LEAST-SQUARES SPECTRAL COLLOCATION PARALLEL METHODS FOR PARABOLIC PROBLEMS

SEO, JEONG-KWEON,SHIN, BYEONG-CHUN The Honam Mathematical Society 2015 호남수학학술지 Vol.37 No.3

In this paper, we study the first-order system least-squares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

LEAST-SQUARE SWITCHING PROCESS FOR ACCURATE AND EFFICIENT GRADIENT ESTIMATION ON UNSTRUCTURED GRID

SEUNGPYO SEO,CHANGSOO LEE,EUNSA KIM,KYEOL YUNE,CHONGAM KIM 한국산업응용수학회 2020 Journal of the Korean Society for Industrial and A Vol.24 No.1

An accurate and efficient gradient estimation method on unstructured grid is presented by proposing a switching process between two Least-Square methods. Diverse test cases show that the gradient estimation by Least-Square methods exhibit better characteristics compared to Green-Gauss approach. Based on the investigation, switching between the two Least-Square methods, whose merit complements each other, is pursued. The condition number of the Least-Square matrix is adopted as the switching criterion, because it shows clear correlation with the gradient error, and it can be easily calculated from the geometric information of the grid. To illustrate switching process on general grid, condition number is analyzed using stencil vectors and trigonometric relations. Then, the threshold of switching criterion is established. Finally, the capability of Switching Weighted Least-Square method is demonstrated through various two- and three-dimensional applications.

NEGATIVE NORM LEAST-SQUARES SPECTRAL METHODS FOR THE ELLIPTIC PROBLEM

BYEONG CHUN SHIN 한국산업응용수학회 2006 Journal of the Korean Society for Industrial and A Vol.10 No.1

In this paper, we develop and analyze an negative norm first-order system least-squares spectral method for the second-order elliptic boundary value problem. We consider a least-squares functional defined by the sum of the L²- and H<SUP>-1</SUP>-norm of the residual equations. We define a discrete negative norm and then define the discrete negative norm least-squares functional for spectral approximation. The spectral convergence is derived for the proposed method and we provide some numerical results.

Least-squares spectral collocation parallel methods for parabolic problems

서정권,신병춘 호남수학회 2015 호남수학학술지 Vol.37 No.3

In this paper, we study the first-order system least-squares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

비선형 최소자승법의 최적화 방법

김민경,김수환 제어·로봇·시스템학회 2024 제어·로봇·시스템학회 논문지 Vol.30 No.7

. In this paper, we explain how to find the optimal solution for the non-linear least squares problem which minimizes the sum of squared errors. As background knowledge, we discuss matrix calculus, which extends calculus for vector functions, and Taylor series, which represents a function with the sum of an infinite number of polynomials. Subsequently, we explore various iterative optimization methods. First, we explain the gradient method, which approximates a nonlinear loss function as a linear function via the first-order Taylor approximation and iteratively finds the optimal solution. Second, we delve into the Newton method, which approximates the loss function as a quadratic function via the second-order Taylor approximation. Moreover, we examine the Gauss-Newton method for the non-linear least squares problem, which approximates the non-linear error as a linear function and thus the loss function as a quadratic function. Finally we investigate the Levenberg-Marquardt method, which combines the gradient method and the Gauss-Newton method to effectively solve the least squares problem.

Analysis and computations of least-squares method for optimal control problems for the Stokes equations

최영미,김상동,이형천 대한수학회 2009 대한수학회지 Vol.46 No.5

First-order least-squares method of a distributed optimal control problem for the incompressible Stokes equations is considered. An optimality system for the optimal solution are reformulated to the equivalent first-order system by introducing the vorticity and then the least-squares functional corresponding to the system is defined in terms of the sum of the squared H^{-1} and L^2 norms of the residual equations of the system. Finite element approximations are studied and optimal error estimates are obtained. Resulting linear system of the optimality system is symmetric and positive definite. The V-cycle multigrid method is applied to the system to test computational efficiency. First-order least-squares method of a distributed optimal control problem for the incompressible Stokes equations is considered. An optimality system for the optimal solution are reformulated to the equivalent first-order system by introducing the vorticity and then the least-squares functional corresponding to the system is defined in terms of the sum of the squared H^{-1} and L^2 norms of the residual equations of the system. Finite element approximations are studied and optimal error estimates are obtained. Resulting linear system of the optimality system is symmetric and positive definite. The V-cycle multigrid method is applied to the system to test computational efficiency.

Development of A Robust and Accurate Meshless Method for 2-D Compressible Flow

Jin Young Huh,Jae Sang Rhee,Jaeyun Byeon,Sungjun Noh,Kyu Hong Kim,Suk Young Jung 한국전산유체공학회 2014 한국전산유체공학회 학술대회논문집 Vol.2014 No.10

A new Meshless method is developed to solve 2-D compressible flow robustly and accurately. By using the method of Lagrange multiplier, least square method is applied with constraints which can remove excessive numerical oscillation. The modified least square method can improve robustness and accuracy of the Meshless method, compared with the original least square method, especially when points are not distributed evenly. Numerical analyses of hypersonic flow over a blunt body with strong shock were carried out using the developed Meshless method, then robustness, accuracy and convergence of their results were compared with those obtained from the original Least Square method.

