2021 EIMS International Conference on Computational Mathematics

Abstracts

• George Exner

Conditional Positive Definiteness Bridging k-hyponormality and n-contractivity

It is well known that there are two standard routes to subnormality, that arising from k-hyponormality for all k and that arising (for contractive operators) from n-contractivity for all n. While these both arrive at subnormality, the connections between these two points of view for finite k and n has remained unclear for some time. By considering the conditional positive definiteness of moment matrices for weighted shifts, we find a bridge which shows these two points of view are extremes of a common family of conditions. (Joint work with Chafiq Benhid and Raul Curto)

• Hyun-Kyoung Kwon

Similarity of Cowen-Douglas Operator Tuples

We present a similarity result that holds for Cowen-Douglas operators in the multivariable setting. Interestingly, one can extend the previous known results concerning a single Cowen-Douglas operator to Cowen-Douglas operator tuples. A generalization of G. Misra’s result of 1984 on curvature inequality is also given. This talk is based on joint work with Kui Ji, Shanshan Ji, and Jing Xu.

• Jongrak Lee

Hyponormal Toeplitz operators with non-harmonic symbols

In this talk, we present some necessary and sufficient conditions for the hyponormality of Toeplitz operator. In particular, we consider the Toeplitz operator with non-harmonic symbols under certain assumptions.

• Ji Eun Lee

Properties of Toeplitz operators on Newton spac

In this paper, we study Newton space N^2 (P) which has Newton polynomials as an orthonormal basis. We first investigate some relations between the orthonormal basis {z^n} of the Hardy space H^2 (D) and the orthonormal basis {N_n} of Newton space N^2 (P). Moreover, we consider the properties of Toeplitz operators on Newton space N^2 (P).

• Norikazu Saito

Analysis of the finite element method for the Stokes problems with singular source terms

We consider an interface problem for the Stokes equations. We introduce two formulations for interpreting the interface condition: the first one is based on an additional singular force term using the delta function and the other one is based on the additional term using the characteristic function. The finite element method is applied after introducing regularizations of singular source terms. Consequently, the error is divided into the regularization and discretization parts which are studied separately. We derive error estimates for the finite element approximation. Finally, we compare these two approaches.

• Lei Zhang

Solution Landscape of Nematic Liquid Crystals

Topological defect plays an important role in the physics of liquid crystals. Although a large amount of previous studies is devoted to compute the stable defect structures in liquid crystals as a consequence of geometric frustration, how do we search for the entire family tree of all possible solutions without unwanted random guesses? Here we introduce a novel concept of “solution landscape”, which is a pathway map consisting of all stationary points and their connections. We then present an efficient numerical algorithm to construct the solution landscape by combining the saddle dynamics and the downward/upward algorithm. As illustration, we solve the Landau-de Gennes energy to construct the defect landscapes of confined nematic liquid crystals.

• Jaemin Shin

Energy stable methods for the phase-field crystal equations

We introduce Runge--Kutta methods to solve the phase-field crystal models in an unconditionally energy stable manner. For the phase-field crystal equation, the convex splitting Runge--Kutta method is available to higher-order accuracy in time. And, it can provide unconditional energy stability as well as unique solvability. Recently, the energy quadratization Runge--Kutta method has been proposed as the high-order energy stable scheme. We discuss the comparison between the two methods. In addition, we introduce the extension of the energy quadratization Runge--Kutta method for the modified phase-field crystal equation.

• Byungjoon Lee

Optimal preconditioners on Solving the Poisson equation with Neumann boundary conditions

MILU preconditioner is well known to be the optimal choice among all the ILU-type preconditioners in solving the Poisson equation with Dirichlet boundary conditions. However, it is less known which is an optimal preconditioner in solving the Poisson equation with Neumann boundary conditions. The condition number of an unpreconditioned matrix is as large as O(h^(-2)) , where h is the step size of grid. Only the optimal preconditioner results in condition number O(h^(-1)) , while the others such as Jacobi and ILU result in O(h^(-2)) . We review Relaxed ILU and Perturbed MILU preconditioners in the case of Neumann boundary conditions, and present empirical results which indicate that the former is optimal in two dimensions and the latter is optimal in two and three dimensions. To the best of our knowledge, these empirical results have not been rigorously verified yet. We present a formal proof for the optimality of Relaxed ILU in rectangular domains, and discuss its possible extension to general smooth domains and Perturbed MILU.

