Pub Date : 2024-02-23DOI: 10.1088/1361-6420/ad2531
B M Afkham, K Knudsen, A K Rasmussen, T Tarvainen
This paper considers a Bayesian approach for inclusion detection in nonlinear inverse problems using two known and popular push-forward prior distributions: the star-shaped and level set prior distributions. We analyze the convergence of the corresponding posterior distributions in a small measurement noise limit. The methodology is general; it works for priors arising from any Hölder continuous transformation of Gaussian random fields and is applicable to a range of inverse problems. The level set and star-shaped prior distributions are examples of push-forward priors under Hölder continuous transformations that take advantage of the structure of inclusion detection problems. We show that the corresponding posterior mean converges to the ground truth in a proper probabilistic sense. Numerical tests on a two-dimensional quantitative photoacoustic tomography problem showcase the approach. The results highlight the convergence properties of the posterior distributions and the ability of the methodology to detect inclusions with sufficiently regular boundaries.
{"title":"A Bayesian approach for consistent reconstruction of inclusions","authors":"B M Afkham, K Knudsen, A K Rasmussen, T Tarvainen","doi":"10.1088/1361-6420/ad2531","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2531","url":null,"abstract":"This paper considers a Bayesian approach for inclusion detection in nonlinear inverse problems using two known and popular push-forward prior distributions: the star-shaped and level set prior distributions. We analyze the convergence of the corresponding posterior distributions in a small measurement noise limit. The methodology is general; it works for priors arising from any Hölder continuous transformation of Gaussian random fields and is applicable to a range of inverse problems. The level set and star-shaped prior distributions are examples of push-forward priors under Hölder continuous transformations that take advantage of the structure of inclusion detection problems. We show that the corresponding posterior mean converges to the ground truth in a proper probabilistic sense. Numerical tests on a two-dimensional quantitative photoacoustic tomography problem showcase the approach. The results highlight the convergence properties of the posterior distributions and the ability of the methodology to detect inclusions with sufficiently regular boundaries.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"4 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140004797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-13DOI: 10.1088/1361-6420/ad22e9
Vincenzo Mottola, Antonio Corbo Esposito, Gianpaolo Piscitelli, Antonello Tamburrino
Inverse problems, which are related to Maxwell’s equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behavior of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.
{"title":"Imaging of nonlinear materials via the Monotonicity Principle","authors":"Vincenzo Mottola, Antonio Corbo Esposito, Gianpaolo Piscitelli, Antonello Tamburrino","doi":"10.1088/1361-6420/ad22e9","DOIUrl":"https://doi.org/10.1088/1361-6420/ad22e9","url":null,"abstract":"Inverse problems, which are related to Maxwell’s equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behavior of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"6 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140004658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-05DOI: 10.1088/1361-6420/ad1e2c
Tiangang Cui, Gianluca Detommaso, Robert Scheichl
We present a non-trivial integration of dimension-independent likelihood-informed (DILI) MCMC (Cui et al 2016) and the multilevel MCMC (Dodwell et al 2015) to explore the hierarchy of posterior distributions. This integration offers several advantages: First, DILI-MCMC employs an intrinsic likelihood-informed subspace (LIS) (Cui et al 2014)—which involves a number of forward and adjoint model simulations—to design accelerated operator-weighted proposals. By exploiting the multilevel structure of the discretised parameters and discretised forward models, we design a Rayleigh–Ritz procedure to significantly reduce the computational effort in building the LIS and operating with DILI proposals. Second, the resulting DILI-MCMC can drastically improve the sampling efficiency of MCMC at each level, and hence reduce the integration error of the multilevel algorithm for fixed CPU time. Numerical results confirm the improved computational efficiency of the multilevel DILI approach.
