• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.125-135
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• 1992

In various viscus flow problems it has been the custom to replace the convective derivative by the ordinary partial derivative in problems for which the data are small. In this paper we consider the Benard Convection problem with small data and compare the solution of this problem (assumed to exist) with that of the linearized system resulting from dropping the nonlinear terms in the expression for the convective derivative. The objective of the present work is to derive an estimate for the error introduced in neglecting the convective inertia terms. In fact, we derive an explicit bound for the L$_{2}$ error. Indeed, if the initial data are O(.epsilon.) where .epsilon. << 1, and the Rayleigh number is sufficiently small, we show that this error is bounded by the product of a term of O(.epsilon.$^{2}$) times a decaying exponential in time. The results of the present paper then give a justification for linearizing the Benard Convection problem. We remark that although our results are derived for classical solutions, extensions to appropriately defined weak solutions are obvious. Throughout this paper we will make use of a comma to denote partial differentiation and adopt the summation convention of summing over repeated indices (in a term of an expression) from one to three. As reference to work of continuous dependence on modelling and initial data, we mention the papers of Payne and Sather [8], Ames [2] Adelson [1], Bennett [3], Payne et al. [9], and Song [11,12,13,14]. Also, a similar analysis of a micropolar fluid problem backward in time (an ill-posed problem) was given by Payne and Straughan [10] and Payne [7].

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.10 no.10
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• pp.5129-5152
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• 2016

Multi-view super-resolution (MVSR) aims to estimate a high-resolution (HR) image from a set of low-resolution (LR) images that are captured from different viewpoints (typically by different cameras). MVSR is usually applied in camera array imaging. Given that MVSR is an ill-posed problem and is typically computationally costly, we super-resolve multi-view LR images of the original scene via image fusion (IF) and blind deblurring (BD). First, we reformulate the MVSR problem into two easier problems: an IF problem and a BD problem. We further solve the IF problem on the premise of calculating the depth map of the desired image ahead, and then solve the BD problem, in which the optimization problems with respect to the desired image and with respect to the unknown blur are efficiently addressed by the alternating direction method of multipliers (ADMM). Our approach bridges the gap between MVSR and BD, taking advantages of existing BD methods to address MVSR. Thus, this approach is appropriate for camera array imaging because the blur kernel is typically unknown in practice. Corresponding experimental results using real and synthetic images demonstrate the effectiveness of the proposed method.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.15 no.12
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• pp.4420-4438
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• 2021

Image dehazing is an ill-posed problem which is far from being solved. Traditional image dehazing methods often yield mediocre effects and possess substandard processing speed, while modern deep learning methods perform best only in certain datasets. The haze removal effect when processed by said methods is unsatisfactory, meaning the generalization performance fails to meet the requirements. Concurrently, due to the limited processing speed, most dehazing algorithms cannot be employed in the industry. To alleviate said problems, a lightweight fast dehazing network based on a multiple scale-patch framework (MSP) is proposed in the present paper. Firstly, the multi-scale structure is employed as the backbone network and the multi-patch structure as the supplementary network. Dehazing through a single network causes problems, such as loss of object details and color in some image areas, the multi-patch structure was employed for MSP as an information supplement. In the algorithm image processing module, the image is segmented up and down for processed separately. Secondly, MSP generates a clear dehazing effect and significant robustness when targeting real-world homogeneous and nonhomogeneous hazy maps and different datasets. Compared with existing dehazing methods, MSP demonstrated a fast inference speed and the feasibility of real-time processing. The overall size and model parameters of the entire dehazing model are 20.75M and 6.8M, and the processing time for the single image is 0.026s. Experiments on NTIRE 2018 and NTIRE 2020 demonstrate that MSP can achieve superior performance among the state-of-the-art methods, such as PSNR, SSIM, LPIPS, and individual subjective evaluation.

• Geophysics and Geophysical Exploration
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• v.7 no.3
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• pp.207-212
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• 2004

This article reviews recent developments in three-dimensional (3-D) magntotelluric (MT) imaging. The inversion of MT data is fundamentally ill-posed, and therefore the resultant solution is non-unique. A regularizing scheme must be involved to reduce the non-uniqueness while retaining certain a priori information in the solution. The standard approach to nonlinear inversion in geophysis has been the Gauss-Newton method, which solves a sequence of linearized inverse problems. When running to convergence, the algorithm minimizes an objective function over the space of models and in the sense produces an optimal solution of the inverse problem. The general usefulness of iterative, linearized inversion algorithms, however is greatly limited in 3-D MT applications by the requirement of computing the Jacobian(partial derivative, sensitivity) matrix of the forward problem. The difficulty may be relaxed using conjugate gradients(CG) methods. A linear CG technique is used to solve each step of Gauss-Newton iterations incompletely, while the method of nonlinear CG is applied directly to the minimization of the objective function. These CG techniques replace computation of jacobian matrix and solution of a large linear system with computations equivalent to only three forward problems per inversion iteration. Consequently, the algorithms are efficient in computational speed and memory requirement, making 3-D inversion feasible.

• Journal of International Area Studies (JIAS)
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• v.14 no.4
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• pp.27-52
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• 2011

The regions of Central Asia have each acquired an elevated strategic importance in the new security paradigm of post-September 1lth. Comprised of five states, Kazakhstan, Kyrgyzstan, Tajikistan, Turkmenistan and Uzbekistan, Central Asia's newly enhanced strategic importance stems from several other factors, ranging from trans-national threats posed by Islamic extremism, drug production and trafficking, to the geopolitical threats inherent in the region's location as a crossroads between Russia, Southwest Asia and China. Although the U.S. military presence in the region began before September 11th, the region became an important platform for the projection of U.S. military power against the Taliban in neighboring Afghanistan. The analysis goes on to warn that 'with US troops already in place to varying extents in Central Asian states, it becomes particularly important to understand the faultlines, geography, and other challenges this part of the world presents'. The Kyrgyz military remains an embryonic force with a weak chain of command, the ground force built to Cold War standards, and an almost total lack of air capabilities. Training, discipline and desertion - at over 10 per cent, the highest among the Central Asian republics - continue to present major problems for the creation of combat-effective armed forces. Kyrgyzstan has a declared policy of national defence and independence without the use of non-conventional weapons. Kyrgyzstan participates in the regional security structures, such as the Collective Security Treaty Organisation (CSTO) and the Shanghai Co-operation Organisation (SCO) but, in security matters at least, it is dependent upon Russian support. The armed forces are poorly trained and ill-equipped to fulfil an effective counter-insurgency or counter-terrorist role. The task of rebuilding is much bigger, and so are the stakes - the integrity and sovereignty of the Kyrgyz state. Only democratization, the fight against corruption, reforms in the military and educational sectors and strategic initiatives promoting internal economic integration and national cohesion hold the key to Kyrgyzstan's lasting future