Pub Date : 2022-01-04DOI: 10.1080/17415977.2021.2018427
I. E. Stepanova, T. Gudkova, A. Salnikov, A. Batov
{"title":"A New Approach to Analytical Modeling of Mars’s Magnetic Field","authors":"I. E. Stepanova, T. Gudkova, A. Salnikov, A. Batov","doi":"10.1080/17415977.2021.2018427","DOIUrl":"https://doi.org/10.1080/17415977.2021.2018427","url":null,"abstract":"","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42066636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-24DOI: 10.1080/17415977.2021.2016738
S. Milewski
{"title":"Recovery of thermal load parameters by means of the Monte Carlo method with fixed and meshless random walks","authors":"S. Milewski","doi":"10.1080/17415977.2021.2016738","DOIUrl":"https://doi.org/10.1080/17415977.2021.2016738","url":null,"abstract":"","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44850173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-23DOI: 10.1080/17415977.2021.1949590
M. Alosaimi, D. Lesnic, B. Johansson
We propose a regularization method based on the iterative conjugate gradient method for the solution of a Cauchy problem for the wave equation in one dimension. This linear but ill-posed Cauchy problem consists of finding the displacement and flux on a hostile and inaccessible part of the medium boundary from Cauchy data measurements of the same quantities on the remaining friendly and accessible part of the boundary. This inverse boundary value problem is recast as a least-squares minimization problem that is solved by using the conjugate gradient method whose iterations are stopped according to the discrepancy principle for obtaining stable reconstructions. The objective functional associated is proved Fréchet differentiable and a formula for its gradient is derived. The well-posed direct, adjoint and sensitivity problems present in the conjugate gradient method are solved by using a finite-difference method. Two numerical examples to illustrate the accuracy and stability of the proposed numerical procedure are thoroughly presented and discussed.
{"title":"Solution of the Cauchy problem for the wave equation using iterative regularization","authors":"M. Alosaimi, D. Lesnic, B. Johansson","doi":"10.1080/17415977.2021.1949590","DOIUrl":"https://doi.org/10.1080/17415977.2021.1949590","url":null,"abstract":"We propose a regularization method based on the iterative conjugate gradient method for the solution of a Cauchy problem for the wave equation in one dimension. This linear but ill-posed Cauchy problem consists of finding the displacement and flux on a hostile and inaccessible part of the medium boundary from Cauchy data measurements of the same quantities on the remaining friendly and accessible part of the boundary. This inverse boundary value problem is recast as a least-squares minimization problem that is solved by using the conjugate gradient method whose iterations are stopped according to the discrepancy principle for obtaining stable reconstructions. The objective functional associated is proved Fréchet differentiable and a formula for its gradient is derived. The well-posed direct, adjoint and sensitivity problems present in the conjugate gradient method are solved by using a finite-difference method. Two numerical examples to illustrate the accuracy and stability of the proposed numerical procedure are thoroughly presented and discussed.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"49 1","pages":"2757 - 2771"},"PeriodicalIF":1.3,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/17415977.2021.1949590","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59997975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-23DOI: 10.1080/17415977.2021.1951722
F. Watson, M. G. Crabb, W. Lionheart
For many inverse parameter problems for partial differential equations in which the domain contains only well-separated objects, an asymptotic solution to the forward problem involving ‘polarization tensors’ exists. These are functions of the size and material contrast of inclusions, thereby describing the saturation component of the non-linearity. In this paper, we show how such an asymptotic series can be applied to non-linear least-squares reconstruction problems, by deriving an approximate diagonal Hessian matrix for the data misfit term. Often, the Hessian matrix can play a vital role in dealing with the non-linearity, generating good update directions which accelerate the solution towards a global minimum, but the computational cost can make direct calculation infeasible. Since the polarization tensor approximation assumes sufficient separation between inclusions, our approximate Hessian does not account for non-linearity in the form of lack of superposition in the inverse problem. It does, however, account for the non-linear saturation of the change in the data with increasing material contrast. We, therefore, propose to use it as an initial Hessian for quasi-Newton schemes. We present numerical experimentation into the accuracy and reconstruction performance of the approximate Hessian for the case of electrical impedance tomography, providing a proof of principle of the reconstruction scheme.