我们提出了一种与维度无关的似然信息（DILI）MCMC（Cui 等人，2016 年）和多级 MCMC（Dodwell 等人，2015 年）的非难整合，以探索后验分布的层次结构。这种整合具有几个优势：首先，DILI-MCMC 采用内在似然信息子空间（LIS）（Cui 等人，2014 年）--其中涉及大量前向和邻接模型模拟--来设计加速算子加权建议。通过利用离散参数和离散前向模型的多层次结构，我们设计了一种 Rayleigh-Ritz 程序，以显著减少构建 LIS 和使用 DILI 建议的计算量。其次，由此产生的 DILI-MCMC 可以大幅提高各层次 MCMC 的采样效率，从而在 CPU 时间固定的情况下降低多层次算法的积分误差。数值结果证实了多级 DILI 方法提高了计算效率。
{"title":"Multilevel dimension-independent likelihood-informed MCMC for large-scale inverse problems","authors":"Tiangang Cui, Gianluca Detommaso, Robert Scheichl","doi":"10.1088/1361-6420/ad1e2c","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1e2c","url":null,"abstract":"We present a non-trivial integration of dimension-independent likelihood-informed (DILI) MCMC (Cui <italic toggle=\"yes\">et al</italic> 2016) and the multilevel MCMC (Dodwell <italic toggle=\"yes\">et al</italic> 2015) to explore the hierarchy of posterior distributions. This integration offers several advantages: First, DILI-MCMC employs an intrinsic <italic toggle=\"yes\">likelihood-informed subspace</italic> (LIS) (Cui <italic toggle=\"yes\">et al</italic> 2014)—which involves a number of forward and adjoint model simulations—to design accelerated operator-weighted proposals. By exploiting the multilevel structure of the discretised parameters and discretised forward models, we design a <italic toggle=\"yes\">Rayleigh–Ritz procedure</italic> to significantly reduce the computational effort in building the LIS and operating with DILI proposals. Second, the resulting DILI-MCMC can drastically improve the sampling efficiency of MCMC at each level, and hence reduce the integration error of the multilevel algorithm for fixed CPU time. Numerical results confirm the improved computational efficiency of the multilevel DILI approach.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"69 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139762540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-01DOI: 10.1088/1361-6420/ad1fe5
Anna Fitzpatrick, Molly Folino, Andrea Arnold
Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters (TVPs) in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic TVPs of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.
{"title":"Fourier series-based approximation of time-varying parameters in ordinary differential equations","authors":"Anna Fitzpatrick, Molly Folino, Andrea Arnold","doi":"10.1088/1361-6420/ad1fe5","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1fe5","url":null,"abstract":"Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters (TVPs) in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic TVPs of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"23 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139762863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in R2. The operators of interest include the longitudinal, transverse, and mixed VLTs, their integral moments, and the star transform. With the exception of the star transform, all these operators are natural generalizations to the broken-ray trajectories of the corresponding well-studied concepts defined for straight-line paths of integration. We characterize the kernels of the VLTs and derive exact formulas for reconstruction of tensor fields from various combinations of these transforms. The star transform on tensor fields is an extension of the corresponding concepts that have been previously studied on vector fields and scalar fields (functions). We describe all injective configurations of the star transform on symmetric 2-tensor fields and derive an exact, closed-form inversion formula for that operator.
{"title":"V-line 2-tensor tomography in the plane","authors":"Gaik Ambartsoumian, Rohit Kumar Mishra, Indrani Zamindar","doi":"10.1088/1361-6420/ad1f83","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1f83","url":null,"abstract":"In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^2$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad1f83ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. The operators of interest include the longitudinal, transverse, and mixed VLTs, their integral moments, and the star transform. With the exception of the star transform, all these operators are natural generalizations to the broken-ray trajectories of the corresponding well-studied concepts defined for straight-line paths of integration. We characterize the kernels of the VLTs and derive exact formulas for reconstruction of tensor fields from various combinations of these transforms. The star transform on tensor fields is an extension of the corresponding concepts that have been previously studied on vector fields and scalar fields (functions). We describe all injective configurations of the star transform on symmetric 2-tensor fields and derive an exact, closed-form inversion formula for that operator.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"8 2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139762535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-16DOI: 10.1088/1361-6420/ad1609
Pierre Maréchal, Faouzi Triki, Walter C Simo Tao Lee
In this paper we study the inverse Laplace transform. We first derive a new global logarithmic stability estimate that shows that the inversion is severely ill-posed. Then we propose a regularization method to compute the inverse Laplace transform using the concept of mollification. Taking into account the exponential instability we derive a criterion for selection of the regularization parameter. We show that by taking the optimal value of this parameter we improve significantly the convergence of the method. Finally, making use of the holomorphic extension of the Laplace transform, we suggest a new PDEs based numerical method for the computation of the solution. The effectiveness of the proposed regularization method is demonstrated through several numerical examples.
{"title":"Regularization of the inverse Laplace transform by mollification","authors":"Pierre Maréchal, Faouzi Triki, Walter C Simo Tao Lee","doi":"10.1088/1361-6420/ad1609","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1609","url":null,"abstract":"In this paper we study the inverse Laplace transform. We first derive a new global logarithmic stability estimate that shows that the inversion is severely ill-posed. Then we propose a regularization method to compute the inverse Laplace transform using the concept of mollification. Taking into account the exponential instability we derive a criterion for selection of the regularization parameter. We show that by taking the optimal value of this parameter we improve significantly the convergence of the method. Finally, making use of the holomorphic extension of the Laplace transform, we suggest a new PDEs based numerical method for the computation of the solution. The effectiveness of the proposed regularization method is demonstrated through several numerical examples.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139506590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-08DOI: 10.1088/1361-6420/ad149f
Hongyu Liu, Catharine W K Lo
In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in partial differential equation (PDE) analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities.
