Dear colleagues and researchers,

The Journal of the Korean Society for Industrial and Applied Mathematics (J-KSIAM) Volume 25 Number 4 (December 2021 issue) has been posed on KSIAM Journal Archive or other information on the journal is available on the KSIAM website http://www.ksiam.org or https://ksiam.org/journal/journal01.asp.

The journal is one of Korea Citation Indexed (KCI) journals since 2007 and is indexed in Emerging Sources Citation Index (ESCI) since 2017.

Readers interested in the following articles may download each of articles free of charge from our website and authors are encouraged to submit a paper via the online submission site.

Sincerely yours,

Hi Jun Choe, Editor-in-Chief

Zhiming Chen, Hyeong-Ohk Bae, Tao Tang, Associate Editors-In-Chief

Min Seok Choi, Jae Hoon Jung, Gun Jin Yun, Managing Editors

------------------------------------------------------------------------------------------------

JKSIAM-v25n5 pp149-161

**Markov Decision Process-based Potential Field Technique for UAV Planning**

CHAEHWAN MOON AND JAEMYUNG AHN

This study proposes a methodology for mission/path planning of an unmanned aerial vehicle (UAV) using an artificial potential field with the Markov Decision Process (MDP). The planning problem is formulated as an MDP. A low-resolution solution of the MDP is obtained and used to define an artificial potential field, which provides a continuous UAV mission plan. A numerical case study is conducted to demonstrate the validity of the proposed technique.

JKSIAM-v25n5 pp162-172

**Stochastic Gradient Methods for L**^{2}-Wasserstein Least Squares Problem of Gaussian Measures

SANGWOON YUN, XIANG SUN AND JUNG-IL CHOI

This paper proposes stochastic methods to find an approximate solution for the L2-Wasserstein least squares problem of Gaussian measures. The variable for the problem is in a set of positive definite matrices. The first proposed stochastic method is a type of classical stochastic gradient methods combined with projection and the second one is a type of variance reduced methods with projection. Their global convergence are analyzed by using the framework of proximal stochastic gradient methods. The convergence of the classical stochastic gradient method combined with projection is established by using diminishing learning rate rule in which the learning rate decreases as the epoch increases but that of the variance reduced method with projection can be established by using constant learning rate. The numerical results show that the present algorithms with a proper learning rate outperforms a gradient projection method.

JKSIAM-v25n5 pp173-195

**Discontinuous Galerkin Spectral Element Method for Elliptic Problems Based on First-Order Hyperbolic System**

DEOKHUN KIM AND HYUNG TAEK AHN

A new implicit discontinuous Galerkin spectral element method (DGSEM) based on the first order hyperbolic system(FOHS) is presented for solving elliptic type partial different equations, such as the Poisson problems. By utilizing the idea of hyperbolic formulation of Nishikawa[1], the original Poisson equation was reformulated in the first-order hyperbolic system. Such hyperbolic system is solved implicitly by the collocation type DGSEM. The steady state solution in pseudotime, which is the solution of the original Poisson problem, was obtained by the implicit solution of the global linear system. The optimal polynomial orders of 𝒪(𝒽𝑝+1)) are obtained for both the solution and gradient variables from the test cases in 1D and 2D regular grids. Spectral accuracy of the solution and gradient variables are confirmed from all test cases of using the uniform grids in 2D.

JKSIAM-v25n5 pp196-223

**Novel Geometric Parameterization Scheme for The Certified Reduced Basis Analysis of a Square Unit Cell**

SON HAI LE, SHINSEONG KANG, TRIET MINH PHAM AND KYUNGHOON LEE

This study formulates a new geometric parameterization scheme to effectively address numerical analysis subject to the variation of the fiber radius of a square unit cell. In particular, the proposed mesh-morphing approach may lead to a parameterized weak form whose bilinear and linear forms are affine in the geometric parameter of interest, i.e. the fiber radius. As a result, we may certify the reduced basis analysis of a square unit cell model for any parameters in a predetermined parameter domain with a rigorous a posteriori error bound. To demonstrate the utility of the proposed geometric parameterization, we consider a two-dimensional, steadystate heat conduction analysis dependent on two parameters: a fiber radius and a thermal conductivity. For rapid yet rigorous a posteriori error evaluation, we estimate a lower bound of a coercivity constant via the min-θ method as well as the successive constraint method. Compared to the corresponding finite element analysis, the constructed reduced basis analysis may yield nearly the same solution at a computational speed about 29 times faster on average. In conclusion, the proposed geometric parameterization scheme is conducive for accurate yet efficient reduced basis analysis.