{"title":"A polarization tensor approximation for the Hessian in iterative solvers for non-linear inverse problems","authors":"F. Watson, M. G. Crabb, W. Lionheart","doi":"10.1080/17415977.2021.1951722","DOIUrl":"https://doi.org/10.1080/17415977.2021.1951722","url":null,"abstract":"For many inverse parameter problems for partial differential equations in which the domain contains only well-separated objects, an asymptotic solution to the forward problem involving ‘polarization tensors’ exists. These are functions of the size and material contrast of inclusions, thereby describing the saturation component of the non-linearity. In this paper, we show how such an asymptotic series can be applied to non-linear least-squares reconstruction problems, by deriving an approximate diagonal Hessian matrix for the data misfit term. Often, the Hessian matrix can play a vital role in dealing with the non-linearity, generating good update directions which accelerate the solution towards a global minimum, but the computational cost can make direct calculation infeasible. Since the polarization tensor approximation assumes sufficient separation between inclusions, our approximate Hessian does not account for non-linearity in the form of lack of superposition in the inverse problem. It does, however, account for the non-linear saturation of the change in the data with increasing material contrast. We, therefore, propose to use it as an initial Hessian for quasi-Newton schemes. We present numerical experimentation into the accuracy and reconstruction performance of the approximate Hessian for the case of electrical impedance tomography, providing a proof of principle of the reconstruction scheme.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"18 1","pages":"2804 - 2830"},"PeriodicalIF":1.3,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76842049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-12DOI: 10.1080/17415977.2021.2011863
Yuemeng Feng, J. Létang, D. Sarrut, Voichia Maxim
The Compton camera is a gamma ray imaging device expected to provide clinically relevant images in the SPECT applications where collimated cameras are sub-optimal. Its imaging performances depend not only on the design of the detection system but also on choices related to tomographic reconstruction. The aim of this work is to show that the accuracy in modelling the acquisition largely influences the quality of the images. For this purpose, we restrict here to Doppler broadening models in conjunction with the list-mode maximum likelihood expectation maximization (LM-MLEM) algorithm. The study was carried out with Monte-Carlo simulation. We show that the reconstructed point spread function is location-dependent when the model is not accurate, and the usual elongation artefacts well-known in Compton camera imaging will appear. The model we propose allows us to reconstruct isolated point sources and more complex non-uniform sources with improved resolution even in the direction orthogonal to the camera.
{"title":"Influence of Doppler broadening model accuracy in Compton camera list-mode MLEM reconstruction","authors":"Yuemeng Feng, J. Létang, D. Sarrut, Voichia Maxim","doi":"10.1080/17415977.2021.2011863","DOIUrl":"https://doi.org/10.1080/17415977.2021.2011863","url":null,"abstract":"The Compton camera is a gamma ray imaging device expected to provide clinically relevant images in the SPECT applications where collimated cameras are sub-optimal. Its imaging performances depend not only on the design of the detection system but also on choices related to tomographic reconstruction. The aim of this work is to show that the accuracy in modelling the acquisition largely influences the quality of the images. For this purpose, we restrict here to Doppler broadening models in conjunction with the list-mode maximum likelihood expectation maximization (LM-MLEM) algorithm. The study was carried out with Monte-Carlo simulation. We show that the reconstructed point spread function is location-dependent when the model is not accurate, and the usual elongation artefacts well-known in Compton camera imaging will appear. The model we propose allows us to reconstruct isolated point sources and more complex non-uniform sources with improved resolution even in the direction orthogonal to the camera.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"3509 - 3529"},"PeriodicalIF":1.3,"publicationDate":"2021-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48370480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-09DOI: 10.1080/17415977.2021.2009476
A. Carasso
The 2D viscous Burgers equation is a system of two nonlinear equations in two unknowns, . This paper considers the data assimilation problem of finding initial values that can evolve into a close approximation to a desired target result , at some realistic T>0. Highly nonsmooth target data are considered, that may not correspond to actual solutions at time T. Such an ill-posed 2D viscous Burgers problem has not previously been studied. An effective approach is discussed and demonstrated based on recently developed stabilized explicit finite difference schemes that can be run backward in time. Successful data assimilation experiments are presented involving 8 bit, pixel grey-scale images, defined by nondifferentiable intensity data. An instructive example of failure is also included.