{"title":"Determining a parabolic system by boundary observation of its non-negative solutions with biological applications","authors":"Hongyu Liu, Catharine W K Lo","doi":"10.1088/1361-6420/ad149f","DOIUrl":"https://doi.org/10.1088/1361-6420/ad149f","url":null,"abstract":"In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in partial differential equation (PDE) analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139506406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-03DOI: 10.1088/1361-6420/ad1a3c
Zhuoxu Cui, Qingyong Zhu, Jing Cheng, Bo Zhang, Dong Liang
Deep unfolding methods have gained significant popularity in the field of inverse problems as they have driven the design of deep neural networks (DNNs) using iterative algorithms. In contrast to general DNNs, unfolding methods offer improved interpretability and performance. However, their theoretical stability or regularity in solving inverse problems remains subject to certain limitations. To address this, we reevaluate unfolded DNNs and observe that their algorithmically-driven cascading structure exhibits a closer resemblance to iterative regularization. Recognizing this, we propose a modified training approach and configure termination criteria for unfolded DNNs, thereby establishing the unfolding method as an iterative regularization technique. Specifically, our method involves the joint learning of a convex penalty function using an input-convex neural network (ICNN) to quantify distance to a real data manifold. Then, we train a DNN unfolded from the proximal gradient descent algorithm, incorporating this learned penalty. Additionally, we introduce a new termination criterion for the unfolded DNN. Under the assumption that the real data manifold intersects the solutions of the inverse problem with a unique real solution, even when measurements contain perturbations, we provide a theoretical proof of the stable convergence of the unfolded DNN to this solution. Furthermore, we demonstrate with an example of MRI reconstruction that the proposed method outperforms original unfolding methods and traditional regularization methods in terms of reconstruction quality, stability, and convergence speed.
{"title":"Deep unfolding as iterative regularization for imaging inverse problems","authors":"Zhuoxu Cui, Qingyong Zhu, Jing Cheng, Bo Zhang, Dong Liang","doi":"10.1088/1361-6420/ad1a3c","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1a3c","url":null,"abstract":"\u0000 Deep unfolding methods have gained significant popularity in the field of inverse problems as they have driven the design of deep neural networks (DNNs) using iterative algorithms. In contrast to general DNNs, unfolding methods offer improved interpretability and performance. However, their theoretical stability or regularity in solving inverse problems remains subject to certain limitations. To address this, we reevaluate unfolded DNNs and observe that their algorithmically-driven cascading structure exhibits a closer resemblance to iterative regularization. Recognizing this, we propose a modified training approach and configure termination criteria for unfolded DNNs, thereby establishing the unfolding method as an iterative regularization technique. Specifically, our method involves the joint learning of a convex penalty function using an input-convex neural network (ICNN) to quantify distance to a real data manifold. Then, we train a DNN unfolded from the proximal gradient descent algorithm, incorporating this learned penalty. Additionally, we introduce a new termination criterion for the unfolded DNN. Under the assumption that the real data manifold intersects the solutions of the inverse problem with a unique real solution, even when measurements contain perturbations, we provide a theoretical proof of the stable convergence of the unfolded DNN to this solution. Furthermore, we demonstrate with an example of MRI reconstruction that the proposed method outperforms original unfolding methods and traditional regularization methods in terms of reconstruction quality, stability, and convergence speed.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"14 6","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139451521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-28DOI: 10.1088/1361-6420/ad14a0
Bangti Jin, Kwancheol Shin, Zhi Zhou
In this work we investigate an inverse problem of recovering a time-dependent potential in a semilinear subdiffusion model from an integral measurement of the solution over the domain. The model involves the Djrbashian–Caputo fractional derivative in time. Theoretically, we prove a novel conditional Lipschitz stability result, and numerically, we develop an easy-to-implement fixed point iteration for recovering the unknown coefficient. In addition, we establish rigorous error bounds on the discrete approximation. These results are obtained by crucially using smoothing properties of the solution operators and suitable choice of a weighted Lp(0,T) norm. The efficiency and accuracy of the scheme are showcased on several numerical experiments in one- and two-dimensions.
