针对结构化的非凸非光滑优化问题,提出了一种改进的惯性近端交替方向乘子法(Modified Inertial Proximal Alternating Direction Method of Multipliers, MID-PADMM)。该问题在多个领域,包括机器学习、信号处理和经济学中具有重要应用...针对结构化的非凸非光滑优化问题,提出了一种改进的惯性近端交替方向乘子法(Modified Inertial Proximal Alternating Direction Method of Multipliers, MID-PADMM)。该问题在多个领域,包括机器学习、信号处理和经济学中具有重要应用。现有算法在处理这类问题时,往往面临收敛速度慢或无法保证收敛的挑战。为了克服这些限制,引入了一种双重松弛项,以增强算法的鲁棒性和灵活性。理论分析表明,MID-PADMM算法在适当的条件下能够实现全局收敛,并且具有O(1/k)的迭代复杂度,其中k代表迭代次数。数值实验结果表明,与现有的状态最优算法相比,MID-PADMM在多个实例中展现出更快的收敛速度和更高的求解质量。展开更多
地震勘探技术在地震学研究的有关领域中起着重要作用,地震数据处理正确与否,直接影响了解释的准确性与精确度。为了验证动校正速度对最终叠加剖面的影响,对实际某地区的地震资料处理分析,最终得到了最佳动校正速度的叠加图像。实验结果...地震勘探技术在地震学研究的有关领域中起着重要作用,地震数据处理正确与否,直接影响了解释的准确性与精确度。为了验证动校正速度对最终叠加剖面的影响,对实际某地区的地震资料处理分析,最终得到了最佳动校正速度的叠加图像。实验结果表明,选择正确的动校正速度改善了叠加剖面图的信噪比。同时,选择最佳的动校正速度是得到信噪比高的图像的必要选择,这对现实中处理野外采集的地震数据处理有一定的借鉴作用。Seismic exploration techniques play a crucial role in seismic research, directly affecting the accuracy and precision of interpretation based on the correctness of seismic data processing. In order to verify the impact of different dynamic correction speeds on the final superimposed profile, seismic data processing analysis was conducted in a specific region. The optimal dynamic correction speed was finally obtained for the superposition image. The experimental results show that selecting the correct dynamic correction speed can improve the signal-to-noise ratio of the superimposed profile image. At the same time, choosing the optimal dynamic correction speed is a necessary choice to obtain high signal-to-noise ratio images, which has certain reference significance for the seismic data processing of field acquisition in reality.展开更多
多元函数的梯度是微积分中的一个重要概念,在分析学中占有举足轻重的地位,它允许我们在多维空间中对函数进行深入的理解和操作。梯度不仅揭示了函数在特定点的局部行为,还为优化问题提供了方向性指导。在数学、物理学、工程学以及其他...多元函数的梯度是微积分中的一个重要概念,在分析学中占有举足轻重的地位,它允许我们在多维空间中对函数进行深入的理解和操作。梯度不仅揭示了函数在特定点的局部行为,还为优化问题提供了方向性指导。在数学、物理学、工程学以及其他科学领域,梯度的概念和应用都极为广泛。The gradient of multivariate functions is an important concept in calculus and occupies a pivotal position in the field of analysis. It allows us to deeply understand and manipulate functions within multidimensional spaces. The gradient not only reveals the local behavior of a function at specific points but also provides directional guidance for optimization problems. The concept and application of the gradient are extremely broad in mathematics, physics, engineering, and other scientific fields.展开更多
在教学过程中,我们发现拉格朗日中值定理是学生学习微积分的巨大障碍,这是因为拉格朗日中值定理是微分中值定理的核心内容,是研究函数与导数之间联系的理论工具,在微积分学中起着至关重要的作用,应用十分广泛。本文重点研究拉格朗日中...在教学过程中,我们发现拉格朗日中值定理是学生学习微积分的巨大障碍,这是因为拉格朗日中值定理是微分中值定理的核心内容,是研究函数与导数之间联系的理论工具,在微积分学中起着至关重要的作用,应用十分广泛。本文重点研究拉格朗日中值定理在证明导数极限定理、求函数极限问题、证明不等式以及证明函数单调性方面的应用,以及拉格朗日中值定理的两个推广。希望本文可以对学生学习微积分有所帮助。During the teaching process, we found that the Lagrange Mean Value Theorem is a significant obstacle for students learning calculus. The Lagrange Mean Value Theorem is the core content of the Mean Value Theorem in differential calculus. It is a theoretical tool for studying the relationship between functions and their derivatives and plays a crucial role in calculus, with a wide range of applications. This paper focuses on the application of the Lagrange Mean Value Theorem in proving the derivative limit theorem, solving limit problems of functions, proving inequalities, and proving the monotonicity of functions, as well as two extensions of the Lagrange Mean Value Theorem. It is hoped that this article can be of assistance to students in their study of calculus.展开更多
