Computational Mathematics and Mathematical Physics最新文献

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Asymptotics of the Solution of a Bisingular Optimal Distributed Control Problem in a Convex Domain with a Small Parameter Multiplying a Highest Derivative 具有乘以最高衍生物的小参数的凸域中比星形最优分布式控制问题解的渐近性

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics

Pub Date : 2024-06-13 DOI: 10.1134/s0965542524700210

A. R. Danilin

Abstract

We consider an optimal distributed control problem in a strictly convex planar domain with a smooth boundary and a small parameter multiplying a highest derivative of an elliptic operator. A zero Dirichlet condition is set on the boundary of the domain, and control is additively involved in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. The solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with some coefficient. Due to this structure of the optimality criterion, the role of the first or second term of the criterion can be strengthen, if necessary. It is more important to achieve a given state in the first case and to minimize the resource cost in the second case. The asymptotics of the problem generated by the sum of a second-order differential operator with a small coefficient at a highest derivative and a zero-order differential operator is studied in detail.

摘要我们考虑了一个严格凸平面域中的最优分布式控制问题，该域具有光滑边界和一个乘以椭圆算子最高导数的小参数。在该域的边界上设置了一个零 Dirichlet 条件，控制与不均匀性相加。可接受的控制集是相应的平方可积分函数空间中的单位球。所得到的边界值问题解在广义上被视为希尔伯特空间的元素。最优性准则是状态偏离给定状态的平方准则与带有一定系数的控制平方准则之和。由于最优化准则的这种结构，必要时可以加强准则第一项或第二项的作用。在第一种情况下，实现给定状态更为重要，而在第二种情况下，资源成本最小化更为重要。本文详细研究了最高导数系数较小的二阶微分算子与零阶微分算子之和所产生问题的渐近性。

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引用次数: 0

Iterative PDE-Constrained Optimization for Seismic Full-Waveform Inversion 地震全波形反演的迭代 PDE 约束优化

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics

Pub Date : 2024-06-13 DOI: 10.1134/s0965542524700192

M. S. Malovichko, A. Orazbayev, N. I. Khokhlov, I. B. Petrov

Abstract

This paper presents a novel numerical method for the Newton seismic full-waveform inversion (FWI). The method is based on the full-space approach, where the state, adjoint state, and control variables are optimized simultaneously. Each Newton step is formulated as a PDE-constrained optimization problem, which is cast in the form of the Karush–Kuhn–Tucker (KKT) system of linear algebraic equitations. The KKT system is solved inexactly with a preconditioned Krylov solver. We introduced two preconditioners: the one based on the block-triangular factorization and its variant with an inexact block solver. The method was benchmarked against the standard truncated Newton FWI scheme on a part of the Marmousi velocity model. The algorithm demonstrated a considerable runtime reduction compared to the standard FWI. Moreover, the presented approach has a great potential for further acceleration. The central result of this paper is that it establishes the feasibility of Newton-type optimization of the KKT system in application to the seismic FWI.

摘要本文提出了一种新的牛顿地震全波形反演（FWI）数值方法。该方法以全空间方法为基础，同时对状态、邻接状态和控制变量进行优化。每个牛顿步骤都被表述为一个 PDE 受限优化问题，以线性代数方程的 Karush-Kuhn-Tucker (KKT) 系统的形式呈现。KKT 系统通过预处理 Krylov 求解器精确求解。我们引入了两种预处理方法：一种是基于分块三角形因式分解的预处理方法，另一种是基于非精确分块求解器的预处理方法。我们以标准的截断牛顿 FWI 方案为基准，对 Marmousi 速度模型的一部分进行了测试。与标准 FWI 相比，该算法大大缩短了运行时间。此外，所提出的方法还有进一步加速的巨大潜力。本文的核心成果是确定了 KKT 系统牛顿型优化在地震 FWI 应用中的可行性。

引用次数: 0

Existence of an Optimal Control for a Semilinear Evolution Equation with Unbounded Operator 带无界算子的半线性演化方程的最优控制的存在性

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics

Pub Date : 2024-06-13 DOI: 10.1134/s0965542524700362

A. V. Chernov

Abstract

An optimal control problem is investigated for an abstract semilinear differential equation of the first order in time in a Hilbert space with an unbounded operator and control involved linearly in the right-hand side. The cost functional is assumed to be additively separated with respect to state and control, with a rather general dependence on the state. For this problem, the existence of an optimal control is proved and the properties of the set of optimal controls are established. The author’s previous results on the total preservation of unique global solvability (totally global solvability) and on solution estimation for such equations are developed in the context of the nonlinearity of the equation under study. The indicated estimate is found important for the present study. A nonlinear heat equation and a nonlinear wave equation are considered as examples.

