检索结果-内蒙古大学图书馆

Gradient-type Approaches to inverse and ill-posed problems of Mathematical Physics

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2024年第9期64卷 1974-1990页

作者： Akindinov, G. D. Gasnikov, A. V. Krivorotko, O. I. Matyukhin, V. V. Pletnev, N. V. Moscow Inst Phys & Technol Dolgoprudnyi 141700 Moscow Russia RAS IITP Moscow 127051 Russia Sobolev Inst Math Novosibirsk 630090 Russia Innopolis Univ Innopolis 420500 Russia

In this paper we are trying to formalize the definition of gradient for inverse problems and other non-classical optimization problems such as gradient approaches to problems having to do with the calculus of variations. We provide the readers with several examples of converting the problem to the gradient-type problem, describe several properties of the obtained gradient problem, provide analysis on its convexity or smoothness, and proving the convergence of the algorithm.

关键词： inverse and ill-posed problems inexact gradient accelerated gradient methods regularization Richardson extrapolation mathematical physics

来源：评论

学校读者我要写书评

暂无评论

An inverse problems Approach to Pulse Wave Analysis in the Human Brain

引用

SIAM JOURNAL ON IMAGING SCIENCES 2025年第1期18卷 60-88页

作者： Weissinger, Lukas Hubmer, Simon Ramlau, Ronny Voss, Henning U. Johann Radon Inst Linz Altenbeigerstr 69 A-4040 Linz Austria Johannes Kepler Univ Linz Inst Ind Math Altenbergerstr 69 A-4040 Linz Austria Cornell Univ Coll Human Ecol Cornell MRI Facil Martha Van Rensselaer Hall Ithaca NY 14870 USA

Cardiac pulsations in the human brain have received recent interest due to their possible role in the pathogenesis of neurodegenerative diseases. Further interest stems from their possible application as an endogenous signal source that can be utilized for brain imaging in general. The (pulse-)wave describing the blood flow velocity along an intracranial artery consists of a forward (anterograde) and a backward (retrograde, reflected) part, but measurements of this wave usually consist of a superposition of these components. In this paper, we provide a mathematical framework for the inverse problem of estimating the pulse wave velocity, as well as the forward and backward components of the pulse wave separately from MRI measurements on intracranial arteries. After a mathematical analysis of this problem, we consider possible reconstruction approaches and derive an alternate direction approach for its solution. The resulting methods provide estimates for anterograde/retrograde wave forms and the pulse wave velocity under specified assumptions on a cerebrovascular model system. Numerical experiments on simulated and experimental data demonstrate the applicability and preliminary in vivo feasibility of the proposed methods.

关键词： pulse wave splitting medical imaging magnetic resonance imaging inverse and ill-posed problems Tikhonov regularization alternate direction method

来源：评论

学校读者我要写书评

暂无评论

Norms in sinogram space and stability estimates for the Radon transform

引用

inverse problems 2025年第2期41卷 025008-025008页

作者： Kindermann, Stefan Hubmer, Simon Johannes Kepler Univ Linz Inst Ind Math A-4040 Linz Austria

We consider different norms for the Radon transform Rf of a function f and investigate under which conditions they can be estimated from above or below by some standard norms for f. We define Fourier-based norms for Rf which can be related to Bessel-potential space norms for f. Furthermore, we define a variant of a total-variation norm for Rf and provide conditions under which it is equivalent to the total-variation norm of f. To illustrate potential applications of these results, we propose a novel nonlinear backprojection method for inverting the Radon transform and present numerical results on simulated and experimental data.

关键词： inverse and ill-posed problems radon transform stability estimates total variation regularization

来源：评论

学校读者我要写书评

暂无评论

A Projected Nesterov-Kaczmarz Approach to Stellar Population-Kinematic Distribution Reconstruction in Extragalactic Archaeology

引用

SIAM JOURNAL ON IMAGING SCIENCES 2023年第1期16卷 192-222页

作者： Hinterer, Fabian Hubmer, Simon Jethwa, Prashin Soodhalter, Kirk M. van de Ven, Glenn Ramlau, Ronny Johannes Kepler Univ Linz Inst Ind Math Altenbergerstr 69 A-4040 Linz Austria Johann Radon Inst Linz Altenbergerstr 69 A-4040 Linz Austria Univ Vienna Dept Astrophys Turkenschanzstr 17 A-1180 Vienna Austria Trinity Coll Dublin Sch Math Dublin Ireland Johannes Kepler Univ Linz Altenbergerstr 69 A-4040 Linz Austria

In this paper, we consider the problem of reconstructing a galaxy's stellar population-kinematic distribution function from optical integral field unit measurements. These quantities are connected via a high-dimensional integral equation. To solve this problem, we propose a projected Nesterov-Kaczmarz reconstruction method, which efficiently leverages the problem structure and incorporates physical prior information such as smoothness and nonnegativity constraints. To test the perfor-mance of our reconstruction approach, we apply it to a dataset simulated from a known ground truth density, and validate it by comparing our recoveries to those obtained by the widely used pPXF software.

