Samat A. Kassabek, Targyn A. Nauryz, Amankeldy Toleukhanov
In this paper, free surface problems of Stefan type for the parabolic heat equation are considered. The analytical solutions of the problems are based on the method of heat polynomials and integral error function in the form of series. Convergence of the series solution is considered and proved. Both one-and two-phase Stefan-type problems are investigated. Numerical results for one-phase inverse Stefan problem are presented and discussed.
{"title":"Analytical solution of Stefan-type problems","authors":"Samat A. Kassabek, Targyn A. Nauryz, Amankeldy Toleukhanov","doi":"10.1515/jiip-2021-0077","DOIUrl":"https://doi.org/10.1515/jiip-2021-0077","url":null,"abstract":"In this paper, free surface problems of Stefan type for the parabolic heat equation are considered. The analytical solutions of the problems are based on the method of heat polynomials and integral error function in the form of series. Convergence of the series solution is considered and proved. Both one-and two-phase Stefan-type problems are investigated. Numerical results for one-phase inverse Stefan problem are presented and discussed.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"728 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Generalized Abel equations have been employed in the recent literature to invert Radon transforms which arise in a number of important imaging applications, including Compton Scatter Tomography (CST), Ultrasound Reflection Tomography (URT), and X-ray CT. In this paper, we present novel injectivity results and inversion methods for generalized Abel operators. We apply our theory to a new Radon transform, Rjmathcal{R}_{j}, of interest in URT, which integrates a square integrable function of compact support, 𝑓, over ellipsoid and hyperboloid surfaces with centers on a plane. Using our newly established theory on generalized Abel equations, we show that Rjmathcal{R}_{j} is injective and provide an inversion method based on Neumann series. In addition, using algebraic methods, we present image phantom reconstructions from Rjfmathcal{R}_{j}f data with added pseudo-random noise.
在最近的文献中，广义阿贝尔方程被用来反演Radon变换，Radon变换出现在许多重要的成像应用中，包括康普顿散射层析成像(CST)、超声反射层析成像(URT)和x射线CT。本文给出了广义Abel算子新的注入性结果和反演方法。我们将我们的理论应用于一个新的Radon变换，R j mathcal{R}_{j}，在URT中，它对紧支撑的平方可积函数𝑓在椭球面和双曲面上的中心在一个平面上进行积分。利用新建立的广义Abel方程理论，证明了rj mathcal{R}_{j}是内射的，并给出了基于Neumann级数的反演方法。此外，利用代数方法，我们提出了从rj _ f mathcal{R}_{j}f数据中加入伪随机噪声重建图像的方法。
{"title":"Generalized Abel equations and applications to translation invariant Radon transforms","authors":"James W. Webber","doi":"10.1515/jiip-2023-0049","DOIUrl":"https://doi.org/10.1515/jiip-2023-0049","url":null,"abstract":"Generalized Abel equations have been employed in the recent literature to invert Radon transforms which arise in a number of important imaging applications, including Compton Scatter Tomography (CST), Ultrasound Reflection Tomography (URT), and X-ray CT. In this paper, we present novel injectivity results and inversion methods for generalized Abel operators. We apply our theory to a new Radon transform, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">R</m:mi> <m:mi>j</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2023-0049_ineq_0001.png\" /> <jats:tex-math>mathcal{R}_{j}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, of interest in URT, which integrates a square integrable function of compact support, 𝑓, over ellipsoid and hyperboloid surfaces with centers on a plane. Using our newly established theory on generalized Abel equations, we show that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">R</m:mi> <m:mi>j</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2023-0049_ineq_0001.png\" /> <jats:tex-math>mathcal{R}_{j}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is injective and provide an inversion method based on Neumann series. In addition, using algebraic methods, we present image phantom reconstructions from <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi mathvariant=\"script\">R</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2023-0049_ineq_0003.png\" /> <jats:tex-math>mathcal{R}_{j}f</jats:tex-math> </jats:alternatives> </jats:inline-formula> data with added pseudo-random noise.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"193 ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138506534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present a new methodology, based on the inverse problem approach, for the determination of an unknown shear force acting on the inaccessible tip of the microcantilever, which is a key component of <jats:italic>transverse dynamic force microscopy</jats:italic> (TDFM). The mathematical modelling of this phenomenon leads to the inverse problem of determining the shear force <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jiip-2023-0021_eq_0228.png" /> <jats:tex-math>{g(t)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> acting on the inaccessible boundary <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>=</m:mo> <m:mi mathvariant="normal">ℓ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jiip-2023-0021_eq_0276.png" /> <jats:tex-math>{x=ell}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in a system governed by the variable coefficient Euler–Bernoulli equation <jats:disp-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>ρ</m:mi> <m:mi>A</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>t</m:mi> <m:mo></m:mo> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>μ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mi>r</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>κ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo rspace="12.5pt">,</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:mrow
