Journal of Inverse and Ill-Posed Problems最新文献

英文中文

Determination of an unknown coefficient in the Korteweg–de Vries equation 确定 Korteweg-de Vries 方程中的未知系数

IF 1.1 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems

Pub Date : 2024-06-25 DOI: 10.1515/jiip-2024-0008

Lin Sang, Yan Qiao, Hua Wu

In this paper, a space-time spectral method for solving an inverse problem in the Korteweg–de Vries equation is considered. Optimal order of convergence of the semi-discrete method is obtained in L 2

{L^{2}}

-norm. The discrete schemes of the method are based on the modified Fourier pseudospectral method in spatial direction and the Legendre-tau method in temporal direction. The nonlinear term is computed via the fast Fourier transform and fast Legendre transform. The method is implemented by the explicit-implicit iterative method. Numerical results are given to show the accuracy and capability of this space-time spectral method.

本文研究了一种求解 Korteweg-de Vries 方程逆问题的时空谱方法。半离散方法在 L 2 {L^{2}} 规范下获得了最佳收敛阶数。 -norm。该方法的离散方案在空间方向上基于改进的傅立叶伪谱法，在时间方向上基于 Legendre-tau 法。非线性项通过快速傅立叶变换和快速 Legendre 变换计算。该方法通过显式-隐式迭代法实现。给出的数值结果表明了这种时空谱方法的准确性和能力。

引用次数: 0

Inverse problems for the eigenparameter Dirac operator with complex weight 具有复权的特征参数狄拉克算子的逆问题

IF 1.1 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems

Pub Date : 2024-06-25 DOI: 10.1515/jiip-2024-0032

Ran Zhang, Kai Wang, Chuan-Fu Yang

Inverse spectral problems are considered for the discontinuous Dirac operator with complex-value weight and the spectral parameter boundary conditions. We investigate some properties of spectral characteristics and show that the potential can be uniquely determined by the Weyl-type function or by two spectra on the whole interval.

我们考虑了具有复值权重和光谱参数边界条件的不连续狄拉克算子的逆光谱问题。我们研究了频谱特征的一些特性，并证明韦尔型函数或整个区间上的两个频谱可以唯一地确定势。

引用次数: 0

Error estimates for simplified Levenberg–Marquardt method for nonlinear ill-posed operator equations in Hilbert Spaces 希尔伯特空间非线性问题算子方程的简化 Levenberg-Marquardt 方法的误差估计

IF 1.1 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems

Pub Date : 2024-06-25 DOI: 10.1515/jiip-2023-0090

Pallavi Mahale, Ankush Kumar

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>F</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mi>y</m:mi> </m:mrow> </m:math> {F(u)=y} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>F</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi mathvariant="script">𝒟</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>⊂</m:mo> <m:mi>𝖴</m:mi> <m:mo>→</m:mo> <m:mi>𝖸</m:mi> </m:mrow> </m:mrow> </m:math> {F:mathcal{D}(F)subsetmathsf{U}tomathsf{Y}} is a nonlinear operator between Hilbert spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖴</m:mi> </m:math> {mathsf{U}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖸</m:mi> </m:math> {mathsf{Y}} . The method is defined as follows: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mi>u</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>δ</m:mi> </m:msubsup> <m:mo>=</m:mo> <m:mrow> <m:msubsup> <m:mi>u</m:mi> <m:mi>n</m:mi> <m:mi>δ</m:mi> </m:msubsup> <m:mo>-</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:msubsup> <m:mi>T</m:mi> <m:mn>0</m:mn> <m:mo>∗</m:mo> </m:msubsup> <m:mo>⁢</m:mo> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>α</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>I</m:mi> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>T</m:mi> <m:mn>0</m:mn> <m:mo>∗</m:mo> </m:msubsup> <m:mo>⁢</m:mo> <m:m

