• Title, Summary, Keyword: Global Error

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ON EXACT CONVERGENCE RATE OF STRONG NUMERICAL SCHEMES FOR STOCHASTIC DIFFERENTIAL EQUATIONS

  • Nam, Dou-Gu
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.1
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    • pp.125-130
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    • 2007
  • We propose a simple and intuitive method to derive the exact convergence rate of global $L_{2}-norm$ error for strong numerical approximation of stochastic differential equations the result of which has been reported by Hofmann and $M{\"u}ller-Gronbach\;(2004)$. We conclude that any strong numerical scheme of order ${\gamma}\;>\;1/2$ has the same optimal convergence rate for this error. The method clearly reveals the structure of global $L_{2}-norm$ error and is similarly applicable for evaluating the convergence rate of global uniform approximations.

Finite Element Analysis and Local a Posteriori Error Estimates for Problems of Flow through Porous Media (다공매체를 통과하는 유동문제의 유한요소해석과 부분해석후 오차계산)

  • Lee, Choon-Yeol
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.21 no.5
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    • pp.850-858
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    • 1997
  • A new a posteriori error estimator is introduced and applied to variational inequalities occurring in problems of flow through porous media. In order to construct element-wise a posteriori error estimates the global error is localized by a special mixed formulation in which continuity conditions at interfaces are treated as constraints. This approach leads to error indicators which provide rigorous upper bounds of the element errors. A discussion of a compatibility condition for the well-posedness of the local error analysis problem is given. Two numerical examples are solved to check the compatibility of the local problems and convergence of the effectivity index both in a local and a global sense with respect to local refinements.

A Long-term Accuracy Analysis of the GPS Klobuchar Ionosphere Model (GPS Klobuchar 전리층 모델의 장기간 정확도 분석)

  • Kim, Mingyu;Kim, Jeongrae
    • Journal of the Korean Society for Aviation and Aeronautics
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    • v.24 no.2
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    • pp.11-18
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    • 2016
  • Global Positioning System (GPS) is currently widely used for aviation applications. Single-frequency GPS receivers are highly affected by the ionospheric delay error, and the ionospheric delay should be corrected for accurate positioning. Single-frequency GPS receivers use the Klobuchar model, whose model parameters are transmitted from GPS satellites. In this paper, the long-term accuracy of the Klobuchar model from 2002 to 2014 is analyzed. The IGS global ionosphere map is considered as true ionospheric delay, and hourly, seasonal, and geographical error variations are analyzed. Histogram of the ionospheric delay error is also analyzed. The influence of solar and geomagnetic activity on the Klobuchar model error is analyzed, and the Klobuchar model error is highly correlated with solar activity. The results show that the Klobuchar model estimates 8 total electron content unit (TECU) over the true ionosphere delay in average. The Klobuchar model error is greater than 12 TECU within $20^{\circ}$ latitude, and the error is less than 6 TECU at high latitude.

HIERARCHICAL ERROR ESTIMATORS FOR LOWEST-ORDER MIXED FINITE ELEMENT METHODS

  • Kim, Kwang-Yeon
    • Korean Journal of Mathematics
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    • v.22 no.3
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    • pp.429-441
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    • 2014
  • In this work we study two a posteriori error estimators of hierarchical type for lowest-order mixed finite element methods. One estimator is computed by solving a global defect problem based on the splitting of the lowest-order Brezzi-Douglas-Marini space, and the other estimator is locally computable by applying the standard localization to the first estimator. We establish the reliability and efficiency of both estimators by comparing them with the standard residual estimator. In addition, it is shown that the error estimator based on the global defect problem is asymptotically exact under suitable conditions.

Simultaneous Unwrapping Phase and Error Recovery from Inhomogeneity (SUPER) for Quantitative Susceptibility Mapping of the Human Brain

  • Yang, Young-Joong;Yoon, Jong-Hyun;Baek, Hyun-Man;Ahn, Chang-Beom
    • Investigative Magnetic Resonance Imaging
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    • v.22 no.1
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    • pp.37-49
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    • 2018
  • Purpose: The effect of global inhomogeneity on quantitative susceptibility mapping (QSM) was investigated. A technique referred to as Simultaneous Unwrapping Phase with Error Recovery from inhomogeneity (SUPER) is suggested as a preprocessing to QSM to remove global field inhomogeneity-induced phase by polynomial fitting. Materials and Methods: The effect of global inhomogeneity on QSM was investigated by numerical simulations. Three types of global inhomogeneity were added to the tissue susceptibility phase, and the root mean square error (RMSE) in the susceptibility map was evaluated. In-vivo QSM imaging with volunteers was carried out for 3.0T and 7.0T MRI systems to demonstrate the efficacy of the proposed method. Results: The SUPER technique removed harmonic and non-harmonic global phases. Previously only the harmonic phase was removed by the background phase removal method. The global phase contained a non-harmonic phase due to various experimental and physiological causes, which degraded a susceptibility map. The RMSE in the susceptibility map increased under the influence of global inhomogeneity; while the error was consistent, irrespective of the global inhomogeneity, if the inhomogeneity was corrected by the SUPER technique. In-vivo QSM imaging with volunteers at 3.0T and 7.0T MRI systems showed better definition in small vascular structures and reduced fluctuation and non-uniformity in the frontal lobes, where field inhomogeneity was more severe. Conclusion: Correcting global inhomogeneity using the SUPER technique is an effective way to obtain an accurate susceptibility map on QSM method. Since the susceptibility variations are small quantities in the brain tissue, correction of the inhomogeneity is an essential element for obtaining an accurate QSM.

