Alexander V. Rogovoy, Tynysbek S. Kalmenov, Sergey I. Kabanikhin
Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions.
{"title":"The overdetermined Cauchy problem for the hyperbolic Gellerstedt equation","authors":"Alexander V. Rogovoy, Tynysbek S. Kalmenov, Sergey I. Kabanikhin","doi":"10.1515/jiip-2024-0037","DOIUrl":"https://doi.org/10.1515/jiip-2024-0037","url":null,"abstract":"Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"33 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182649","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}
The probe and singular sources methods are two well-known classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. In this paper, by considering an inverse obstacle problem governed by the Laplace equation in a bounded domain as a prototype case, an integrated theory of the probe and singular sources methods is proposed. The theory consists of three parts: (i) introducing the singular sources method combined with the notion of the probe method; (ii) finding a third indicator function whose two ways decomposition yields the indicator functions in the probe and singular sources methods; (iii) finding the completely integrated version of the probe and singular sources methods.
{"title":"Integrating the probe and singular sources methods","authors":"Masaru Ikehata","doi":"10.1515/jiip-2024-0006","DOIUrl":"https://doi.org/10.1515/jiip-2024-0006","url":null,"abstract":"The probe and singular sources methods are two well-known classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. In this paper, by considering an inverse obstacle problem governed by the Laplace equation in a bounded domain as a prototype case, an integrated theory of the probe and singular sources methods is proposed. The theory consists of three parts: (i) introducing the singular sources method combined with the notion of the probe method; (ii) finding <jats:italic>a third indicator function</jats:italic> whose two ways decomposition yields the indicator functions in the probe and singular sources methods; (iii) finding the completely integrated version of the probe and singular sources methods.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182612","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}
Leyla Soudani, Abdelkader Amara, Khaled Zennir, Junaid Ahmad
The main goal of this paper is to investigate a newly proposed hybrid and hybrid inclusion problem consisting of fractional differential problems involving two different fractional derivatives of order μ, Caputo and Liouville–Riemann operators, with multi-order mixed Riemann–Liouville integro-derivative conditions. Although α is between one and two, we need three boundary value conditions to find the integral equation. The study investigates the results of existence for hybrid, hybrid inclusion, and non-hybrid inclusion problems by employing several analytical approaches, including Dhage’s technique, α-ψ{alpha-psi}-contractive mappings, fixed points, and endpoints of the product operators. To further illustrate our findings, we present three examples.
{"title":"Duality of fractional derivatives: On a hybrid and non-hybrid inclusion problem","authors":"Leyla Soudani, Abdelkader Amara, Khaled Zennir, Junaid Ahmad","doi":"10.1515/jiip-2023-0098","DOIUrl":"https://doi.org/10.1515/jiip-2023-0098","url":null,"abstract":"The main goal of this paper is to investigate a newly proposed hybrid and hybrid inclusion problem consisting of fractional differential problems involving two different fractional derivatives of order μ, Caputo and Liouville–Riemann operators, with multi-order mixed Riemann–Liouville integro-derivative conditions. Although α is between one and two, we need three boundary value conditions to find the integral equation. The study investigates the results of existence for hybrid, hybrid inclusion, and non-hybrid inclusion problems by employing several analytical approaches, including Dhage’s technique, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>α</m:mi> <m:mo>-</m:mo> <m:mi>ψ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2023-0098_eq_0355.png\"/> <jats:tex-math>{alpha-psi}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-contractive mappings, fixed points, and endpoints of the product operators. To further illustrate our findings, we present three examples.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"19 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949013","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 formulate the modified iteratively regularized Landweber iteration method in Banach spaces to solve the inverse problems for which the forward operator may be smooth or non-smooth. We study the convergence analysis of the modified method for both the perturbed as well as unperturbed data by utilizing the Hölder stability estimates. In the presence of perturbed data, we terminate the method via a discrepancy principle and show that it is in fact a convergence regularization method that terminates within a few iterations. In the presence of unperturbed data, we show that the iterates converge to the exact solution. Additionally, we deduce the convergence rates in the presence of perturbed as well as unperturbed data. Finally, we discuss two inverse problems on which the method is applicable.