• Ernest K. Ryu

WGAN with an Infinitely Wide Generator Has No Spurious Stationary Points

Generative adversarial networks (GAN) are a widely used class of deep generative models, but their minimax training dynamics are not understood very well. In this work, we show that GANs with a 2-layer infinite-width generator and a 2-layer finite-width discriminator trained with stochastic gradient ascent-descent have no spurious stationary points. We then show that when the width of the generator is finite but wide, there are no spurious stationary points within a ball whose radius becomes arbitrarily large (to cover the entire parameter space) as the width goes to infinity.

• Feihu Huang

Stochastic Alternating Direction Method of Multipliers for Nonconvex Optimization

In this talk, I will first introduce the basic mini-batch stochastic Alternating Direction Method of Multipliers (ADMM) for solving nonconvex-nonsmooth composite problems. Then I will further introduce a class of accelerated stochastic ADMM methods by using the variance-reduced techniques of SAG, SAGA, SVRG and SPIDER/SARAH. Moreover, I will demonstrate the convergence properties of these ADMM methods for finding stationary points. Finally, I will show the performances of these methods in some machine learning applications.

• Steven T. Dougherty

The Foundations of Algebraic Coding Theory over Frobenius Rings

We shall describe the foundational results for coding theory over Frobenius rings giving special attention to the MacWilliams relations and the generating character of the ring. We shall describe families of codes, their structures and various alphabets of interest. We shall also give open areas of research.

• Qin Yue

Binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

Goppa codes are particularly appealing for cryptographic applications. Every improvement of our knowledge of Goppa codes is of particular interest. In this paper, we present a sufficient and necessary condition for an irreducible monic polynomial $g(x)$ of degree $r$ over $\mathbb{F}_{q}$ satisfying $\gamma g(x)=(x+d)^rg({A}(x))$, where $q=2^n$, $A=\left(\begin{array}{cc} a&b\\1&d\end{array}\right)\in PGL_2(\Bbb F_{q})$, $\mathrm{ord}(A)$ is a prime, $g(a)\ne 0$, and $0\ne \gamma\in \Bbb F_q$. In some conditions, we give a complete characterization of irreducible polynomials $g(x)$ as above. Moreover, we construct some binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes.

• Gretchen Matthews

Reed-Solomon projections and fractional decoding of codes from the Hermitian curve

Fractional decoding is a method by which an original codeword may be obtained from a received word using only a proportion of symbols of the received word, provided not too many errors have occurred. In this talk, we make use of Reed-Solomon projections of codes from the Hermitian curve and apply a method of W. Santos to achieve fractional decoding for these codes. (This is joint work with A. Murphy and W. Santos.)

• Hyun Kwang Kim

Classification of weighted posets and digraphs admitting the extended Hamming code to be a perfect code

Recently T. Etzion, M. Firer and R. Machado introduced metrics on F2n based on directed graphs on n vertices and developed some basic coding theory on directed graph metric spaces. In this talk, we consider the problem of classifying weighted posets and directed graphs which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between weighted poset metrics and directed graph based metrics. In the next, we classify weighted posets and directed graphs which admit the extended Hamming code of length 8 or 16 to be a 2-perfect code.

• Denis Krotov

Completely regular codes in halved hypercubes

A set of vertices of a graph is called a completely regular code if the distance partition with respect to it is equitable, that is, each cell of the partition induces a regular subgraph and the edges between any two cells form a biregular subgraph. The halved hypercube is the graph on the even-weight vertices of the binary Hamming space, where two vertices are adjacent if the Hamming distance between them is 2. We consider constructions of completely regular codes in halved hypercubes, mainly focused on codes with covering radius 2 (in the graph metrics).

