Fall 2016

High dimensional computation -- that is, numerical computation in which there are very many or even infinitely many continuous variables -- is a new frontier in scientific computing, with applications ranging from financial mathematics such as option pricing or risk management, to groundwater flow, heat transport, and wave propagation. Often the difficulties come from uncertainty or randomness in the data, e.g., in groundwater flow from permeability that is rapidly varying and uncertain, or in heat transport from uncertainty in the conductivity. These high dimensional problems present major challenges to computational resources, and requires serious mathematical efforts in devising new and effective methods.

This talk will provide a contemporary review of quasi-Monte Carlo (QMC) methods for approximating high dimensional integrals. I will highlight some recent developments on "lattice rules" and "higher order digital nets". One key element is the "fast component-by-component construction" which yields QMC methods with a prescribed rate of convergence for sufficiently smooth functions. Another key element is the careful selection of parameters called "weights" to ensure that the worst case errors in an appropriately weighted function space are bounded independently of the dimension. Then I will showcase how this modern QMC theory can be tuned for a number of applications, including PDEs with random coefficients.

References
[1] J. Dick, F. Y. Kuo, and I. H. Sloan, High-dimensional integration: the quasi-Monte Carlo way, Acta Numerica, 22, 133--288 (2013).
[2] F. Y. Kuo and D. Nuyens, Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients -- a survey of analysis and implementation, Foundations of Computational Mathematics, 67 pages, in press (2016).

Wedge product on deRham complex of a Riemannian manifold $M$ can be pulled back to $H^*(M)$ via explicit homotopy, constructed using Green's operator, to give higher product structures. Fukaya conjectures the Witten deformation of these higher product structures have semi-classical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of deRham differential. We will describe briefly the proof of Fukaya's conjecture, and an application to Mirror symmetry which realizes the scattering diagram as semi-classical limit of solution to the Maurer-Cartan equation.

混沌動態系統和複動態系統是近十幾年來動態系統理論中非常活躍的分支。在此次講座中我將對混沌動態系統和複動態系統的起源和歷史作一簡短的回顧和分析，並對複動態系統中一些已知及待解的重要問題進行介紹和評論。

The IFS (iterated function system) is a usual way to generate fractals. We will talk about the dimensions of self-affine sets generated by the affine IFSs. An open problem on the dimension drop of self-affine sets will be raised and partially answered in the talk.

醫學影像的技術發展有數十年的歷史，隨著人類的科技更新與醫學診斷要求提高，對於醫學影像的需求也越來越多，目前醫學影像的主流有採用游離輻射的一大類, 如 CT, PET, SPECT, DR 另一大類是採用非游離輻射的MRI與Ultrasound; MRI由於可以提供優秀的軟組織對比與特定分子的分佈影像，搭配無輻射的特性，隨著儀器價格的降低，設備的普及，臨床應用越來越廣泛; 華源磁振科技專注在臨床超導MRI系統，願景成為台灣第一家大型影像設備系統公司，要成就此願景需要更多優秀的人才投入產業，因此希望透過與學校的交流，吸引更多教授與同學的加入影像產業。

This study experimentally examines the resonance of mechanically vibrated sessile drops. Shape, frequency and amplitude responses of the drops are investigated for individual modes. The observations are characterized by relating to Rayleigh-Lamb (RL), Bostwick-Steen invisid (BS inviscid) and Bostwick-Steen viscous potential flow (VPF) theories. Observed mode shapes are compared to predictions from RL and BS inviscid theories via ray-tracing simulation. The comparison of frequency response suggests that VPF theory most adequately predicts resonance frequency of observed modes. The adequacy implies the necessity of considering both substrate constraint and viscosity, and hence distinguishes viscous sessile drops from and inviscid free spherical drops. The amplitude responses of modes are explored from the growth and decay of the lowest axisymmetric mode. Evident nonlinearity is observed. The amplitude study thus exposes the surprising extent to which our nonlinear observations can be understood in the context of linear theories. Further exploration reveals the interactions of modes such as spectral crossing and mode mixing. For subhemispherical drops, typical observations are mixtures of a half-frequency subharmonic non-zonal mixing a harmonic zonal mode. For superhemispherical drops, more diverse mixing phenomena are discovered. From the scientific perspective, the study reveals a rich collection of resonance modes catagorized according to shapes and harmonic types for future investigation. From engineers, the study provides guidelines for applications relevant to pattern selection of surface waves, such as ordered self-assembly of particles, droplet transport, drop atomization, enhanced mixing, and suspension collection.