{"title":"A modified iteratively regularized Landweber iteration method: Hölder stability and convergence rates","authors":"Gaurav Mittal, Ankik Kumar Giri","doi":"10.1515/jiip-2023-0070","DOIUrl":"https://doi.org/10.1515/jiip-2023-0070","url":null,"abstract":"In this paper, we formulate the modified iteratively regularized Landweber iteration method in Banach spaces to solve the inverse problems for which the forward operator may be smooth or non-smooth. We study the convergence analysis of the modified method for both the perturbed as well as unperturbed data by utilizing the Hölder stability estimates. In the presence of perturbed data, we terminate the method via a discrepancy principle and show that it is in fact a convergence regularization method that terminates within a few iterations. In the presence of unperturbed data, we show that the iterates converge to the exact solution. Additionally, we deduce the convergence rates in the presence of perturbed as well as unperturbed data. Finally, we discuss two inverse problems on which the method is applicable.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"94 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969823","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}
A problem of identification of the set of a thermal-conductivity coefficients for the nonlinear heat equation in the case of non-uniqueness is considered. Classes of inverse heat conduction problems (IHCP) with a non-unique solution are defined. Explicit descriptions of sets of thermal-conductivity coefficients for these classes are obtained. For solving the identification problem the functional identification approach is used. Unlike traditional methods, the proposed algorithm does not utilize approximations of the coefficient with a finite system of basis functions. The results of computational experiments are presented. It is shown that the functional identification approach makes it possible to numerically identify the non-uniqueness of the solution of the inverse problem of heat conduction.
{"title":"The inverse problem of heat conduction in the case of non-uniqueness: A functional identification approach","authors":"Valentin Terentievich Borukhov, Galina M. Zayats","doi":"10.1515/jiip-2022-0056","DOIUrl":"https://doi.org/10.1515/jiip-2022-0056","url":null,"abstract":"A problem of identification of the set of a thermal-conductivity coefficients for the nonlinear heat equation in the case of non-uniqueness is considered. Classes of inverse heat conduction problems (IHCP) with a non-unique solution are defined. Explicit descriptions of sets of thermal-conductivity coefficients for these classes are obtained. For solving the identification problem the functional identification approach is used. Unlike traditional methods, the proposed algorithm does not utilize approximations of the coefficient with a finite system of basis functions. The results of computational experiments are presented. It is shown that the functional identification approach makes it possible to numerically identify the non-uniqueness of the solution of the inverse problem of heat conduction.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"78 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969920","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}
Artem Mikhaylov, Fedor Meshchaninov, Vasily Ivanov, Igor Labutin, Nikolai Stulov, Evgeny Burnaev, Vladimir Vanovskiy
At present, computationally intensive numerical weather prediction systems based on physics equations are widely used for short-term weather forecasting. In this paper, we investigate the potential of accelerating the Weather Research and Forecasting (WRF-ARW) model using machine learning techniques. Two main approaches are considered. First, we assess the viability of complete replacing the numerical weather model with deep learning models, capable of predicting the full range forecast directly from basic initial data. Second, we consider a “super-resolution” technique involving low-resolution WRF computation and a machine learning based downscaling using coarse-grid forecast for conditioning. The process of downscaling is intrinsically an ill-posed problem. In both categories, several prominent and promising machine learning methods are evaluated and compared on real data from a variety of sources. for the Moscow region Namely, in addition to the ground truth WRF forecasts that were utilized for training, we compare the model predictions against ERA5 reanalysis and measurements from local weather stations. We show that deep learning approaches can be successfully applied to accelerate a numerical model and even produce more realistic forecasts in other aspects. As a practical outcome, this study offers empirically validated guidance for the selection and application of deep learning methods to accelerate the computation of detailed short-term atmospheric forecasts tailored to specific needs.