• Minjia Shi

On isodual double Toeplitz codes

Double Toeplitz (shortly DT) codes are introduced here as a generalization of double circulant codes. We show that such a code is isodual, hence formally self-dual. Self-dual DT codes are characterized as double circulant or double negacirculant. Likewise, even DT binary codes are characterized as double circulants. Numerical examples obtained by exhaustive search show that the codes constructed have best-known minimum distance, up to one unit, amongst formally self-dual codes, and sometimes improve on the known values. Over F4 an explicit construction of DT codes, based on quadratic residues in a prime field, performs equally well. We show that DT codes are asymptotically good over Fq. Speci_cally, we construct DT codes arbitrarily close to the asymptotic varshamov-Gilbert bound for codes of rate one half.

• Jong Yoon Hyun

Optimal linear codes from simplicial complexes

A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes C∆c constructed from simplicial complexes in F n 2 , where ∆ is a simplicial complex in F n 2 and ∆c the complement of ∆. We first find an explicit computable criterion for C∆c to be optimal; this criterion is given in terms of the 2-adic valuation of Ps i=1 2 |Ai|−1 , where the Ai’s are maximal elements of ∆. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of ∆. In particular, we find that C∆c is a Griesmer code if and only if the maximal elements of ∆ are pairwise disjoint and their sizes are all distinct. Specially, when F has exactly two maximal elements, we explicitly determine the weight distribution of C∆c . This is a joint work with Yoonjin Lee (Ewha Womans University) and Jungyun Lee (Kangwon National University).

• Lingfei Jin

Construction of maximally recoverable local reconstruction codes

Local Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. They have emerged as the codes of choice for large scale distributed storage systems due to the very efficient repair of failed storage nodes in the typical scenario of a single or few nodes failing, while also offering fault tolerance against worst-case scenarios with more erasures. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. In this talk, we introduce an approach to construct MR LRCs.

• Hyun Jin Kim

Quasi-cyclic Self-dual Codes over Finite Fields using three factors

We study l-quasi-cyclic self-dual codes of length ml over a finite field F_q, provided that the polynomial X^m-1 has exactly three distinct irreducible factors in F_q [X], where F_q is the finite field of order q. In this case, there are two types of the ring R=F_q [X]/(X^m-1). We show that every self-dual code over the ring R of the first type with length ≥6 has free rank ≥2. This implies that we can obtain all self-dual codes over R of the first type. There exists a self-dual code of free rank ≤1 over the ring R of the second type. This is a joint work with Yoonjin Lee (Ewha Womans University).

• Jeong Rye Park

On Q-integral graphs with Q-spectral radius 6

$Q$-integral graphs are graphs whose signless Laplacians have only integral eigenvalues. The edge-degree of an edge is the number of edges adjacent to the edge. In [1], S. K. Simić and Z. Stanić studied connected $Q$-integral graphs with edge-degrees at most 5. In [2], J. Y. Park and Y. Sano proved that a connected $Q$-integral graph with $Q$-spectra radius 6 is known or has maximum edge-degree 5, or is a bipartite graph containing $S_{3,3}$ as an induced subgraph, where $S_{3,3}$ is the double star which is obtained from two disjoint claws $K_{1,3}$ by adding one edge between the two vertices of degree 3. We will slightly improve this result by showing that there is no connected $Q$-integral bipartite graph with $Q$-spectral radius 6 containing $S_{3,3}$ as an induced subgraph. This is joint work with Prof. Jongyook Park, Prof. Yoshio Sano and Dr. Semin Oh.

[1] Simić, S. K., & Stanić, Z. (2008). Q-integral graphs with edge-degrees at most five. Discrete Mathematics, 308(20), 4625–4634. https://doi.org/10.1016/j.disc.2007.08.055

[2] Park, J. Y., & Sano, Y. (2019). On Q-integral graphs with edge-degrees at most six. Linear Algebra and Its Applications, 577, 384–411. https://doi.org/10.1016/j.laa.2019.04.015

• Whan-Hyuk Choi

Construction of DNA codes for DNA computing and DNA data-storage

At first, we present an explicit method for designing DNA codes from reversible self-dual codes over the finite field GF(4). We obtain an efficient and feasible algorithm for designing DNA codes from RSD codes over GF(4). Using this algorithm, we construct many new DNA codes with better parameters compared to the previously known results, and improve the lower bounds on the maximum size of DNA codes of a fixed code length and dimension.