Error Control Policy for Initial Value Problems with Discontinuities and Delays

  • Khader, Abdul Hadi Alim A.
    • Kyungpook Mathematical Journal
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    • v.48 no.4
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    • pp.665-684
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    • 2008
  • Runge-Kutta-Nystr$\"{o}$m (RKN) methods provide a popular way to solve the initial value problem (IVP) for a system of ordinary differential equations (ODEs). Users of software are typically asked to specify a tolerance ${\delta}$, that indicates in somewhat vague sense, the level of accuracy required. It is clearly important to understand the precise effect of changing ${\delta}$, and to derive the strongest possible results about the behaviour of the global error that will not have regular behaviour unless an appropriate stepsize selection formula and standard error control policy are used. Faced with this situation sufficient conditions on an algorithm that guarantee such behaviour for the global error to be asympotatically linear in ${\delta}$ as ${\delta}{\rightarrow}0$, that were first derived by Stetter. Here we extend the analysis to cover a certain class of ODEs with low-order derivative discontinuities, and the class of ODEs with constant delays. We show that standard error control techniques will be successful if discontinuities are handled correctly and delay terms are calculated with sufficient accurate interpolants. It is perhaps surprising that several delay ODE algorithms that have been proposed do not use sufficiently accurate interpolants to guarantee asymptotic proportionality. Our theoretical results are illustrated numerically.

Control of Outdoor Robot Based on Error Compensation in Global Coordinates of Two Low-cost GPS (두개의 저가형 GPS의 전역 좌표 오차 보상에 의한 실외 이동로봇의 목표점 도달 제어)

  • Park, Ki-Hun;Hong, Yun-Ki;Yoon, Tae-Sung;Park, Seung-Kyu
    • Proceedings of the KIEE Conference
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    • pp.1778-1779
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    • 2011
  • The location of outdoor mobile robot is obtained using global coordinates from GPS. However, the error generated by GPS is about 10m ~ 100m, so the precise control is difficult. D-GPS has the error value of 1m and it is very accurate, but the price is very expensive. In this paper, a method to reduce the error in global coordinates is proposed using two low-cost GPS for the autonomous navigation control of outdoor mobile robot.

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Analysis and Calculation of Global Hourly Solar Irradiation Based on Sunshine Duration for Major Cities in Korea (국내 주요도시의 일조시간데이터를 이용한 시간당전일사량 산출 및 분석)

  • Lee, Kwan-Ho;Sim, Kwang-Yeal
    • Journal of the Korean Solar Energy Society
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    • v.30 no.2
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    • pp.16-21
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    • 2010
  • Computer simulation of buildings and solar energy systems are being used increasingly in energy assessments and design. This paper discusses the possibility of using sunshine duration data instead of global hourly solar irradiation (GHSI) data for localities with abundant data on sunshine duration. For six locations in South Korea where global radiation is currently measured, the global radiation was calculated using Sunshine Duration Radiation Model (SDRM), compared and analyzed. Results of SDRM has been compared with the measured data on the coefficients of determination (R2), root-mean-square error (RMSE) and mean bias error (MBE). This study recommends the use of sunshine duration based irradiation models if measured solar radiation data is not available.

Autonomous Navigation of Mobile Robot Using Global Ultrasonic System (전역 초음파 시스템을 이용한 이동 로봇의 자율 주행)

  • 황병훈;이수영
    • Journal of Institute of Control, Robotics and Systems
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    • v.10 no.6
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    • pp.529-536
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    • 2004
  • Autonomous navigation of an indoor mobile robot using the global ultrasonic system is presented in this paper. Since the trajectory error of the dead-reckoning navigation grows with time and distance, the autonomous navigation of a mobile robot requires to localize the current position of the robot, so that to compensate the trajectory error. The global ultrasonic system consisting of four ultrasonic generators fixed at a priori known positions in the work space and two receivers on the mobile robot has the similar structure with the well-known satellite GPS(Global Positioning System), and it is useful for the self-localization of an indoor mobile robot. The EKF(Extended Kalman Filter) algorithm for the self-localization is proposed and the autonomous navigation based on the self-localization is verified by experiments.

Local A Posteriori Error Estimates for Obstacle Contact Problems (장애물 접촉문제에서의 지역 A Posteriori 오차계산)

  • 이춘열
    • Journal of the Korean Society for Precision Engineering
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    • v.15 no.5
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    • pp.120-127
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    • 1998
  • Differential inequalities occurring in problems of obstacle contact problems are recast into variational inequalities and analyzed by finite element methods. A new a posteriori error estimator, which is essential in adaptive finite element method, is introduced to capture the errors in finite element approximations of these variational inequalities. In order to construct a posteriori error estimates, saddle point problems are introduced using Lagrange parameters and upper bounds are provided. The global upper bound is localized by a special mixed formulation, which leads to upper bounds of the element errors. A numerical experiment is performed on an obstacle contact problem to check the effectivity index both in a local and a global sense.

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