{"title":"Data assimilation in 2D viscous Burgers equation using a stabilized explicit finite difference scheme run backward in time","authors":"A. Carasso","doi":"10.1080/17415977.2021.2009476","DOIUrl":"https://doi.org/10.1080/17415977.2021.2009476","url":null,"abstract":"The 2D viscous Burgers equation is a system of two nonlinear equations in two unknowns, . This paper considers the data assimilation problem of finding initial values that can evolve into a close approximation to a desired target result , at some realistic T>0. Highly nonsmooth target data are considered, that may not correspond to actual solutions at time T. Such an ill-posed 2D viscous Burgers problem has not previously been studied. An effective approach is discussed and demonstrated based on recently developed stabilized explicit finite difference schemes that can be run backward in time. Successful data assimilation experiments are presented involving 8 bit, pixel grey-scale images, defined by nondifferentiable intensity data. An instructive example of failure is also included.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"3475 - 3489"},"PeriodicalIF":1.3,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44831309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-09DOI: 10.1080/17415977.2021.2009475
C. Daveau, S. Bornhofen, A. Khelifi, Brice Naisseline
In this paper, we use asymptotic expansion of the velocity field to reconstruct small deformable droplets (i.e. their forms and locations) immersed in an incompressible Newtonian fluid. Here the fluid motion is assumed to be governed by the non-stationary linear Stokes system. Taking advantage of the smallness of the droplets, our asymptotic formula and identification methods extend those already derived for rigid inhomogeneity and for stationary Stokes system. Our derivations, based on dynamical boundary measurements, are rigorous and proved by involving the notion of viscous moment tensor VMT. The viability of our reconstruction approach is documented by numerical results.
{"title":"Identification of deformable droplets from boundary measurements: the case of non-stationary Stokes problem","authors":"C. Daveau, S. Bornhofen, A. Khelifi, Brice Naisseline","doi":"10.1080/17415977.2021.2009475","DOIUrl":"https://doi.org/10.1080/17415977.2021.2009475","url":null,"abstract":"In this paper, we use asymptotic expansion of the velocity field to reconstruct small deformable droplets (i.e. their forms and locations) immersed in an incompressible Newtonian fluid. Here the fluid motion is assumed to be governed by the non-stationary linear Stokes system. Taking advantage of the smallness of the droplets, our asymptotic formula and identification methods extend those already derived for rigid inhomogeneity and for stationary Stokes system. Our derivations, based on dynamical boundary measurements, are rigorous and proved by involving the notion of viscous moment tensor VMT. The viability of our reconstruction approach is documented by numerical results.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"3451 - 3474"},"PeriodicalIF":1.3,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49542437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-21DOI: 10.1080/17415977.2021.2000609
L. Su, V. Vasil'ev, T. Jiang, G. Wang
In this paper, we consider the initial-boundary value problem of determining the stationary right-hand side function in the anomalous diffusion equation with a Caputo fractional derivative with respect to time. The value of the solution of the problem at the final time moment is set as the overdetermination condition. In order to carry out the numerical solution, the iterative conjugate gradient method is used, while at each iteration a direct problem is solved by the finite-difference method using a purely implicit difference scheme. The computational experiment results for the model problem are presented to confirm the efficiency of this new method.
{"title":"Identification of stationary source in the anomalous diffusion equation","authors":"L. Su, V. Vasil'ev, T. Jiang, G. Wang","doi":"10.1080/17415977.2021.2000609","DOIUrl":"https://doi.org/10.1080/17415977.2021.2000609","url":null,"abstract":"In this paper, we consider the initial-boundary value problem of determining the stationary right-hand side function in the anomalous diffusion equation with a Caputo fractional derivative with respect to time. The value of the solution of the problem at the final time moment is set as the overdetermination condition. In order to carry out the numerical solution, the iterative conjugate gradient method is used, while at each iteration a direct problem is solved by the finite-difference method using a purely implicit difference scheme. The computational experiment results for the model problem are presented to confirm the efficiency of this new method.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"3406 - 3422"},"PeriodicalIF":1.3,"publicationDate":"2021-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41907222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-18DOI: 10.1080/17415977.2021.2000606
Asim Ilyas, S. Malik, Summaya Saif
This paper focuses on considering two inverse source problems (ISPs) for a multi-term time-fractional evolution equation with an involution term, interpolating the heat and wave equations. The fractional derivatives are defined in Caputo's sense. The ISPs are proved to be ill-posed in the sense of Hadamard. Recovering a space dependent source term from over-specified data given at some time constitute the first ISP, while in the second ISP determination of a time dependent component of the source term is considered when over-specified condition of integral type is given. The solution of ISPs are constructed by using Fourier's method. The time-dependent components of the solutions are presented in terms of the multinomial Mittag-Leffler function. Under certain conditions, the solutions of ISPs for the multi-term time-fractional evolution equation are shown to be classical solutions. In addition, some particular examples are formulated to illustrate the obtained results for the ISPs.