{"title":"Accelerating regional weather forecasting by super-resolution and data-driven methods","authors":"Artem Mikhaylov, Fedor Meshchaninov, Vasily Ivanov, Igor Labutin, Nikolai Stulov, Evgeny Burnaev, Vladimir Vanovskiy","doi":"10.1515/jiip-2023-0078","DOIUrl":"https://doi.org/10.1515/jiip-2023-0078","url":null,"abstract":"At present, computationally intensive numerical weather prediction systems based on physics equations are widely used for short-term weather forecasting. In this paper, we investigate the potential of accelerating the Weather Research and Forecasting (WRF-ARW) model using machine learning techniques. Two main approaches are considered. First, we assess the viability of complete replacing the numerical weather model with deep learning models, capable of predicting the full range forecast directly from basic initial data. Second, we consider a “super-resolution” technique involving low-resolution WRF computation and a machine learning based downscaling using coarse-grid forecast for conditioning. The process of downscaling is intrinsically an ill-posed problem. In both categories, several prominent and promising machine learning methods are evaluated and compared on real data from a variety of sources. for the Moscow region Namely, in addition to the ground truth WRF forecasts that were utilized for training, we compare the model predictions against ERA5 reanalysis and measurements from local weather stations. We show that deep learning approaches can be successfully applied to accelerate a numerical model and even produce more realistic forecasts in other aspects. As a practical outcome, this study offers empirically validated guidance for the selection and application of deep learning methods to accelerate the computation of detailed short-term atmospheric forecasts tailored to specific needs.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"10 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141948898","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}
We recover the conductivity σ at the boundary of a domain from a combination of Dirichlet and Neumann boundary data and generalized power/current density data at the boundary, from a single quite arbitrary set of data, in AET or CDII. The argument is elementary, algebraic and local. More generally, we consider the variable exponent p(⋅){p(,cdot,)}-Laplacian as a forward model with the interior density data σ|∇u|q{sigma|nabla u|^{q}}, and find out that single measurement specifies the boundary conductivity when p-q≥1{p-qgeq 1}, and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both p and q may depend on the spatial variable x, but they are assumed to be a priori known. We illustrate the practical situations with numerical examples with the code available.
我们从 AET 或 CDII 中的一组相当任意的数据中，结合 Dirichlet 和 Neumann 边界数据以及边界处的广义功率/电流密度数据，恢复域边界处的电导率 σ。论证是基本的、代数的和局部的。更一般地说，我们将可变指数 p ( ⋅ ) {p(,cdot,)} - 拉普拉斯视为一个前向模型，其内部密度数据为 σ | ∇ u | q {sigma|nabla u|^{q}} ，并发现单次测量就能指定边界的功率/电流密度数据。 并发现当 p - q ≥ 1 {p-qgeq 1} 时，单次测量指定了边界电导率，否则测量指定了两个备选方案。 否则，测量会指定两个备选方案。我们提出了在这些备选方案中进行选择的启发式方法。p 和 q 都可能取决于空间变量 x，但假设它们是先验已知的。我们将用可用代码中的数值示例来说明实际情况。
{"title":"Boundary determination for hybrid imaging from a single measurement","authors":"Tommi Brander, Torbjørn Ringholm","doi":"10.1515/jiip-2019-0083","DOIUrl":"https://doi.org/10.1515/jiip-2019-0083","url":null,"abstract":"We recover the conductivity σ at the boundary of a domain from a combination of Dirichlet and Neumann boundary data and generalized power/current density data at the boundary, from a single quite arbitrary set of data, in AET or CDII. The argument is elementary, algebraic and local. More generally, we consider the variable exponent <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo></m:mo> <m:mrow> <m:mo rspace=\"4.2pt\" stretchy=\"false\">(</m:mo> <m:mo rspace=\"4.2pt\">⋅</m:mo> <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-2019-0083_eq_0208.png\"/> <jats:tex-math>{p(,cdot,)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-Laplacian as a forward model with the interior density data <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>σ</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mi>q</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2019-0083_eq_0184.png\"/> <jats:tex-math>{sigma|nabla u|^{q}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and find out that single measurement specifies the boundary conductivity when <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2019-0083_eq_0222.png\"/> <jats:tex-math>{p-qgeq 1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both <jats:italic>p</jats:italic> and <jats:italic>q</jats:italic> may depend on the spatial variable <jats:italic>x</jats:italic>, but they are assumed to be a priori known. We illustrate the practical situations with numerical examples with the code available.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"18 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949015","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}
Dinh Nho Hào, Nguyen Trung Thành, Nguyen Van Duc, Nguyen Van Thang
The inverse problem of reconstructing two space-varying coefficients in a system of one-dimensional (1-d) time-dependent advection-diffusion-reaction (ADR) equations is considered. The ADR system can be used as a water quality model which describes the evolution of the biochemical oxygen demand (BOD) and dissolved oxygen (DO) in a river or stream. The coefficients to be reconstructed represents the effect of the deoxygenation and superficial reaeration processes on the DO and BOD concentration in water. Hölder stability estimates for the coefficients of interest are established using the Carleman estimate technique.