In this talk, we will study the Campanato spaces $\Lambda^{\kappa}_{q, \mathcal P}$ and Hardy spaces $H^p_{\mathcal P}$ associated with a family $\mathcal P$ of parabolic sections which is closely related to the parabolic Monge-Amp\`ere equation. Moreover, we show the Campanato spaces are the duals of the corresponding Hardy spaces.

The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products $\{\alpha, -\alpha\}$, $\alpha\in[0,1)$, are called equiangular. The problem of determining the maximal size of $s$-distance sets in various spaces has a long history in mathematics. We determine a new method of bounding the size of an $s$-distance set in two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in $\mathbb{R}^n$ is $\frac{n(n+1)}2$ with possible exceptions for some $n=(2k+1)^2-3$, $k \in \mathbb{N}$. We also prove the universal upper bound $\sim \frac 2 3 n a^2$ for equiangular sets with $\alpha=\frac 1 a$ and, employing this bound, prove a new upper bound on the size of equiangular sets in an arbitrary dimension. Finally, we classify all equiangular sets reaching this new bound.

In this talk, we would like to introduce a reaction–diffusion–advection equation with a free boundary. This model may be used to describe the spreading of an invasive species in one-dimensional heterogeneous environments. We assume that the species has a tendency to move upward along the resource gradient in addition to random dispersal, and the spreading mechanism of species is determined by a Stefan-type condition. We are mainly interested in the effect of the inhomogeneous advection. Some biological implications will be discussed. This is a joint work with Harunori Monobe.

We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem. This is joint work with Gwendal Collet and Lukas Klausner.

Asymptotics of recurrences is the key to get the typical properties of combinatorial structures, and thus the complexity of many algorithms relying on these structures. The associated generating function often follows a linear differential equation: we are here in the so-called "D-finite" world. For the matters of asymptotics, this case of linear recurrences (with polynomial coefficients) is well covered by the "Analytic Combinatorics" book of Flajolet and Sedgewick (though the computations of constants is still a challenge, related to the theory of Kontsevich-Zagier periods and evaluation of G-functions and E-functions). At the border of this D-finite world, lies "algebraic-differential functions". The terminology is not yet fixed and similar terms are used, up to a permutation, by several authors: let dz^m be the m-th derivative of F(z), the function is said "algebraic-differential" if there a exists a polynomial P such that P(z,F,F',..., dz^m F)=0. For all these worlds, having some positive (integer) coefficients leads to some strong constraints on the asymptotics (these is now well understood for algebraic function), and we try to see what happens in a more general setting.

We will give examples of such functions (motivated by some combinatorial problems), and show how a symbolic combinatorics approach can help for automatic asymptotics of their coefficients, and some open related open questions/challenges for computer algebra (joint works with Michael Drmota and Hsien-Kuei Hwang)

We live in an era of ubiquitous network-connected sensors. We use cameras, sensors and switches to make buildings smarter, and places more responsive. However, it remains a huge challenge to design, configure and extend a system such that we can gather relevant information from these devices. How do we make devices communicate and work together, in this era of rapidly changing standards and technologies? How do we design a coherent user experience such that these devices do not place additional burden to the users? And do we need a new robotic intelligence manifesto for these smart places? In this talk we discuss the status and challenges in the general field of video surveillance, and our current progress toward answering these questions at Skywatch. These include technologies and services to enable reliable capture of both video and IoT sensor data, and algorithms for extracting sensible information from these unstructured data. We will also provide a hint about future directions we are moving toward to.