JKSIAM-v25n5 pp224-261

**Implicit-Explicit Second Derivative LMM For Stiff Ordinary Differential Equations**

SEUN EVANS OGUNFEYITIMI AND MONDAY NDIDI IKHILE

The interest in implicit-explicit (*IMEX*) integration methods has emerged as an alternative for dealing in a computationally cost-effective way with stiff ordinary differential equations arising from practical modeling problems. In this paper, we introduce implicitexplicit second derivative linear multi-step methods (*IMEX *SDLMM) with error control. The proposed *IMEX *SDLMM is based on second derivative backward differentiation formulas (SDBDF) and recursive SDBDF. The *IMEX *second derivative schemes are constructed with order *p *ranging from *p *= 1 to 8. The methods are numerically validated on well-known stiff equations.

JKSIAM-v25n5 pp262-295

**FSAL Mono-Implicit Nordsieck General Linear Methods with Inherent Runge-Kutta Stability for DAEs**

PETER OLATUNJI AND MONDAY IKHILE

This paper introduces mono-implicit general linear methods, a special class of general linear methods, which are implicit in the output solution for the numerical integration of differential algebraic equations. We show how L-stable inherent Runge-Kutta members can be derived. The procedures for implementation have been discussed. The numerical test on the problem considered shows that the methods have improved accuracy when compared to RADAU IIA and the results from MATLAB ode15s, which have been taken as our reference solution.

JKSIAM-v25n5 pp296-311

**Instability of the Betti Sequence for Persistent Homology and A Stabilized Version of The Betti Sequence**

MEGAN JOHNSON AND JAE-HUN JUNG

Topological Data Analysis (TDA), a relatively new field of data analysis, has proved very useful in a variety of applications. The main persistence tool from TDA is persistent homology in which data structure is examined at many scales. Representations of persistent homology include persistence barcodes and persistence diagrams, both of which are not straightforward to reconcile with traditional machine learning algorithms as they are sets of intervals or multisets. The problem of faithfully representing barcodes and persistent diagrams has been pursued along two main avenues: kernel methods and vectorizations. One vectorization is the Betti sequence, or Betti curve, derived from the persistence barcode. While the Betti sequence has been used in classification problems in various applications, to our knowledge, the stability of the sequence has never before been discussed. In this paper we show that the Betti sequence is unstable under the 1-Wasserstein metric with regards to small perturbations in the barcode from which it is calculated. In addition, we propose a novel stabilized version of the Betti sequence based on the Gaussian smoothing seen in the Stable Persistence Bag of Words for persistent homology. We then introduce the normalized cumulative Betti sequence and provide numerical examples that support the main statement of the paper.

JKSIAM-v25n5 pp312-326

**A Modified Prey-Predator Model with Coupled Rates of Change**

HYEJI HAN, GWANGIL KIM, AND SEOYOUNG OH

The prey–predator model is one of the most influential mathematical models in ecology and evolutionary biology. In this study, we considered a modified prey–predator model, which describes the rate of change for each species. The effects of modifications to the classical prey–predator model are investigated here. The conditions required for the existence of the first integral and the stability of the fixed points are studied. In particular, it is shown that the first integral exists only for a subset of the model parameters, and the phase portraits around the fixed points exhibit physically relevant phenomena over a wide range of the parameter space. The results show that adding coupling terms to the classical model widely expands the dynamics with great potential for applicability in real-world phenomena.

JKSIAM-v25n5 pp327-336

**A Study on Pupil Detection and Tracking Methods Based on Image Data Analysis**

HANA CHOI, MINJUNG GIM AND SANGWON YOON

In this paper, we will introduce the image processing methods for the remote pupillary light reflex measurement using the video taken by a general smartphone camera without a special device such as an infrared camera. We propose an algorithm for estimate the size of the pupil that changes with light using image data analysis without a learning process. In addition, we will introduce the results of visualizing the change in the pupil size by removing noise from the recorded data of the pupil size measured for each frame of the video. We expect that this study will contribute to the construction of an objective indicator for remote pupillary light reflex measurement in the situation where non-face-to-face communication has become common due to COVID-19 and the demand for remote diagnosis is increasing.