在本文中，我们考虑了希尔伯特空间中非线性失当逆问题的简化 Levenberg-Marquardt 方法，以获得形式为 F ( u ) = y {F(u)=y} 的失当非线性方程的稳定近似解，其中 F : 𝒟 ( F ) ⊂ 𝖴 → 𝖸 {F:mathcardt =y} 。其中 F : 𝒟 ( F ) ⊂ 𝖴 → 𝖸 {F:mathcal{D}(F)subsetmathsf{U}tomathsf{Y}} 是希尔伯特空间 𝖴 {mathsf{U}} 和 𝖸 {mathsf{Y}} 之间的非线性算子。 .该方法定义如下： u n + 1 δ = u n δ - ( T 0 ∗ T 0 + α n I ) - 1 T 0 ∗ ( F ( u n δ ) - y δ ) 。 , u_{n+1}^{delta}=u_{n}^{delta}-(T_{0}^{ast}T_{0}+alpha_{n}I)^{-1}T_{0}^{% ast}(F(u_{n}^{delta})-y^{delta}), 其中 T 0 = F ′ ( u 0 ) {T_{0}=F^{prime}(u_{0})} and T 0 ∗ = F ′ ( u 0 )∗ {T_{0}^{ast}=F^{prime}(u_{0})^{ast}} . .这里 F ′ ( u 0 ) {F^{prime}(u_{0})} 表示 F 在初始猜测 u 0 ∈ 𝒟 ( F ) {u_{0}inmathcal{D}(F)} 的精确解 u † {u^{dagger}} 时的弗雷谢特导数。 F ′ ( u 0 )∗ {F^{prime}(u_{0})^{ast}} 是 F ′ ( u 0 ) {F^{prime}(u_{0})} 的矢量，{ α n } 是 F ′ ( u 0 ) {F^{prime}(u_{0})} 的矢量。 {{α_{n}}}是一个先验选择的非负实数序列，满足适当的属性。我们使用莫罗佐夫型停止规则来终止迭代。在算子 F 的适当非线性条件下，我们证明了该方法的收敛性，并在元素 u 0 - u † {u_{0}-u^{dagger}} 的荷尔德型源条件下获得了收敛率结果。 .此外，我们还推导出在不使用源条件的情况下方法的收敛性，研究最后通过数值示例验证了理论结论。

引用次数: 0

Ensemble Kalman inversion based on level set method for inverse elastic scattering problem 基于水平集方法的集合卡尔曼反演用于反弹性散射问题

IF 1.1 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems

Pub Date : 2024-06-25 DOI: 10.1515/jiip-2023-0060

Jiangfeng Huang, Quanfeng Wang, Zhaoxing Li

We consider an ensemble Kalman inversion scheme for inverse elastic scattering problems in which the unknown quantity is the shape of the scatterer. Assume that the scatterer is a piecewise constant function with known value inside inhomogeneities. The level set method is described as an implicit representation of the scatterer boundary, with Gaussian random fields serving as prior to provide information on the level set functions. The ensemble Kalman filter method is then employed based on the level set functions to reconstruct the shape of the scatterer. We demonstrate the effectiveness of the proposed method using several numerical examples.

我们考虑了一种用于反弹性散射问题的集合卡尔曼反演方案，其中的未知量是散射体的形状。假设散射体是一个片断常数函数，在非均质体内部具有已知值。水平集方法被描述为散射体边界的隐式表示，高斯随机场作为先验，提供水平集函数的信息。然后根据水平集函数采用集合卡尔曼滤波法来重建散射体的形状。我们通过几个数值示例证明了所提方法的有效性。

引用次数: 0

Application of locally regularized extremal shift to the problem of realization of a prescribed motion 局部正则化极值移动在实现规定运动问题中的应用

IF 1.1 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems

Pub Date : 2024-06-25 DOI: 10.1515/jiip-2024-0018

Yury S. Osipov, Vyacheslav I. Maksimov

A controlled system of differential equations under the action of an unknown disturbance is considered. The problem discussed in the paper consists in constructing algorithms for forming a control that provides the realization of a prescribed motion for any admissible disturbance. Namely these algorithms should provide the closeness in the metric of the space of differentiable functions of a phase trajectory of a given controlled system and some etalon trajectory of an analogous system functioning when any outer actions are absent. As the space of admissible disturbances, we take the space of measurable square integrable (with respect to the Euclidean norm) functions. The cases of inaccurate measurements of phase trajectories of both systems at all times and at discrete times are under study. Two computer oriented algorithms for solving the problem are designed. The algorithms are based on the (well-known in the theory of guaranteed control) method of extremal shift. In the process, its local (at each time of control correction) regularization is performed by the method of smoothing functional (the Tikhonov method). In addition, estimates for algorithm’s convergence rate are presented.

