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심사위원장: 백형렬, 심사위원: 남경식, 최서영, Kasra Rafi(University of Toronto), Giulio Tiozzo(University of Toronto)

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심사위원장: 이창옥, 심사위원: 김동환, 임미경, 예종철(겸임교수), 한송희(삼성전자)

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심사위원장: 김재경, 심사위원: 김용정, 정연승, 황강욱, 이승규(경상대학교)

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심사위원장: 김용정, 심사위원: 권순식, 강문진, 김재경, 윤창욱(충남대학교)

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심사위원장: 안드레아스 홈슨, 심사위원: 김동수, 김재훈, 엄상일, 김민기(광주과학기술원)

This study is concerned with multivariate approximation by non-polynomial functions with internal shape parameters. The main topics of this presentation are two folds. First, interpolation by radial basis function (RBF) is considered. We especially discuss the convergence behavior of the RBF interpolants when the basis function is scaled to be increasingly flat. Moreover, we investigate the advantages of interpolation methods based on exponential polynomials. The second topic of this presentation is the approximation method based on sparse grids in $[0,1]^d \subset \RR^d$. The goal of sparse grid methods is to approximate high dimensional functions with good accuracy using as few grid points as possible. In this study, we present a new class of quasi-interpolation schemes for the approximation of multivariate functions on sparse grids. Each scheme in this class is based on shifts of kernels constructed from one-dimensional RBFs such as multiquadrics. The kernels are modified near the boundaries to prevent deterioration of the fidelity of the approximation. We show that our methods provide significantly better rates of approximation, compared to another quasi-interpolation scheme in the literature based on the Gaussian kernel using the multilevel technique. Some numerical results are presented to demonstrate the performance of the proposed schemes.

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Online: https://kaist.zoom.us/j/81807153144

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심사위원장: 엄상일, 심사위원: 안드레아스 홈슨, 김재훈, 권오정(한양대학교), Hong Liu(기초과학연구원)

Metal artifact reduction has become a challenging issue for practical X-ray CT applications since metal artifacts severely cause contrast degradation and the misinterpretation of information about the property and structure of a scanned object. In this talk, we propose a methodology to reduce metal artifacts by extending the method proposed by Jeon and Lee (2018) to a three-dimensional industrial cone beam CT system. We develop a registration technique managing the three dimensional data in order to find accurate segmentation regions needed for the proposed algorithm. Through various simulations and experiments, we verify that the proposed algorithm reduces metal artifacts successfully.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87958862292

In a region closer to the boundary compared to Prandtl layer, an inviscid disturbance can be manifested by the interaction with viscous mode via the no-slip boundary condition due to resonance. In some unstable range of parameters, this leads to instability in the transition regime from laminar flow to turbulence. This instability phenomenon was observed by physicists long time ago, such as Heisenberg, Tollmien and C.C. Lin, etc. And it was justified rigorously in mathematics by Grenier-Guo-Nguyen using the incompressible Navier-Stokes equation. In this talk, we will present some results on this phenomenon in some other physical situations in which the governing system is either MHD or compressible Navier-Stokes equation. The talk is based on some recent joint work with Chengjie Liu and Zhu Zhang.

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심사위원장: 백형렬, 심사위원:최서영, 응우옌 응옥 쿠옹, 김상현(KIAS), 이계선(서울대학교)

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심사위원장: 백형렬, 심사위원:최서영, 응우옌 응옥 쿠옹, 김상현(KIAS), 이계선(서울대학교)

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, however, are very complex and sophisticated, commonly requiring many trial-and-error and tricks. These result in a lack of robustness and interpretability, which are critical factors for scientific applications. This talk centers around mathematical approaches for SciML, promoting trustworthiness. The first part is about how to embed physics into neural networks (NNs). I will present a general framework for designing NNs that obey the first and second laws of thermodynamics. The framework not only provides flexible ways of leveraging available physics information but also results in expressive NN architectures. The second part is about the training of NNs, one of the biggest challenges in ML. I will present an efficient training method for NNs - Active Neuron Least Squares (ANLS). ANLS is developed from the insight gained from the analysis of gradient descent training.

Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $\lambda$-adic representations of the \'etale fundamental group of $X$ with connected algebraic monodromy groups $\bG_\lambda$, we construct a common $E$-form $\bG$ of all the groups $\bG_\lambda$ and in the absolutely irreducible case, a common $E$-form $\bG\hookrightarrow\GL_{n,E}$ of all the tautological representations $\bG_\lambda\hookrightarrow\GL_{n,E_\lambda}$. Analogous rationality results in characteristic $p$ assuming the existence of crystalline companions in $\mathrm{\textbf{F-Isoc}}^{\dagger}(X)\otimes E_{v}$ for all $v|p$ and in characteristic zero assuming ordinariness are also obtained. Applications include a construction of $\bG$-compatible system from some $\GL_n$-compatible system and some results predicted by the Mumford-Tate conjecture. (If you would like to join this seminar please contact Bo-Hae Im to get the zoom link.)