In many applications, including option pricing, integrals of $d$-variate functions with ``kinks'' or ``jumps'' are encountered. (Kinks describe simple discontinuities in the derivative, jumps describe simple discontinuities in function values.) The standard analyses of sparse grid or Quasi Monte Carlo methods fail completely in the presence of kinks or jumps, yet the observed performance of these methods can remain reasonable.

In recent joint papers with Michael Griebel and Frances Kuo we sought an explanation by showing that many terms of the ANOVA expansion of a function with kinks can be smooth, because of the smoothing effect of integration. The problem settings have included both the unit cube and $d$-dimensional Euclidean space. The underlying idea is that integration of a non-smooth function with respect to a well chosen variable, say $x_j$, can result in a smooth function of $d-1$ variables.

In still more recent joint work with Andreas Griewank, Hernan Leovey and Frances Kuo we have extended the theoretical results from kinks to jumps, and have turned ``preintegration'' into a practical method for evaluating integrals of some non-smooth functions over $d$-dimensional Euclidean space. In this talk I will explain the concept of the ANOVA decomposition, the reason for smoothness in the ANOVA decomposition, and the method of ``smoothing by preintegration''.

How to recognize and model domain knowledge from data, and how to generate data representations comprehensive and helpful for experts and learners are challenging issues in combining artificial intelligence technology and expertise. Abundant information related to domain knowledge is usually interpreted and utilized in specific manners according to a small amount of observations, and the observation usually contains overlapped patterns and non-steady behaviors. Such tasks are hardly scalable because of the complexity of ground truth and the insufficiency of rigorously labeled datasets. Periodicity is arguably one of the most fundamental features in data streams. In particular, finding and extracting every oscillating component which has non-steady periods and oscillating patterns from multi-component data streams is still an open problem. In this talk, a newly proposed approach for mining non-steady periodicities is introduced. This approach includes the combined frequency and periodicity (CFP) method, the de-shaped short-time Fourier transform (STFT), together with an optimization approach for pattern extraction. The approach is reported useful and efficient in challenging problems including automatic music transcription for music information retrieval (MIR), motion artifact reduction in plethysmograph (PPG) signals, the separation of fetal electrocardiogram (ECG) signals, and other problems in future interdisciplinary technologies.

混沌動態系統和複動態系統是近十幾年來動態系統理論中非常活躍的分支。在此次講座中我將對混沌動態系統和複動態系統的起源和歷史作一簡短的回顧和分析，並對複動態系統中一些已知及待解的重要問題進行介紹和評論。

Segmentation of cells in time-lapse bright-field microscopic images is crucial in understanding cell behaviors for medical research. However, the complex nature of the cells, together with poor contrast, broken cell boundaries and the halo artifact, pose nontrivial challenges to this problem. In this talk, we present two robust mathematical models based on linear algebra techniques to segment bright-field cells automatically. These models treat cell image segmentation as a background subtraction problem, which can be formulated as a robust Principal Component Pursuit (PCP) problem which minimizes the rank of the image matrix. Our first segmentation model is formulated as a PCP with nonnegative constraints. In this approach, we exploit the sparse component of the PCP solution for identifying the cell pixels. However, the sparse component and the nonzero entries can scatter all over the image, resulting in a noisy segmentation. The second model is an improvement of the first model by combining PCP with spectral clustering. Spectral clustering makes use of the eigenvectors of the graph Laplacian matrix to classify data. Seemingly unrelated approaches, we combine the two techniques by incorporating normalized-cut in the PCP as a measure for the quality of the segmentation. Experimental results demonstrate that the proposed models are effective in segmenting cells obtained from bright-field images.