在本文中，我们提出了一种基于反问题方法的新方法，用于确定作用于微悬臂梁不可接近尖端的未知剪切力，这是横向动态力显微镜(TDFM)的关键组成部分。这种现象的数学建模导致了一个反问题，即在一个由变系数欧拉-伯努利方程控制的系统中，确定作用在不可达边界{x= x=}{ell}上的剪切力 g(t) g(t)， ρ a (x)减去u t减去t + μ (x)减去u t + (r (x)减去u x减去x + κ (x)减去u x减去x减去t) x减去x= 0， (x, t)∈(0,l) × (0, t)，rho ＿A{(x)}u＿tt{+ }mu (x){u＿t}+(r(x){u＿xx}+ kappa (x){u＿xxt}){＿xx}=0，quad(x,t)% in(0,ell)times(0,T), subject to the homogeneous initial conditions and the boundary conditions u ( 0 , t ) = u 0 ( t ) , u x ( 0 , t ) = 0 , ( u x x ( x , t ) + κ ( x ) u x x t ) x = ℓ = 0 , ( - ( r ( x ) u x x + κ ( x ) u x x t ) x ) x = ℓ = g ( t ) , u(0,t)=u_{0}(t),quad u_{x}(0,t)=0,quad(u_{xx}(x,t)+kappa(x)u_{xxt})_{x=ell% }=0,quadbigl{(}-(r(x)u_{xx}+kappa(x)u_{xxt})_{x}bigr{)}_{x=ell}=g(t), from the final time measured output (displacement) u T ( x ) := u ( x , T ) {u_{T}(x):=u(x,T)} . We introduce the input-output map ( Φ g ) ( x ) := u ( x , T ; g ) {(Phi g)(x):=u(x,T;g)} , g ∈ 𝒢 {ginmathcal{G}} , and prove that it is a compact and Lipschitz continuous linear operator. Then we introduce the Tikhonov functional J ( F ) = 1 2 ∥ Φ g - u T ∥ L 2 ( 0 , ℓ ) 2 J(F)=frac{1}{2}lVertPhi g-u_{T}rVert_{L^{2}(0,ell)}^{2} and prove the existence of a quasi-solution of the inverse problem. We derive a gradient formula for the Fréchet gradient of the Tikhonov functional through the corresponding adjoint problem solution and prove that it is a Lipschitz continuous functional. The results of the numerical experiments clearly illustrate the effectiveness and feasibility of the proposed approach.
{"title":"Determination of unknown shear force in transverse dynamic force microscopy from measured final data","authors":"Onur Baysal, Alemdar Hasanov, Sakthivel Kumarasamy","doi":"10.1515/jiip-2023-0021","DOIUrl":"https://doi.org/10.1515/jiip-2023-0021","url":null,"abstract":"In this paper, we present a new methodology, based on the inverse problem approach, for the determination of an unknown shear force acting on the inaccessible tip of the microcantilever, which is a key component of <jats:italic>transverse dynamic force microscopy</jats:italic> (TDFM). The mathematical modelling of this phenomenon leads to the inverse problem of determining the shear force <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2023-0021_eq_0228.png\" /> <jats:tex-math>{g(t)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> acting on the inaccessible boundary <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>x</m:mi> <m:mo>=</m:mo> <m:mi mathvariant=\"normal\">ℓ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2023-0021_eq_0276.png\" /> <jats:tex-math>{x=ell}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in a system governed by the variable coefficient Euler–Bernoulli equation <jats:disp-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>ρ</m:mi> <m:mi>A</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>t</m:mi> <m:mo></m:mo> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>μ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mrow> <m:mi>r</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>κ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>t</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo rspace=\"12.5pt\">,</m:mo> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:mrow","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"13 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P Asensio, J-M Badier, J Leblond, J-P Marmorat, M Nemaire
Abstract We study the inverse source localisation problem using the electric potential measured point-wise inside the head with stereo-ElectroEncephaloGraphy (sEEG), the electric potential measured point-wise on the scalp with ElectroEncephaloGraphy (EEG) or the magnetic flux density measured point-wise outside the head with MagnetoEncephaloGraphy (MEG). We present a method that works on a wide range of models of primary currents; in particular, we give details for primary currents that are assumed to be smooth vector fields that are supported on and normally oriented to the grey/white matter interface. Irrespective of the data used, we also solve the transmission problem of the electric potential associated with a recovered source; hence we solve the cortical mapping problem. To ensure that the electric potential and normal currents are continuous in the head, the electric potential is expressed as a linear combination of double layer potentials and the magnetic flux density is expressed as a linear combination of single layer potentials. Numerically, we solve the problems on meshed surfaces of the grey/white matter interface, cortical surface, skull and scalp. A main feature of the numerical approach we take is that, on the meshed surfaces, we can compute the double and single layer potentials exactly and at arbitrary points. Because we explicitly study the transmission of the electric potential in head when using magnetic data, the coupling of electric and magnetic data in the source recovery problem is made explicit and shows the advantage of using simultaneous electric and magnetic data. We provide numerical examples of the source recovery and inverse cortical mapping using synthetic data.