摘要研究了希尔伯特空间中一个抽象的一阶半线性微分方程的最优控制问题，该方程具有一个无界算子，控制与右侧线性相关。假定成本函数在状态和控制方面是加法分离的，与状态有相当普遍的依赖关系。对于这个问题，证明了最优控制的存在，并建立了最优控制集的属性。作者之前关于唯一全局可解性（全局可解性）和解估计的成果在所研究方程的非线性背景下得到了发展。所指出的估计值对本研究非常重要。本研究以非线性热方程和非线性波方程为例。

引用次数: 0

A Shannon Wavelet-Based Approximation Scheme for Thomas–Fermi Models of Confined Atoms and Ions 基于香农小波的密闭原子和离子托马斯-费米模型近似方案

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics

Pub Date : 2024-06-13 DOI: 10.1134/s0965542524700350

Sharda Kumari, Pratik Majhi, M. M. Panja

Abstract

An efficient numerical scheme based on the Shannon wavelet basis has been presented here for obtaining highly accurate approximate solutions of Thomas–Fermi equations (TFE) in the finite domain with various initial/boundary conditions (IC/BCs). A point transformation followed by a finite Whittaker Cardinal function approximation (FWCFA) is employed here. The formula relating exponent (n) in the desired order of accuracy ((O{{(10}^{{ - n}}}))) with the resolution (J), the lower and upper limits in the sum of FWCFA have been provided. Examples of TFE with various IC/BCs have been exercised to exhibit the elegance and efficiency of the present scheme.

摘要本文提出了一种基于香农小波基的高效数值方案，用于在有限域中获得具有各种初始/边界条件（IC/BC）的托马斯-费米方程（TFE）的高精度近似解。这里采用的是点变换后的有限惠特克卡迪纳函数近似（FWCFA）。提供了所需精度等级（(O{(10}^{-n}}))的指数(n)与分辨率(J)、FWCFA 总和的下限和上限的相关公式。为了展示本方案的优雅和高效，我们还使用了不同 IC/BC 的 TFE 示例。

引用次数: 0

On the Stability of a Central Difference Scheme with a Stabilizing Correction for the 3D Transport Equation 论带稳定修正的中央差分方案在三维传输方程中的稳定性

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics

Pub Date : 2024-06-13 DOI: 10.1134/s0965542524700271

V. P. Zhukov

Abstract

It is generally accepted that the central differences scheme with a stabilizing correction for the transport equation in the 3D case is conditionally stable. This article shows that, strictly speaking, this scheme is absolutely unstable. However, the region of unstable harmonics in the wave vector space and their increments quickly tend to zero as the Courant parameter tends to zero, which makes it possible to successfully use this scheme. Therefore, it is more correct to talk about the practically conditional stability of this scheme.

摘要一般认为，在三维情况下，对输运方程进行稳定修正的中心差分方案是有条件稳定的。本文指出，严格来说，该方案是绝对不稳定的。然而，随着库朗参数趋于零，波矢量空间中的不稳定谐波区域及其增量很快趋于零，这使得成功使用该方案成为可能。因此，更正确的说法是这种方案实际上的条件稳定性。

引用次数: 0

Multizonal Internal Layers in a Stationary Piecewise–Smooth Reaction-Diffusion Equation in the Case of the Difference of Multiplicity for the Roots of the Degenerate Solution 稳态片状光滑反应-扩散方程中的多层内层与退化解根的多重性差异

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics

Pub Date : 2024-06-13 DOI: 10.1134/s0965542524700179

Qian Yang, Mingkang Ni

Abstract

A singularly perturbed stationary problem for a one-dimensional reaction-diffusion equation in the case when the degenerate equation has multiple roots is studied. This is a new class of problems with discontinuous reactive terms along some curve that is independent of the small parameter. The existence of a smooth solution with the transition from the triple root of one degenerate equation to the double root of the other degenerate equation in the neighborhood of some point on the discontinuous curve is studied. Based on the existence theorem of classical boundary value problems and the technique of matching asymptotic expansion, the existence of a smooth solution is proved. And the point itself and the asymptotic representation of this solution are constructed by the matching technique and modified boundary layer function method.

摘要研究了一维反应-扩散方程在退化方程有多个根的情况下的奇异扰动静止问题。这是一类新问题，其反应项沿着与小参数无关的曲线不连续。研究了在不连续曲线上的某一点附近，是否存在从一个退化方程的三重根过渡到另一个退化方程的双重根的平稳解。基于经典边界值问题的存在定理和匹配渐近展开技术，证明了平稳解的存在性。并通过匹配技术和修正边界层函数法构建了点本身和该解的渐近表示。

引用次数: 0

Numerical-Analytical Decomposition-Autocompensation Method for Signal Recognition from Incorrect Observations 从错误观测数据中识别信号的数值-分析分解-自动补偿方法

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics

Pub Date : 2024-06-13 DOI: 10.1134/s0965542524700180

Yu. G. Bulychev

Abstract

A numerical-analytical method is developed for solving the problem of optimal recognition of a set of possible signals observed in the form of an additive mixture involving not only fluctuation measurement errors (with an unknown statistical distribution law), but also a singular disturbance (with parametric uncertainty). The method not only detects signals in the mixture, but also estimates their parameters as based on a given cost functional and accompanying constraints. Based on the idea of generalized invariant unbiased estimation of linear functionals, the method ensures decomposition of the numerical procedure and autocompensation of the singular disturbance without resorting to conventional state space extension. Parametric finite-dimensional representations of the signals and the disturbance are obtained using linear spectral decompositions in given functional bases. The measurement error is described using only its correlation matrix. The random and systematic errors are analyzed, and an illustrative example is given.