本文考虑了在未知干扰作用下的微分方程受控系统。本文讨论的问题包括构建算法，以形成一种控制，为任何可接受的干扰提供规定运动的实现。也就是说，这些算法应在给定受控系统的相位轨迹的可微分函数空间的度量中，提供与在没有任何外部作用时运行的类似系统的某些等值线轨迹的接近度。作为可容许干扰的空间，我们采用可测量的平方可积分（关于欧几里得规范）函数的空间。我们正在研究两个系统在任何时间和离散时间的相位轨迹测量不准确的情况。设计了两种面向计算机的算法来解决这个问题。这些算法基于（保证控制理论中著名的）极值移动方法。在此过程中，通过平滑函数法（Tikhonov 法）对其进行局部（每次控制修正时）正则化。此外，还提出了算法收敛速率的估计值。

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引用次数: 0

Craniorachischisis in a 33-week-old Female Fetus: A Case Report. 一名 33 周大女性胎儿的颅底裂伤：病例报告。

4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems

Pub Date : 2024-04-15 eCollection Date: 2024-01-01 DOI: 10.47895/amp.vi0.6712

Clarisse Veronica L Mirhan, Cecile C Dungog, Karen Cybelle J Sotalbo

We report the case of a 33-week-old female fetus born with craniorachischisis to a gravida 5, para 4 (3104) mother with no previous history of conceiving a child with a neural tube defect. Craniorachischisis is characterized by anencephaly and an open defect extending from the brain to the spine and is the most severe and fatal type of neural tube defect. Although the cause of neural tube defects is hypothesized to be multifactorial and is usually sporadic, the risk is increased in neonates born to mothers with a family history or a previous pregnancy with neural tube defect, both of which are not present in the index case. This case is unique in that only during the fifth pregnancy did the couple conceive a child with a neural tube defect, emphasizing that folic acid supplementation, the sole preventive measure proven to decrease the risk of neural tube defects, remains to be important in the periconceptual period for all women of childbearing age.

我们报告了一例 33 周大的颅咽管裂伤女胎，其母亲为孕酮 5，4 段（3104），既往无神经管缺陷孕育史。颅底裂的特点是无脑畸形和从大脑延伸到脊柱的开放性缺损，是神经管缺陷中最严重和最致命的类型。虽然神经管畸形的病因被认为是多因素的，而且通常是偶发性的，但有家族史或曾妊娠过神经管畸形的母亲所生的新生儿患神经管畸形的风险会增加，而本病例不存在这两种情况。本病例的独特之处在于，这对夫妇在第五次怀孕时才怀上了一个有神经管缺陷的孩子，这就强调了叶酸补充剂作为唯一被证实能降低神经管缺陷风险的预防措施，在所有育龄妇女的围受孕期仍然非常重要。

引用次数: 0

Well-posedness and Tikhonov regularization of an inverse source problem for a parabolic equation with an integral constraint 具有积分约束条件的抛物方程的反源问题的良好拟合和 Tikhonov 正则化

IF 1.1 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems

Pub Date : 2024-04-02 DOI: 10.1515/jiip-2023-0050

Sedar Ngoma

We investigate a time-dependent inverse source problem for a parabolic partial differential equation with an integral constraint and subject to Neumann boundary conditions in a domain of R d

mathbb{R}^{d}

, d ≥ 1

dgeq 1

. We prove the well-posedness as well as higher regularity of solutions in Hölder spaces. We then develop and implement an algorithm that we use to approximate solutions of the inverse problem by means of a finite element discretization in space. Due to instability in inverse problems, we apply Tikhonov regularization combined with the discrepancy principle for selecting the regularization parameter in order to obtain a stable reconstruction. Our numerical results show that the proposed scheme is an accurate technique for approximating solutions of this inverse problem.

我们研究了在 R d mathbb{R}^{d}, d ≥ 1 dgeq 1 的域中，具有积分约束条件并受诺伊曼边界条件限制的抛物线偏微分方程的时变反源问题。我们证明了在赫尔德空间中解的可求性及高正则性。然后，我们开发并实现了一种算法，通过有限元空间离散化来近似求解逆问题。由于逆问题的不稳定性，我们采用提霍诺夫正则化结合差异原则来选择正则化参数，以获得稳定的重构。我们的数值结果表明，所提出的方案是近似求解该逆问题的精确技术。

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引用次数: 0

Adaptive neural network surrogate model for solving the nonlinear elastic inverse problem via Bayesian inference 通过贝叶斯推理解决非线性弹性逆问题的自适应神经网络代用模型

IF 1.1 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems

Pub Date : 2024-03-25 DOI: 10.1515/jiip-2022-0050

Fuchang Huo, Kai Zhang, Yu Gao, Jingzhi Li

In this paper, we consider a Bayesian method for nonlinear elastic inverse problems. As a working model, we are interested in the inverse problem of restoring elastic properties from measured tissue displacement. In order to reduce the computational cost, we will use the following multi-fidelity model approach. First, we construct a surrogate low-fidelity DNNs-based model in the prior distribution, then use a certain number of simulations of high fidelity model associated with an adaptive strategy online to update the low-fidelity model locally. Numerical examples show that the proposed method can solve nonlinear elastic inverse problems efficiently and accurately.