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서울대학교 산업공학과 학사 행정고시 37회 재경직 합격 KDI 국제정책대학원 석사 서울지방국세청 조사과장 국세청 기획재정담당관

The development and analysis of efficient methods and techniques for solving an inverse scattering problem are of great interest due to their potential in various applications, such as nondestructive testing, biomedical imaging, radar imaging, and structural imaging, among others. Sampling-type imaging methods allow us to non-iteratively retrieve the support of (possibly multiconnected) targets with low computational cost, assuming no a priori information about the targets. A sampling method tests a region of interest with its associated indicator function; the indicator function blows up if a test location is in support of inhomogeneities. Even though the sampling methods show promising results in ideal (multistatic, full-aperture, sufficiently many receivers) measurement configuration, one can frequently encounter limited measurement cases in practical applications. This presentation introduces the sampling-type imaging methods in two-dimensional limited-aperture, monostatic, and bistatic measurement cases. We identify the asymptotic structure of imaging methods to explore the applicability and intrinsic properties.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87958862292

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서울대학교 산업공학과 학/석사 삼성전자 연구원 조지아텍 박사 삼성전자 종합기술원 전문연구원 현) 삼성전자 종합기술원 머신러닝 랩장

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this talk, we will present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we will also present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions.

In this talk, we discuss the Cauchy problem for the Vlasov-Riesz system, which is a Vlasov equation featuring interaction potentials generalizing various previously studied cases, including the Coulomb and Manev potentials. For the first time, we extend the local theory of classical solutions to interaction potentials which are more singular than that for the Manev. Then, we obtain finite-time singularity formation for solutions with various attractive interaction potentials, extending the well-known singularity formation result for attractive Vlasov-Poisson. Our local well-posedness and singularity formation results extend to cases with linear diffusion and damping in velocity.

In recent years, community detection has been an active research area in various fields including machine learning and statistics. While a plethora of works has been published over the past few years, most of the existing methods depend on a predetermined number of communities. Given the situation, determining the proper number of communities is directly related to the performance of these methods. Currently, there does not exist a golden rule for choosing the ideal number, and people usually rely on their background knowledge of the domain to make their choices. To address this issue, we propose a community detection method that is equipped with data-adaptive methods of finding the number of the underlying communities. Central to our method is fused l-1 penalty applied on an induced graph from the given data. The proposed method shows promising results.

A theorem of Khare-Wintenberger and Kisin (once Serre’s modularity conjecture) says that every two-dimensional odd absolutely irreducible representation \bar\rho of the Galois group of the rationals over a finite field comes from a modular form f, that is, \bar\rho ~ \bar\rho_f. The conjecture even provides a recipe for the weight, level and character of f, but does not give any information about the slope of f. In this talk, based on joint work with Kumar, we provide conditions on f - the main one being that the weight of f is close to 0 - which guarantee that the slope of a modular form g giving rise to the twist of \bar\rho_f by the cyclotomic character has slope one more than the slope of f. This provides a global explanation of some local patterns mentioned in our first talk. The proof uses the theta operator and Coleman-Hida families of overconvergent forms. (This is the second of the two KAIX Invited Lectures.)

The zig-zag conjecture predicts that the reductions of two-dimensional irreducible p-adic crystalline representations of half-integral slope and exceptional weights - weights which are two more than twice the slope modulo (p-1) - have reductions which are given by an alternating sequence of irreducible and reducible representations. Some partial progress was made towards this conjecture over the years by Buzzard-Gee (slope 1/2), Bhattacharya-G-Rozensztajn (slope 1) and G-Rai (slope 3/2). In this talk I shall use work of Breuil-Mézard and Guerberoff-Park in the semi-stable case and a limiting argument connecting crystalline and semi-stable representations due to Chitrao-G-Yasuda to show that zig-zag holds for half-integal slopes bounded by (p-1)/2, at least on the inertia subgroup, if the weight is sufficiently close to a weight bounded by p+1. (This is the first of the two KAIX Invited Lectures.)

Shift workers experience profound circadian disruption due to the nature of their work, which often has them working at times when their internal clock is sending a strong signal for sleep. Mathematical models can be used to generate recommendations for shift workers that shift their body’s clock to better align with their work schedules, to help them sleep, feel, and perform better. In this talk, I will discuss our recent mobile app, Shift, which pulls wearable data from user’s devices and generates personalized recommendations to help them manage shift work schedules. I will also discuss how this product was designed, how it can interface with Internet of Things devices, and how its insights can be useful for other groups beyond shift workers.

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온라인(ZOOM) https://kaist.zoom.us/j/7011567816