Secondly, we present a method for constructing DNA codes with single-deletion-correcting capability. We present an explicit algorithm for the construction of the q-ary single-deletion-correcting codes using a class of the complementary information set codes. Applying our algorithm to the generated DNA codes with appropriate modification, we obtain the DNA codes with single- deletion-correcting capability. We present various examples of such DNA codes, and lower bounds on the maximum size of the single-deletion-correcting DNA codes. This is a joint work with Hyun Jin Kim and Yoonjin Lee (Ewha Womans University).

• Jae-kwang Kim

Statistical Inference with Neural Network Imputation for Item Nonresponse

We consider the problem of nonparametric imputation using a neural network model. Neural network models can capture complex nonlinear trends and interaction effects, making it a powerful tool for predicting missing values under minimum assumptions on the missingness mechanism. Statistical inference with neural network imputation, including variance estimation, is challenging because the basis for function estimation is estimated rather than known. In this talk, we tackle the problem of statistical inference with neural network imputation by treating the hidden nodes in a neural network as data-driven basis functions. We show that the uncertainty in estimating the basis functions can be safely ignored and hence the linearisation method for neural network imputation can be greatly simplified.

• Eun Sug Park

Unified Bayesian Multivariate Receptor Modeling by Using Markov chain Monte Carlo

For the development of effective air pollution control strategies, it is crucial to identify major sources and quantify their impacts on air pollution. Multivariate receptor modeling aims to address these problems by unfolding ambient concentrations of multiple air pollutants into components associated with different source types. Estimation of source profiles and contributions by multivariate receptor modeling is hindered, however, due to model uncertainty caused by the unknown number of major contributing sources and model identification conditions. Bayesian multivariate receptor modeling based on Markov chain Monte Carlo methods is an attractive approach as it offers a great deal of flexibility in modeling and the capability to estimate both parameter uncertainty and model uncertainty. With the expanded monitoring efforts established over the past decades, multipollutant data from multiple monitoring sites are now available, enabling the prediction of source contributions at unmonitored locations by modeling spatial correlation in those multipollutant spatial data. We present spatially enhanced Bayesian multivariate receptor modeling that can predict unobserved source contributions along with their uncertainty estimates at any location while simultaneously dealing with model uncertainty. The proposed method is illustrated with real multipollutant data obtained from multiple monitoring sites.

• Soyoung Kim

Improving the efficiency of estimation in the stratified Cox model for case-cohort studies with multiple diseases

The case-cohort study design is widely used to reduce cost when collecting expensive covariates in large cohort studies with survival outcomes, consists of a random sample of the full cohort, called the subcohort and all cases of failures from a specific cause of interest. The key advantage of case-cohort design is that the same subcohort can be used when studying multiple diseases. In case-cohort studies, an inverse probability weighted estimating equation approach has been used commonly but it is inefficient. In this paper, we propose the augmented estimators in the stratified Cox model for case-cohort studies with multiple diseases to improve parameter estimation efficiency, which is based on the doubly robust estimation. The proposed estimators are shown to be consistent and asymptotically normal. Simulation studies show that the proposed augmented estimators using the full use of all covariate information lead to gains in efficiency. We apply our proposed method to data from the Atherosclerosis Risk in Communities study.

• Soyoung Park

An algorithm to compare two-dimensional footwear outsole images using maximum cliques and speeded-up robust feature

Footwear examiners are tasked with comparing an outsole impression (Q) left at a crime scene with an impression (K) from a database or from the suspect's shoe. We propose a method for comparing two shoe outsole impressions, that relies on robust features (SURF) on each impression and aligns them using a maximum clique (MC). After alignment, an algorithm we denote MC-COMP is used to extract additional features that are then combined into a univariate similarity score using a random forest (RF). We use a database of shoe outsole impressions that includes images from two models of athletic shoes that were purchased new and then worn by study participants for about six months. The shoes share class characteristics such as outsole pattern and size, and thus the comparison is challenging. We find that the RF implemented on SURF outperforms other methods recently proposed in the literature in terms of classification precision. In more realistic scenarios where crime scene impressions may be degraded and smudged, the algorithm we propose -- denoted MC-COMP-SURF -- shows the best classification performance by detecting unique features better than other methods. The algorithm can be implemented with the R package shoeprintr.