{"title":"A layer potential approach to inverse problems in brain imaging","authors":"P Asensio, J-M Badier, J Leblond, J-P Marmorat, M Nemaire","doi":"10.1515/jiip-2023-0041","DOIUrl":"https://doi.org/10.1515/jiip-2023-0041","url":null,"abstract":"Abstract We study the inverse source localisation problem using the electric potential measured point-wise inside the head with stereo-ElectroEncephaloGraphy (sEEG), the electric potential measured point-wise on the scalp with ElectroEncephaloGraphy (EEG) or the magnetic flux density measured point-wise outside the head with MagnetoEncephaloGraphy (MEG). We present a method that works on a wide range of models of primary currents; in particular, we give details for primary currents that are assumed to be smooth vector fields that are supported on and normally oriented to the grey/white matter interface. Irrespective of the data used, we also solve the transmission problem of the electric potential associated with a recovered source; hence we solve the cortical mapping problem. To ensure that the electric potential and normal currents are continuous in the head, the electric potential is expressed as a linear combination of double layer potentials and the magnetic flux density is expressed as a linear combination of single layer potentials. Numerically, we solve the problems on meshed surfaces of the grey/white matter interface, cortical surface, skull and scalp. A main feature of the numerical approach we take is that, on the meshed surfaces, we can compute the double and single layer potentials exactly and at arbitrary points. Because we explicitly study the transmission of the electric potential in head when using magnetic data, the coupling of electric and magnetic data in the source recovery problem is made explicit and shows the advantage of using simultaneous electric and magnetic data. We provide numerical examples of the source recovery and inverse cortical mapping using synthetic data.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"29 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135041582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The turbulent exchange in boundary layer models is usually characterized by a scalar eddy viscosity coefficient assumed to be a positive function of the vertical variable. We introduce a more general form for the turbulence exchange description, which includes two functions that describe the turbulence without any assumption about their positivity. We construct a model of the Akerblom–Ekman type, but with a complex coefficient of turbulent exchange. The basic quality criterion for these models and algorithms is the maximal agreement with meteorological observations. We optimize the agreement between the global meteorological archive of high-resolution wind observations that are provided by World Meteorological Organization (WMO) in Binary Universal Form for the Representation (BUFR). The main result of our work is that agreement between model solutions and observations will be much better if the turbulent exchange coefficient is optimized in the space of all complex-valued functions, and not limited to the cone of real positive functions.
{"title":"Complex turbulent exchange coefficient in Akerblom–Ekman model","authors":"Philipp L. Bykov, Vladimir A. Gordin","doi":"10.1515/jiip-2021-0039","DOIUrl":"https://doi.org/10.1515/jiip-2021-0039","url":null,"abstract":"Abstract The turbulent exchange in boundary layer models is usually characterized by a scalar eddy viscosity coefficient assumed to be a positive function of the vertical variable. We introduce a more general form for the turbulence exchange description, which includes two functions that describe the turbulence without any assumption about their positivity. We construct a model of the Akerblom–Ekman type, but with a complex coefficient of turbulent exchange. The basic quality criterion for these models and algorithms is the maximal agreement with meteorological observations. We optimize the agreement between the global meteorological archive of high-resolution wind observations that are provided by World Meteorological Organization (WMO) in Binary Universal Form for the Representation (BUFR). The main result of our work is that agreement between model solutions and observations will be much better if the turbulent exchange coefficient is optimized in the space of all complex-valued functions, and not limited to the cone of real positive functions.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"136 S236","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135776404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method.