{"title":"A coefficient identification problem for a system of advection-diffusion-reaction equations in water quality modeling","authors":"Dinh Nho Hào, Nguyen Trung Thành, Nguyen Van Duc, Nguyen Van Thang","doi":"10.1515/jiip-2024-0030","DOIUrl":"https://doi.org/10.1515/jiip-2024-0030","url":null,"abstract":"The inverse problem of reconstructing two space-varying coefficients in a system of one-dimensional (1-d) time-dependent advection-diffusion-reaction (ADR) equations is considered. The ADR system can be used as a water quality model which describes the evolution of the biochemical oxygen demand (BOD) and dissolved oxygen (DO) in a river or stream. The coefficients to be reconstructed represents the effect of the deoxygenation and superficial reaeration processes on the DO and BOD concentration in water. Hölder stability estimates for the coefficients of interest are established using the Carleman estimate technique.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"13 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949012","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}
We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the Hausdorff-, Cesàro-, integration operator, and their adjoints, as well as some combinations of those. For the composition of the Hausdorff- and the Cesàro-operator, we give estimates of the decay of the corresponding singular values. As a curiosity, this provides also an example of two practically relevant non-compact operators, for which their composition is compact. Furthermore, we characterize those operators for which a composition with a non-compact operator gives a compact one.
{"title":"Curious ill-posedness phenomena in the composition of non-compact linear operators in Hilbert spaces","authors":"Stefan Kindermann, Bernd Hofmann","doi":"10.1515/jiip-2024-0007","DOIUrl":"https://doi.org/10.1515/jiip-2024-0007","url":null,"abstract":"We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the Hausdorff-, Cesàro-, integration operator, and their adjoints, as well as some combinations of those. For the composition of the Hausdorff- and the Cesàro-operator, we give estimates of the decay of the corresponding singular values. As a curiosity, this provides also an example of two practically relevant non-compact operators, for which their composition is compact. Furthermore, we characterize those operators for which a composition with a non-compact operator gives a compact one.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"20 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141948899","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}
Sparse identification of nonlinear dynamics is a popular approach to system identification. In this approach system identification is reformulated as a sparse regression problem, and the use of a good sparse regression method is crucial. Sparse Bayesian learning based on collaborative neurodynamic optimization is a recent method that consistently produces high-quality solutions. In this article, we extensively assess how this method performs for ordinary differential equation identification. We find that it works very well compared with sparse regression algorithms currently used for this task in terms of the tradeoff between the approximation accuracy and the complexity of the identified system. We also propose a way to substantially reduce the computational complexity of this algorithm compared with its original implementation, thus making it even more practical.
{"title":"Nonlinear system identification via sparse Bayesian regression based on collaborative neurodynamic optimization","authors":"Alexey Okunev, Evgeny Burnaev","doi":"10.1515/jiip-2023-0077","DOIUrl":"https://doi.org/10.1515/jiip-2023-0077","url":null,"abstract":"Sparse identification of nonlinear dynamics is a popular approach to system identification. In this approach system identification is reformulated as a sparse regression problem, and the use of a good sparse regression method is crucial. Sparse Bayesian learning based on collaborative neurodynamic optimization is a recent method that consistently produces high-quality solutions. In this article, we extensively assess how this method performs for ordinary differential equation identification. We find that it works very well compared with sparse regression algorithms currently used for this task in terms of the tradeoff between the approximation accuracy and the complexity of the identified system. We also propose a way to substantially reduce the computational complexity of this algorithm compared with its original implementation, thus making it even more practical.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"5 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949014","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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