• Eun Jeong Min

Sparse multiple co-inertia analysis for integrative analysis of multiple -omics data

Multiple co-inertia analysis (mCIA) is a multivariate analysis method that can assess relationships and trends in multiple datasets. Recently it has been used for integrative analysis of multiple high-dimensional -omics datasets. However, its estimated loading vectors are non-sparse, which presents challenges for identifying important features and interpreting analysis results. We propose two new mCIA methods: 1) a sparse mCIA method that produces sparse loading estimates and 2) a structured sparse mCIA method that further enables incorporation of structural information among variables such as those from functional genomics. Our extensive simulation studies demonstrate the superior performance of the sparse mCIA and structured sparse mCIA methods compared to the existing mCIA in terms of feature selection and estimation accuracy. Application to the integrative analysis of transcriptomics data and proteomics data from a cancer study identified biomarkers that are suggested in the literature related with cancer disease.

• Jungsoon Choi

Bayesian spatially-dependent clustering model in regression coefficients for space-time data

The effects of explanatory variable on response variable in space-time data may vary across space, and their effects may have spatial dependency structure. In addition, the regression coefficient value within spatial cluster may be a constant, but the values across spatial clusters may vary. In this work, we will propose a new Bayesian hierarchical spatially-dependent cluster modeling within the regression framework for space-time data. We will investigate the proposed models by using the range of simulation studies. Finally, we will conduct the real COVID-19 data analysis in USA to estimate the spatially-clustered effects of risk factor of interest on COVID-19 death cases.

• Rosy Oh

Similarity Search on Wafer Bin Map through Nonparametric and Hierarchical Clustering

Searching for and comparing similar wafer maps can provide crucial information for root cause analysis in the manufacturing process of integrated circuits. Owing to the high dimensionality and complexity of defect patterns, comparison of similar maps in their entirety is inefficient. This paper proposes an automated similarity ranking system with a novel feature set as a reduced representation of wafer maps. We use nonparametric Bayesian clustering based on the Dirichlet process Gaussian mixture model and hierarchical clustering to detect systematic failure patterns across wafer maps. The proposed features are efficient because they require minimal computation and storage; furthermore, they allow for highly discriminative rankings of similar failure patterns. Thus, they are suitable for large-scale analysis of wafer maps. The proposed method is experimentally verified using a real wafer map dataset from a semiconductor manufacturing company.

• Francisco German Badia Blasco

Log concavity preservation by beta operator based on probability tools

Beta-type operators are probabilistic approximation operators based on beta or inverted beta distributions. The aim of this paper is to study the preservation of log concavity by beta operator. A positive answer is given in the case that the function is monotone. Proofs are based on coupling probabilistic tools and the bivariate characterization of likelihood stochastic order.

• Inma T. Castro

Condition-based maintenance for systems subject to multiple degradation processes

Condition-Based Maintenance (CBM) is a maintenance strategy based on monitoring the operating condition of a system. Compared to time-based maintenance, and thanks to the development of sensor technologies, CBM usually results in lower maintenance costs, avoiding unnecessary preventive maintenance activities and reducing unexpected failures. Since industrial systems are becoming increasingly more complex and they are likely to suffer from multiple degradation processes, further research is needed in modelling the maintenance when a system possesses more than one degradation path. In this talk, we focus on systems subject to different degradation processes. These degradation processes initiate at random times and then grow depending on the environment and conditions of the system. Hence, two stochastic processes have to be combined: the initiation process and the growth process. Maintenance strategies and the analysis of the reliability of these systems is analyzed in this talk.

• Maxim Finkelstein

Remaining Lifetimes with Random Initial Age in Reliability and Population Dynamics

We revisit the “life lived equals life left” property for stationary populations and discuss it from a more general perspective in reliability and population dynamics contexts. Specifically, we show that identically distributed random age and the remaining lifetime in stationary populations have the same distribution as the equilibrium distribution of the renewal theory. Then we consider specific non-stationary populations that are closed to migration and have a constant birth rate. We obtain some useful inequalities between random age and remaining lifetime for different instants of calendar time. We also focus on relevant aging properties of populations using different stochastic orders. Some examples and general discussion conclude the presentation.