A split least-squares characteristic mixed finite element method for the convection dominated Sobolev equations

엄미례,Jun Yong Shin 한국전산응용수학회 2016 Journal of applied mathematics & informatics Vol.34 No.1

In this paper, we present a split least-squares characteristic mixed finite element method(MFEM) to get the approximate solutions of the convection dominated Sobolev equations. First, to manage both convection term and time derivative term efficiently, we apply a least-squares characteristic MFEM to get the system of equations in the primal unknown and the flux unknown. Then, we obtain a split least-squares characteristic MFEM to convert the coupled system in two unknowns derived from the least-squares characteristic MFEM into two uncoupled systems in the unknowns. We theoretically prove that the approximations constructed by the split least-squares characteristic MFEM converge with the optimal order in $L^2$ and $H^1$ normed spaces for the primal unknown and with the optimal order in $L^2$ normed space for the flux unknown. And we provide some numerical results to confirm the validity of our theoretical results.

감쇠 최소자승법을 이용한 지표 탄성파 탐사자료의 전파형 역산

변중무,주용환,설순지 한국자원공학회 2009 한국자원공학회지 Vol.46 No.3

The gradient method has been used for full waveform inversion more often than the least-squares method that requires the huge computational memory in solving matrix equation. However, since misfits of the gradient method show an oscillating tendency and updated velocities converge to true values very slowly, the gradient method requires many iterations. In this study, we have presented an inversion algorithm to find the subsurface velocity structure using the damped least-squares method in the frequency domain. To verify our waveform inversion scheme, we applied the algorithm to the synthetic data, and found that comparatively correct solutions were acquired even with a few iterations. To reduce computer memory requirements and computing time, we utilized inversion blocks consisting of 2×2 elements, which are larger than that of forward modelling, and parallel processing technique based on MPI (Message Passing Interface). For investigating the effects of the initial model on inversion results, we compared and analyzed inversion results for two initial velocity models. One is a gradually increasing velocity model and the other is a smoothed horizontally layered velocity model. The latter was constructed based on the assumption that we have horizontally layered velocity information from zero-offset VSP or well logging data. The inversion result obtained by using the smoothed horizontally layered velocity model shows interfaces of subsurface velocity structures more clearly and provides more accurate velocity values than those using the gradually increasing velocity model. In addition, since the damped least-squares method calculates the sensitivity matrix unlike the steepest-descent method, we can identify the influence of frequency range on inversion results by comparing the sensitivity matrices for a couple of frequencies. 전파형을 이용한 역산에서는 행렬식의 계산을 위해 대용량의 메모리를 요구하는 최소자승법보다는 대부분 경사법이 사용되어왔다. 하지만 경사법은 잔차가 진동하는 경향을 보이며 수렴 속도가 느려 상대적으로 많은 연산횟수를 필요로 하는 단점을 가진다. 본 연구에서는 주파수 영역에서 감쇠 최소자승법을 사용하여 지하의 속도모델을 효과적으로 찾아가는 역산 알고리듬을 구현하고, 이 알고리듬의 타당성 및 적용성 검토를 위해 수치모형 실험을 수행하여 적은 반복 연산 횟수에도 비교적 정확한 해를 찾아가는 것을 확인하였다. 감쇠 최소자승법의 메모리 문제를 해결하기 위하여 모델링에 이용된 셀의 2 × 2 크기로 역산블록을 설정하고 계산시간의 단축을 위하여 병렬처리 프로그램을 이용하였다. 또한 초기모델의 선정이 역산에 미치는 영향을 확인하기 위해 심도에 따라 속도가 증가하는 초기모델과 제로-오프셋 VSP나 음파검층 자료가 있을 때 이로부터 획득한 구간속도를 평활화한 초기모델을 사용하여 역산한 결과를 비교·분석하였다. 그 결과, VSP 자료나 음파검층 등의 자료를 초기속도모델로 사용한 경우 지하 속도구조의 층 경계들을 잘 영상화하고 보다 정확한 속도 정보를 제공함을 확인할 수 있었다. 또한 감쇠 최소자승법의 경우 최대경사법과 달리 민감도 행렬을 계산하기 때문에 이러한 정보를 이용하여 주파수에 따른 영향범위를 확인할 수 있다.

A SPLIT LEAST-SQUARES CHARACTERISTIC MIXED FINITE ELEMENT METHOD FOR THE CONVECTION DOMINATED SOBOLEV EQUATIONS

OHM, MI RAY,SHIN, JUN YONG The Korean Society for Computational and Applied M 2016 Journal of applied mathematics & informatics Vol.34 No.1

In this paper, we present a split least-squares characteristic mixed finite element method(MFEM) to get the approximate solutions of the convection dominated Sobolev equations. First, to manage both convection term and time derivative term efficiently, we apply a least-squares characteristic MFEM to get the system of equations in the primal unknown and the flux unknown. Then, we obtain a split least-squares characteristic MFEM to convert the coupled system in two unknowns derived from the least-squares characteristic MFEM into two uncoupled systems in the unknowns. We theoretically prove that the approximations constructed by the split least-squares characteristic MFEM converge with the optimal order in L² and H¹ normed spaces for the primal unknown and with the optimal order in L² normed space for the flux unknown. And we provide some numerical results to confirm the validity of our theoretical results.