摘要本文提出了一种数值分析方法，用于解决以加法混合物形式观测到的一组可能信号的最佳识别问题，这组信号不仅涉及波动测量误差（具有未知的统计分布规律），还涉及奇异干扰（具有参数不确定性）。该方法不仅能检测混合物中的信号，还能根据给定的成本函数和相应的约束条件估算其参数。基于线性函数广义不变无偏估计的思想，该方法确保了数值程序的分解和奇异干扰的自动补偿，而无需诉诸传统的状态空间扩展。在给定的函数基础上，利用线性谱分解获得信号和干扰的参数有限维表示。测量误差仅使用其相关矩阵进行描述。分析了随机误差和系统误差，并给出了一个示例。

引用次数: 0

Spectral Methods for Solution of Differential and Functional Equations 求解微分方程和函数方程的谱方法

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics

Pub Date : 2024-06-13 DOI: 10.1134/s0965542524700222

V. P. Varin

Abstract

An operational approach developed earlier for the spectral method that uses Legendre polynomials is generalized here for arbitrary systems of basis functions (not necessarily orthogonal) that satisfy only two conditions: the result of multiplication by (x) or of differentiation with respect to (x) is expressed in the same functions. All systems of classical orthogonal polynomials satisfy these conditions. In particular, we construct a spectral method that uses Chebyshev polynomials, which is most effective for numerical computations. This method is applied for numerical solution of the linear functional equations that appear in problems of generalized summation of series as well as in the problems of analytical continuation of discrete maps. We also demonstrate how these methods are used for solution of nonstandard and nonlinear boundary value problems for which ordinary algorithms are not applicable.

摘要早先为使用 Legendre 多项式的光谱法开发的一种运算方法在此得到了推广，适用于只满足两个条件的任意基函数系统（不一定是正交的）：与 (x) 相乘或与(x) 相乘或与(x) 相乘或与(x) 相乘或与(x) 相乘的结果用相同的函数表示。所有经典正交多项式系统都满足这些条件。我们特别构建了一种使用切比雪夫多项式的谱方法，这对数值计算最为有效。这种方法可用于线性函数方程的数值求解，这些方程出现在广义数列求和问题以及离散映射的解析延续问题中。我们还演示了这些方法如何用于解决普通算法无法解决的非标准和非线性边界值问题。

引用次数: 0

The Simultaneous Reduction of a Pair of Unitoid Matrices to Diagonal Form Revisited 将一对单元矩阵同时还原为对角线形式的再探讨

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics

Pub Date : 2024-06-13 DOI: 10.1134/s0965542524700234

Kh. D. Ikramov

Abstract

This note is an addendum to the paper on the same subject published by the author somewhat earlier. Its aim is to more precisely characterize pairs of unitoids that admit simultaneous reduction to diagonal form.

摘要本说明是作者稍早发表的同一主题论文的增补。其目的是更精确地描述允许同时还原为对角线形式的单位体对的特征。

引用次数: 0

Application of the CABARET and WENO Schemes for Solving the Nonlinear Transport Equation in the Problem of Simulating the Propagation of a Sonic Boom Wave in the Atmosphere 应用 CABARET 和 WENO 方案求解大气中声波传播模拟问题中的非线性传输方程

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics

Pub Date : 2024-06-13 DOI: 10.1134/s096554252470026x

P. A. Mishchenko, T. A. Gimon, V. A. Kolotilov

Abstract

The most convenient model describing the propagation of a sonic boom wave in the atmosphere is the augmented Burgers equation. In this work, we studied the influence of a numerical scheme on the result of solving an equation that takes into account the nonlinear nature of the propagation of sonic boom waves in the atmosphere. This equation is a key component of the augmented Burgers equation and determines the nature of the transformation of the disturbed pressure profile during its propagation. Two numerical schemes were used for solving: CABARET and WENO—quasi-monotonic end-to-end computing schemes, which make it possible to obtain a solution without significant numerical oscillations. The applicability of these schemes for solving the problem under consideration is analyzed.

摘要描述音爆波在大气中传播的最便捷模型是增强伯格斯方程。在这项工作中，我们研究了数值方案对方程求解结果的影响，该方程考虑到了音爆波在大气中传播的非线性性质。该方程是增强伯格斯方程的关键组成部分，决定了扰动压力剖面在传播过程中的变化性质。求解时使用了两种数值方案：CABARET 和 WENO--准单调端到端计算方案，这两种方案可以获得无明显数值振荡的解决方案。分析了这些方案对解决所考虑问题的适用性。

引用次数: 0

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