在2021年，Fu Z.， Y. Chen和B. Han利用非平稳迭代Tikhonov正则化方案的思想引入了求解非线性不适定算子方程的非精确牛顿正则化(regin - it)。在本文中，我们提出了regin - it方案的简化版本，使用Bregman距离、对偶映射和合适的参数选择策略来产生近似解。该方法由内部和外部迭代步骤组成。外部迭代通过morozov类型停止规则停止，而内部迭代通过使用非平稳迭代Tikhonov方案执行。我们利用凸分析的工具，在一些标准假设下研究了该方法的收敛性。该方法的新颖之处在于，它只需要在对精确解的初步猜测时计算fr切特导数，因此可以确定为比zz . Fu, Y. Chen和B. Han给出的方法更有效。此外，在论文的最后一部分，我们讨论了测试实例来检验该方法的熟练程度。
{"title":"Simplified REGINN-IT method in Banach spaces for nonlinear ill-posed operator equations","authors":"Pallavi Mahale, Farheen M. Shaikh","doi":"10.1515/jiip-2023-0045","DOIUrl":"https://doi.org/10.1515/jiip-2023-0045","url":null,"abstract":"Abstract In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136158930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A fast numerical algorithm for solving the Cauchy problem for elliptic equations with variable coefficients in standard calculation domains (rectangles, circles, or rings) is proposed. The algorithm is designed to calculate the heat flux at the inaccessible boundary. It is based on the separation of variables method. This approach employs a finite difference approximation and allows obtaining a solution to a discrete problem in arithmetic operations of the order of NlnN Noperatorname{ln}N , where 𝑁 is the number of grid points. As a rule, iterative procedures are needed to solve the Cauchy problem for elliptic equations. The currently available direct algorithms for solving the Cauchy problem have been developed only for (Laplace, Helmholtz) operators with constant coefficients and for use of analytical solutions for problems with such operators. A novel feature of the results of the present paper is that the direct algorithm can be used for an elliptic operator with variable coefficients (of a special form). It is important that in this case no analytical solution to the problem can be obtained. The algorithm significantly increases the range of problems that can be solved. It can be used to create devices for determining in real time heat fluxes on the parts of inhomogeneous constructions that cannot be measured. For example, to determine the heat flux on the inner radius of a pipe made of different materials.
{"title":"Direct numerical algorithm for calculating the heat flux at an inaccessible boundary","authors":"Sergey B. Sorokin","doi":"10.1515/jiip-2022-0032","DOIUrl":"https://doi.org/10.1515/jiip-2022-0032","url":null,"abstract":"Abstract A fast numerical algorithm for solving the Cauchy problem for elliptic equations with variable coefficients in standard calculation domains (rectangles, circles, or rings) is proposed. The algorithm is designed to calculate the heat flux at the inaccessible boundary. It is based on the separation of variables method. This approach employs a finite difference approximation and allows obtaining a solution to a discrete problem in arithmetic operations of the order of <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>N</m:mi> <m:mo lspace=\"0.167em\"></m:mo> <m:mrow> <m:mi>ln</m:mi> <m:mo lspace=\"0.167em\"></m:mo> <m:mi>N</m:mi> </m:mrow> </m:mrow> </m:math> Noperatorname{ln}N , where 𝑁 is the number of grid points. As a rule, iterative procedures are needed to solve the Cauchy problem for elliptic equations. The currently available direct algorithms for solving the Cauchy problem have been developed only for (Laplace, Helmholtz) operators with constant coefficients and for use of analytical solutions for problems with such operators. A novel feature of the results of the present paper is that the direct algorithm can be used for an elliptic operator with variable coefficients (of a special form). It is important that in this case no analytical solution to the problem can be obtained. The algorithm significantly increases the range of problems that can be solved. It can be used to create devices for determining in real time heat fluxes on the parts of inhomogeneous constructions that cannot be measured. For example, to determine the heat flux on the inner radius of a pipe made of different materials.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136318670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this study, singular Sturm–Liouville operators on a star graph with edges are investigated. First, the behavior of sufficiently large eigenvalues is learned. Then the solution of the inverse problem is given to determine the potential functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points. Lastly, a constructive solution to the inverse problems of this class is obtained.