令 G 是一个简单连通图。 图 G 的 Kirchhoff 指标是图 G 中所有顶点对之间的电阻距离之和。 图 G 的电阻距离等效于将图 G 中的每条边替换为一个单位电阻后得到的电网络 N 中任意节点对之间的 有效电阻。 一个包含 n + 2 个多边形和 n ...令 G 是一个简单连通图。 图 G 的 Kirchhoff 指标是图 G 中所有顶点对之间的电阻距离之和。 图 G 的电阻距离等效于将图 G 中的每条边替换为一个单位电阻后得到的电网络 N 中任意节点对之间的 有效电阻。 一个包含 n + 2 个多边形和 n + 1 个四边形的链,使得其中每个四边形的两条平行边各 与一个多边形有一条公共边,这样的链被称为多边形链。 本文利用电网络的标准技术和 S, T -同分 异构体的 Kirchhoff 指标的比较结果,刻画了 Kirchhoff 指标达到最大的极值多边形链为线性多 边形链 Ln, Kirchhoff 指标达到最小的极值多边形链为螺旋多边形链 Dn。 此结果推广了杨玉军 等人以及张雷雷刻画的基于 Kirchhoff 指标的极值亚苯基链的结果。展开更多
文摘针对结构化的非凸非光滑优化问题,提出了一种改进的惯性近端交替方向乘子法(Modified Inertial Proximal Alternating Direction Method of Multipliers, MID-PADMM)。该问题在多个领域,包括机器学习、信号处理和经济学中具有重要应用。现有算法在处理这类问题时,往往面临收敛速度慢或无法保证收敛的挑战。为了克服这些限制,引入了一种双重松弛项,以增强算法的鲁棒性和灵活性。理论分析表明,MID-PADMM算法在适当的条件下能够实现全局收敛,并且具有O(1/k)的迭代复杂度,其中k代表迭代次数。数值实验结果表明,与现有的状态最优算法相比,MID-PADMM在多个实例中展现出更快的收敛速度和更高的求解质量。
文摘地震勘探技术在地震学研究的有关领域中起着重要作用,地震数据处理正确与否,直接影响了解释的准确性与精确度。为了验证动校正速度对最终叠加剖面的影响,对实际某地区的地震资料处理分析,最终得到了最佳动校正速度的叠加图像。实验结果表明,选择正确的动校正速度改善了叠加剖面图的信噪比。同时,选择最佳的动校正速度是得到信噪比高的图像的必要选择,这对现实中处理野外采集的地震数据处理有一定的借鉴作用。Seismic exploration techniques play a crucial role in seismic research, directly affecting the accuracy and precision of interpretation based on the correctness of seismic data processing. In order to verify the impact of different dynamic correction speeds on the final superimposed profile, seismic data processing analysis was conducted in a specific region. The optimal dynamic correction speed was finally obtained for the superposition image. The experimental results show that selecting the correct dynamic correction speed can improve the signal-to-noise ratio of the superimposed profile image. At the same time, choosing the optimal dynamic correction speed is a necessary choice to obtain high signal-to-noise ratio images, which has certain reference significance for the seismic data processing of field acquisition in reality.
文摘多元函数的梯度是微积分中的一个重要概念,在分析学中占有举足轻重的地位,它允许我们在多维空间中对函数进行深入的理解和操作。梯度不仅揭示了函数在特定点的局部行为,还为优化问题提供了方向性指导。在数学、物理学、工程学以及其他科学领域,梯度的概念和应用都极为广泛。The gradient of multivariate functions is an important concept in calculus and occupies a pivotal position in the field of analysis. It allows us to deeply understand and manipulate functions within multidimensional spaces. The gradient not only reveals the local behavior of a function at specific points but also provides directional guidance for optimization problems. The concept and application of the gradient are extremely broad in mathematics, physics, engineering, and other scientific fields.
文摘在教学过程中,我们发现拉格朗日中值定理是学生学习微积分的巨大障碍,这是因为拉格朗日中值定理是微分中值定理的核心内容,是研究函数与导数之间联系的理论工具,在微积分学中起着至关重要的作用,应用十分广泛。本文重点研究拉格朗日中值定理在证明导数极限定理、求函数极限问题、证明不等式以及证明函数单调性方面的应用,以及拉格朗日中值定理的两个推广。希望本文可以对学生学习微积分有所帮助。During the teaching process, we found that the Lagrange Mean Value Theorem is a significant obstacle for students learning calculus. The Lagrange Mean Value Theorem is the core content of the Mean Value Theorem in differential calculus. It is a theoretical tool for studying the relationship between functions and their derivatives and plays a crucial role in calculus, with a wide range of applications. This paper focuses on the application of the Lagrange Mean Value Theorem in proving the derivative limit theorem, solving limit problems of functions, proving inequalities, and proving the monotonicity of functions, as well as two extensions of the Lagrange Mean Value Theorem. It is hoped that this article can be of assistance to students in their study of calculus.