在本文中，我们考虑了一种用于非线性弹性逆问题的贝叶斯方法。作为一个工作模型，我们感兴趣的是根据测量到的组织位移恢复弹性特性的逆问题。为了降低计算成本，我们将采用以下多保真度模型方法。首先，我们在先验分布中构建一个基于 DNNs 的代理低保真模型，然后使用一定数量的高保真模型模拟与在线自适应策略相关联，对低保真模型进行局部更新。数值实例表明，所提出的方法可以高效、准确地解决非线性弹性逆问题。

引用次数: 0

Artificial intelligence for COVID-19 spread modeling 用于 COVID-19 传播建模的人工智能

IF 1.1 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems

Pub Date : 2024-03-19 DOI: 10.1515/jiip-2024-0013

Olga Krivorotko, Sergey Kabanikhin

This paper presents classification and analysis of the mathematical models of the spread of COVID-19 in different groups of population such as family, school, office (3–100 people), town (100–5000 people), city, region (0.5–15 million people), country, continent, and the world. The classification covers major types of models (time-series, differential, imitation ones, neural networks models and their combinations). The time-series models are based on analysis of time series using filtration, regression and network methods. The differential models are those derived from systems of ordinary and stochastic differential equations as well as partial differential equations. The imitation models include cellular automata and agent-based models. The fourth group in the classification consists of combinations of nonlinear Markov chains and optimal control theory, derived by methods of the mean-field game theory. COVID-19 is a novel and complicated disease, and the parameters of most models are, as a rule, unknown and estimated by solving inverse problems. The paper contains an analysis of major algorithms of solving inverse problems: stochastic optimization, nature-inspired algorithms (genetic, differential evolution, particle swarm, etc.), assimilation methods, big-data analysis, and machine learning.

本文对 COVID-19 在家庭、学校、办公室（3-100 人）、城镇（100-5000 人）、城市、地区（50-1500 万人）、国家、大陆和世界等不同人群中传播的数学模型进行了分类和分析。分类包括主要的模型类型（时间序列模型、微分模型、模仿模型、神经网络模型及其组合）。时间序列模型是基于使用过滤、回归和网络方法对时间序列进行分析。微分模型是从常微分方程、随机微分方程和偏微分方程系统中导出的模型。模仿模型包括细胞自动机和基于代理的模型。分类中的第四组包括非线性马尔可夫链和最优控制理论的组合，由均值场博弈论的方法得出。COVID-19 是一种复杂的新型疾病，大多数模型的参数通常是未知的，需要通过求解逆问题来估计。论文分析了解决逆问题的主要算法：随机优化、自然启发算法（遗传、微分进化、粒子群等）、同化方法、大数据分析和机器学习。

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引用次数: 0

On the mean field games system with lateral Cauchy data via Carleman estimates 通过卡勒曼估计论有横向考奇数据的均值场博弈系统

IF 1.1 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems

Pub Date : 2024-01-31 DOI: 10.1515/jiip-2023-0089

Michael V. Klibanov, Jingzhi Li, Hongyu Liu

The second-order mean field games system (MFGS) in a bounded domain with the lateral Cauchy data are considered. This means that both Dirichlet and Neumann boundary data for the solution of the MFGS are given. Two Hölder stability estimates for two slightly different cases are derived. These estimates indicate how stable the solution of the MFGS is with respect to the possible noise in the lateral Cauchy data. Our stability estimates imply uniqueness. The key mathematical apparatus is the apparatus of two new Carleman estimates.

研究考虑了有界域中的二阶均值场博弈系统（MFGS）与横向考奇数据。这意味着 MFGS 解的 Dirichlet 和 Neumann 边界数据均已给出。针对两种略有不同的情况，得出了两个霍尔德稳定性估计值。这些估计值表明了 MFGS 的解对于横向考奇数据中可能存在的噪声有多稳定。我们的稳定性估计值意味着唯一性。关键的数学工具是两个新的卡勒曼估计。

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全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

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Journal of Inverse and Ill-Posed Problems

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