{"title":"Inverse problems for a multi-term time fractional evolution equation with an involution","authors":"Asim Ilyas, S. Malik, Summaya Saif","doi":"10.1080/17415977.2021.2000606","DOIUrl":"https://doi.org/10.1080/17415977.2021.2000606","url":null,"abstract":"This paper focuses on considering two inverse source problems (ISPs) for a multi-term time-fractional evolution equation with an involution term, interpolating the heat and wave equations. The fractional derivatives are defined in Caputo's sense. The ISPs are proved to be ill-posed in the sense of Hadamard. Recovering a space dependent source term from over-specified data given at some time constitute the first ISP, while in the second ISP determination of a time dependent component of the source term is considered when over-specified condition of integral type is given. The solution of ISPs are constructed by using Fourier's method. The time-dependent components of the solutions are presented in terms of the multinomial Mittag-Leffler function. Under certain conditions, the solutions of ISPs for the multi-term time-fractional evolution equation are shown to be classical solutions. In addition, some particular examples are formulated to illustrate the obtained results for the ISPs.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"3377 - 3405"},"PeriodicalIF":1.3,"publicationDate":"2021-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46445081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-16DOI: 10.1080/17415977.2021.2000977
S. Kassabek, S. Kharin, D. Suragan
In this paper, solutions of one-phase inverse Stefan problems are studied. The approach presented in the paper is an application of the heat polynomials method (HPM) for solving one- and two-dimensional inverse Stefan problems, where the boundary data is reconstructed on a fixed boundary. We present numerical results illustrating an application of the heat polynomials method for several benchmark examples. We study the effects of accuracy and measurement error for different degree of heat polynomials. Due to ill-conditioning of the matrix generated by HPM, optimization techniques are used to obtain regularized solution. Therefore, the sensitivity of the method to the data disturbance is discussed. Theoretical properties of the proposed method, as well as numerical experiments, demonstrate that to reach accurate results it is quite sufficient to consider only a few of the polynomials. The heat flux for two-dimensional inverse Stefan problem is reconstructed and coefficients of a solution function are found approximately.
{"title":"A heat polynomial method for inverse cylindrical one-phase Stefan problems","authors":"S. Kassabek, S. Kharin, D. Suragan","doi":"10.1080/17415977.2021.2000977","DOIUrl":"https://doi.org/10.1080/17415977.2021.2000977","url":null,"abstract":"In this paper, solutions of one-phase inverse Stefan problems are studied. The approach presented in the paper is an application of the heat polynomials method (HPM) for solving one- and two-dimensional inverse Stefan problems, where the boundary data is reconstructed on a fixed boundary. We present numerical results illustrating an application of the heat polynomials method for several benchmark examples. We study the effects of accuracy and measurement error for different degree of heat polynomials. Due to ill-conditioning of the matrix generated by HPM, optimization techniques are used to obtain regularized solution. Therefore, the sensitivity of the method to the data disturbance is discussed. Theoretical properties of the proposed method, as well as numerical experiments, demonstrate that to reach accurate results it is quite sufficient to consider only a few of the polynomials. The heat flux for two-dimensional inverse Stefan problem is reconstructed and coefficients of a solution function are found approximately.","PeriodicalId":54926,"journal":{"name":"Inverse Problems in Science and Engineering","volume":"29 1","pages":"3423 - 3450"},"PeriodicalIF":1.3,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47064441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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