonlinear Schrodinger equations in one dimension. We give the asymptotic behavior of the solution depending on the coefficients of the nonlinearity. In particular, we show that the solution decays slower in algebraic order compared with the free solution. This is a joint work with N. Kita (Kumamoto Univ), S. Masaki (Hokkaido Univ), J. Segata (Kyushu Univ).

Colloquium, Professor Koichi Komada, Chukyo University Thursday, February 22, 16:10—17:00 數學系3174 Title: Existence of Blow-Up Solutions for the Quantum Zakharov System Abstract:We consider the quantum Zakharov system in dimensions. The quantum Zakharov system describes the propagation of Langmuir waves in an ionized plasma. In this talk, we prove that all radially symmetric solutions with negative energy blow up in finite time or in infinite time when 6 \le d \le 9.

Colloquium, Professor Jack Xin, University of California Friday, February 23, 11:10—12:00 數學系3174 Title: Deepparticle: Learning Multiscale Pdes with Data Generated from Interacting Particle Methods Abstract:Multiscale time dependent partial differential equations (PDE) are challenging to compute by traditional mesh based methods especially when their solutions develop large gradients or concentrations at unknown locations. Particle methods, based on microscopic aspects of the PDEs, are mesh free and self-adaptive, yet still expensive when a long time or a resolved computation is necessary. We present DeepParticle, an integrated deep learning, optimal transport (OT), and interacting particle (IP) approach, to speed up generation and prediction of PDE dynamics through two case studies on transport in fluid flows with chaotic streamlines: 1) large time front speeds of Fisher-Kolmogorov-Petrovsky-Piskunov equation (FKPP); 2) Keller-Segel (KS) chemotaxis system modeling bacteria evolution in the presence of a chemical attractant. Analysis of FKPP reduces the problem to a computation of principal eigenvalue of an advection-diffusion operator. A normalized Feynman-Kac representation makes possible a genetic IP algorithm to evolve the initial uniform particle distribution to a large time invariant measure from which to extract front speeds. The invariant measure is parameterized by a physical parameter (the Peclet number). We train a light weight deep neural network with local and global skip connections to learn this family of invariant measures. The training data come from IP computation in three dimensions at a few sample Peclet numbers. The training objective being minimized is a discrete Wasserstein distance in OT theory. The trained network predicts a more concentrated invariant measure at a larger Peclet number and also serves as a warm start to accelerate IP computation. The KS is formulated as a McKean-Vlasov equation (macroscopic limit) of a stochastic IP system. The DeepParticle framework extends and learns to generate various finite time bacterial aggregation patterns. Joint work with Zhongjian Wang (Nanyang Tech Univ, Singapore) and Zhiwen Zhang (University of Hong Kong).

Colloquium, 台灣大學數學系 劉子齊博士 Thursday, February 29, 16:10—17:00 數學系3174 Title: Removal of Electrical Stimulus Artifact in Local Field Potential Recorded from Subthalamic Nucleus by Using Manifold Denoising Abstract:Deep brain stimulation (DBS) is an effective treatment for movement disorders such as Parkinson’s disease (PD). However, local field potentials (LFPs) recorded through lead externalization during high-frequency stimulation (HFS) are contaminated by stimulus artifacts, which require to be removed before further analysis. In this study, a novel stimulus artifact removal algorithm based on manifold denoising, termed Shrinkage and Manifold-based Artifact Removal using Template Adaptation (SMARTA), was proposed to remove artifacts by deriving a template for each stimulus artifact and subtracting it from the signal. Under a low-dimensional manifold assumption, a matrix denoising technique called optimal shrinkage was applied to design a similarity metric such that the template for stimulus artifacts could be accurately recovered. SMARTA was evaluated using semirealistic and realistic LFP signals. The results indicated that SMARTA removes stimulus artifacts with a modest distortion in LFP estimates. The proposed SMARTA algorithm helps the exploration of the neurophysiological mechanisms of DBS effects.