{"title":"Inverse nodal problem for singular Sturm–Liouville operator on a star graph","authors":"Rauf Amirov, Merve Arslantaş, Sevim Durak","doi":"10.1515/jiip-2023-0055","DOIUrl":"https://doi.org/10.1515/jiip-2023-0055","url":null,"abstract":"Abstract In this study, singular Sturm–Liouville operators on a star graph with edges are investigated. First, the behavior of sufficiently large eigenvalues is learned. Then the solution of the inverse problem is given to determine the potential functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points. Lastly, a constructive solution to the inverse problems of this class is obtained.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136318667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cristian Vega, Cesare Molinari, Lorenzo Rosasco, Silvia Villa
Abstract Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to select a meaningful solution is to introduce a regularizer. While for most applications the regularizer is convex, in many cases it is neither smooth nor strongly convex. In this paper, we propose and study two new iterative regularization methods, based on a primal-dual algorithm, to regularize inverse problems efficiently. Our analysis, in the noise free case, provides convergence rates for the Lagrangian and the feasibility gap. In the noisy case, it provides stability bounds and early stopping rules with theoretical guarantees. The main novelty of our work is the exploitation of some a priori knowledge about the solution set: we show that the linear equations determined by the data can be used more than once along the iterations. We discuss various approaches to reuse linear equations that are at the same time consistent with our assumptions and flexible in the implementation. Finally, we illustrate our theoretical findings with numerical simulations for robust sparse recovery and image reconstruction. We confirm the efficiency of the proposed regularization approaches, comparing the results with state-of-the-art methods.
{"title":"Fast iterative regularization by reusing data","authors":"Cristian Vega, Cesare Molinari, Lorenzo Rosasco, Silvia Villa","doi":"10.1515/jiip-2023-0009","DOIUrl":"https://doi.org/10.1515/jiip-2023-0009","url":null,"abstract":"Abstract Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to select a meaningful solution is to introduce a regularizer. While for most applications the regularizer is convex, in many cases it is neither smooth nor strongly convex. In this paper, we propose and study two new iterative regularization methods, based on a primal-dual algorithm, to regularize inverse problems efficiently. Our analysis, in the noise free case, provides convergence rates for the Lagrangian and the feasibility gap. In the noisy case, it provides stability bounds and early stopping rules with theoretical guarantees. The main novelty of our work is the exploitation of some a priori knowledge about the solution set: we show that the linear equations determined by the data can be used more than once along the iterations. We discuss various approaches to reuse linear equations that are at the same time consistent with our assumptions and flexible in the implementation. Finally, we illustrate our theoretical findings with numerical simulations for robust sparse recovery and image reconstruction. We confirm the efficiency of the proposed regularization approaches, comparing the results with state-of-the-art methods.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136318489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Extracting the useful information has been used almost everywhere in many fields of mathematics and applied mathematics. It is a classical ill-posed problem due to the unstable dependence of approximations on small perturbation of the data. The traditional regularization methods depend on the choice of the regularization parameter, which are closely related to an available accurate upper bound of noise level; thus it is not appropriate for the randomly distributed noise with big or unknown variance. In this paper, a purely data driven statistical regularization method is proposed, effectively extracting the information from randomly noisy observations. The rigorous upper bound estimation of confidence interval of the error in L 2 L^{2} norm is established, and some numerical examples are provided to illustrate the effectiveness and computational performance of the method.
{"title":"Extract the information via multiple repeated observations under randomly distributed noise","authors":"Min Zhong, Xinyan Li, Xiaoman Liu","doi":"10.1515/jiip-2022-0063","DOIUrl":"https://doi.org/10.1515/jiip-2022-0063","url":null,"abstract":"Abstract Extracting the useful information has been used almost everywhere in many fields of mathematics and applied mathematics. It is a classical ill-posed problem due to the unstable dependence of approximations on small perturbation of the data. The traditional regularization methods depend on the choice of the regularization parameter, which are closely related to an available accurate upper bound of noise level; thus it is not appropriate for the randomly distributed noise with big or unknown variance. In this paper, a purely data driven statistical regularization method is proposed, effectively extracting the information from randomly noisy observations. The rigorous upper bound estimation of confidence interval of the error in L 2 L^{2} norm is established, and some numerical examples are provided to illustrate the effectiveness and computational performance of the method.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"4300 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136318491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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