关键词： astrophysics galactic archaeology inverse and ill-posed problems Kaczmarz method Nesterov acceleration large scale problems

来源：评论

学校读者我要写书评

暂无评论

A NUMERICAL COMPARISON OF SOME HEURISTIC STOPPING RULES FOR NONLINEAR LANDWEBER ITERATION

引用

ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2022年 57卷 216-241页

作者： Hubmer, Simon Sherina, Ekaterina Kindermann, Stefan Raik, Kemal Johann Radon Inst Linz Altenbergerstr 69 A-4040 Linz Austria Univ Vienna Dept Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Johannes Kepler Univ Linz Ind Math Inst Altenbergerstr 69 A-4040 Linz Austria

The choice of a suitable regularization parameter is an important part of most regularization methods for inverse problems. In the absence of reliable estimates of the noise level, heuristic parameter choice rules can be used to accomplish this task. While they are already fairly well understood and tested in the case of linear problems, not much is known about their behaviour for nonlinear problems and even less in the respective case of iterative regularization. Hence, in this paper, we numerically study the performance of some of these rules when used to determine a stopping index for Landweber iteration for various nonlinear inverse problems. These are chosen from different practically relevant fields such as integral equations, parameter estimation, and tomography.

关键词： heuristic parameter choice rules Landweber iteration inverse and ill-posed problems nonlinear operator equations numerical comparison

来源：评论

学校读者我要写书评

暂无评论

On regularization via frame decompositions with applications in tomography

引用

inverse problems 2022年第5期38卷 055003-055003页

作者： Hubmer, Simon Ramlau, Ronny Weissinger, Lukas Johann Radon Inst Linz Altenbergerstr 69 A-4040 Linz Austria Johannes Kepler Univ Linz Inst Ind Math Altenbergerstr 69 A-4040 Linz Austria Johannes Kepler Univ Linz Doctoral Program Computat Math Altenbergerstr 69 A-4040 Linz Austria

In this paper, we consider linear ill-posed problems in Hilbert spaces and their regularization via frame decompositions, which are generalizations of the singular-value decomposition. In particular, we prove convergence for a general class of continuous regularization methods and derive convergence rates under both a priori and a posteriori parameter choice rules. Furthermore, we apply our derived results to a standard tomography problem based on the Radon transform.

关键词： frame decomposition singular-value decomposition inverse and ill-posed problems regularization theory computerized tomography

来源：评论

学校读者我要写书评

暂无评论

Constructing a variational quasi-reversibility method for a Cauchy problem for elliptic equations

引用

MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2021年第5期44卷 3334-3355页

作者： Anh Khoa, Vo Truong Hoang Nhan, Pham Univ N Carolina Dept Math & Stat Charlotte NC 28223 USA VNUHCM Univ Sci Dept Math & Comp Sci Ho Chi Minh City Vietnam Univ Verona Dept Comp Sci Verona Italy

In the recent developments of regularization theory for inverse and ill-posed problems, a variational quasi-reversibility (QR) method has been designed to solve a class of time-reversed quasi-linear parabolic problems. Known as a PDE-based approach, this method relies on adding a suitable perturbing operator to the original problem and consequently, on gaining the corresponding fine stabilized operator, which leads us to a forward-like problem. In this work, we establish new conditional estimates for such operators to solve a prototypical Cauchy problem for elliptic equations. This problem is based on the stationary case of the inverse heat conduction problem, where one wants to identify the heat distribution in a certain medium, given the partial boundary data. Using the new QR method, we obtain a second-order initial value problem for a wave-type equation, whose weak solvability can be deduced using a priori estimates and compactness arguments. Weighted by a Carleman-like function, a new type of energy estimates is explored in a variational setting when we investigate the Holder convergence rate of the proposed scheme. Besides, a linearized version of this scheme is analyzed. Numerical examples are provided to corroborate our theoretical analysis.

关键词： Carleman weight convergence rates energy estimates inverse and ill-posed problems quasi-reversibility method

来源：评论

学校读者我要写书评

暂无评论

Numerical Solution of an inverse Problem for a General Hyperbolic Heat Equation 14th

Numerical Solution of an Inverse Problem for a General H...

引用

14th International Conference on Optimization and Applications, OPTIMA 2023

作者： Akindinov, George Krivorotko, Olga Matyukhin, Vladislav Moscow Institute of Physics and Technology 9 Institute lane Dolgoprudny141701 Russia IITP RAS Bolshoy Karetny per. 19 build.1 Moscow127051 Russia Sobolev Institute of Mathematics 4 Acad. Koptyug avenue Novosibirsk630090 Russia

ISBN: (纸本)9783031487507

In this paper we describe an algorithm of numerical solving of an inverse problem on a general case of a hyperbolic heat equation with additional second time derivative with a small parameter. The problem in this case is finding an initial distribution with given final distribution. This algorithm allows finding a solution to the problem for any admissible given precision. Algorithm allows evading difficulties analogous to the case of heat equation with inverted time. Furthermore, it allows finding an optimal grid size by learning on a relatively big grid size and small amount of iterations of a gradient method and later extrapolates to the required grid size using Richardson’s method. This algorithm allows finding an adequate estimate of Lipschitz constant for the gradient of the target functional. Finally, this algorithm may easily be applied to the problems with similar structure, for example in solving equations for plasma, social processes and various biological problems. The theoretical novelty of the paper consists in the developing of an optimal procedure of finding of the required grid size using Richardson extrapolation in context of ill-posed problems, and accelerated methods were applied to this problem. © 2024, The Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： Accelerated gradient methods Hyperbolic heat equation Inexact gradient inverse and ill-posed problems Regularization Richardson’s method

来源：评论

学校读者我要写书评

暂无评论

A frame decomposition of the atmospheric tomography operator

引用

inverse problems 2020年第9期36卷 094001-094001页

作者： Hubmer, Simon Ramlau, Ronny Johann Radon Inst Linz Altenbergerstr 69 A-4040 Linz Austria Johannes Kepler Univ Linz Inst Ind Math Altenbergerstr 69 A-4040 Linz Austria

We consider the problem of atmospheric tomography, as it appears for example in adaptive optics systems for extremely large telescopes. We derive a frame decomposition, i.e., a decomposition in terms of a frame, of the underlying atmospheric tomography operator, extending the singular-value-type decomposition results of Neubauer and Ramlau (2017SIAM J. Appl. Math.77838-853) by allowing a mixture of both natural and laser guide stars, as well as arbitrary aperture shapes. Based on both analytical considerations as well as numerical illustrations, we provide insight into the properties of the derived frame decomposition and its building blocks.

关键词： adaptive optics atmospheric tomography singular value decomposition frame decomposition inverse and ill-posed problems

来源：评论

学校读者我要写书评

暂无评论

A note on the minimization of a Tikhonov functional withl¹-penalty

引用

inverse problems 2020年第7期36卷 074001-074001页

作者： Hinterer, Fabian Hubmer, Simon Ramlau, Ronny Johannes Kepler Univ Linz Inst Ind Math Altenbergerstr 69 A-4040 Linz Austria Johann Radon Inst Linz Altenbergerstr 69 A-4040 Linz Austria

In this paper, we consider the minimization of a Tikhonov functional with anl(1)penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to transform the Tikhonov functional into one with anl(2)penalty term but a nonlinear operator. The transformed problem can then be analyzed and minimized using standard methods. However, by the nature of this transform, the resulting functional is only once continuously differentiable, which prohibits the use of second order methods. Hence, in this paper, we propose a different transformation, which leads to a twice differentiable functional that can now be minimized using efficient second order methods like Newton's method. We provide a convergence analysis of our proposed scheme, as well as a number of numerical results showing the usefulness of our proposed approach.

关键词： inverse and ill-posed problems Tikhonov regularization sparsity second-order methods Newton's method

来源：评论

学校读者我要写书评

暂无评论

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案：

请选择收藏分类：

通借通还

建议与咨询 留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案： 新增检索档案 确定 取消

请选择收藏分类： 新增自定义分类 确定 取消

通借通还

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

请选择保存的检索档案：

请选择收藏分类：