{"title":"Numerical recovery of a time-dependent potential in subdiffusion *","authors":"Bangti Jin, Kwancheol Shin, Zhi Zhou","doi":"10.1088/1361-6420/ad14a0","DOIUrl":"https://doi.org/10.1088/1361-6420/ad14a0","url":null,"abstract":"In this work we investigate an inverse problem of recovering a time-dependent potential in a semilinear subdiffusion model from an integral measurement of the solution over the domain. The model involves the Djrbashian–Caputo fractional derivative in time. Theoretically, we prove a novel conditional Lipschitz stability result, and numerically, we develop an easy-to-implement fixed point iteration for recovering the unknown coefficient. In addition, we establish rigorous error bounds on the discrete approximation. These results are obtained by crucially using smoothing properties of the solution operators and suitable choice of a weighted <inline-formula>\u0000<tex-math><?CDATA $L^p(0,T)$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>\u0000<inline-graphic xlink:href=\"ipad14a0ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> norm. The efficiency and accuracy of the scheme are showcased on several numerical experiments in one- and two-dimensions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139095803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-27DOI: 10.1088/1361-6420/ad12e0
Housen Li, Frank Werner
We consider statistical linear inverse problems in separable Hilbert spaces and filter-based reconstruction methods of the form <inline-formula> <tex-math><?CDATA $widehat f_alpha = q_alpha left(T,^*Tright)T,^*Y$?></tex-math> <mml:math overflow="scroll"><mml:msub><mml:mover><mml:mi>f</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mi>T</mml:mi></mml:mrow></mml:mfenced><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mi>Y</mml:mi></mml:math> <inline-graphic xlink:href="ipad12e0ieqn1.gif" xlink:type="simple"></inline-graphic> </inline-formula>, where <italic toggle="yes">Y</italic> is the available data, <italic toggle="yes">T</italic> the forward operator, <inline-formula> <tex-math><?CDATA $left(q_alpharight)_{alpha in mathcal A}$?></tex-math> <mml:math overflow="scroll"><mml:msub><mml:mfenced close=")" open="("><mml:msub><mml:mi>q</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mfenced><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math> <inline-graphic xlink:href="ipad12e0ieqn2.gif" xlink:type="simple"></inline-graphic> </inline-formula> an ordered filter, and <italic toggle="yes">α</italic> > 0 a regularization parameter. Whenever such a method is used in practice, <italic toggle="yes">α</italic> has to be appropriately chosen. Typically, the aim is to find or at least approximate the best possible <italic toggle="yes">α</italic> in the sense that mean squared error (MSE) <inline-formula> <tex-math><?CDATA $mathbb{E} [Vert widehat f_alpha - f^daggerVert^2]$?></tex-math> <mml:math overflow="scroll"><mml:mrow><mml:mi mathvariant="double-struck">E</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mo>∥</mml:mo><mml:msub><mml:mover><mml:mi>f</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>f</mml:mi><mml:mo>†</mml:mo></mml:msup><mml:mrow><mml:msup><mml:mo>∥</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:math> <inline-graphic xlink:href="ipad12e0ieqn3.gif" xlink:type="simple"></inline-graphic> </inline-formula> w.r.t. the true solution <inline-formula> <tex-math><?CDATA $f^dagger$?></tex-math> <mml:math overflow="scroll"><mml:msup><mml:mi>f</mml:mi><mml:mo>†</mml:mo></mml:msup></mml:math> <inline-graphic xlink:href="ipad12e0ieqn4.gif" xlink:type="simple"></inline-graphic> </inline-formula> is minimized. In this paper, we introduce the Sharp Optimal Lepskiĭ-Inspired Tuning (SOLIT) method, which yields an <italic toggle="yes">a posteriori</italic> parameter choice rule ensuring adaptive minimax rates of convergence. It depends only on <italic toggle="yes">Y</italic> and the noise level <italic toggle="yes">σ</italic> as well as the operator <italic toggle="yes">T</italic> and th
{"title":"Adaptive minimax optimality in statistical inverse problems via SOLIT—Sharp Optimal Lepskiĭ-Inspired Tuning","authors":"Housen Li, Frank Werner","doi":"10.1088/1361-6420/ad12e0","DOIUrl":"https://doi.org/10.1088/1361-6420/ad12e0","url":null,"abstract":"We consider statistical linear inverse problems in separable Hilbert spaces and filter-based reconstruction methods of the form <inline-formula>\u0000<tex-math><?CDATA $widehat f_alpha = q_alpha left(T,^*Tright)T,^*Y$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msub><mml:mover><mml:mi>f</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mi>T</mml:mi></mml:mrow></mml:mfenced><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mi>Y</mml:mi></mml:math>\u0000<inline-graphic xlink:href=\"ipad12e0ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>, where <italic toggle=\"yes\">Y</italic> is the available data, <italic toggle=\"yes\">T</italic> the forward operator, <inline-formula>\u0000<tex-math><?CDATA $left(q_alpharight)_{alpha in mathcal A}$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>q</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mfenced><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math>\u0000<inline-graphic xlink:href=\"ipad12e0ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> an ordered filter, and <italic toggle=\"yes\">α</italic> > 0 a regularization parameter. Whenever such a method is used in practice, <italic toggle=\"yes\">α</italic> has to be appropriately chosen. Typically, the aim is to find or at least approximate the best possible <italic toggle=\"yes\">α</italic> in the sense that mean squared error (MSE) <inline-formula>\u0000<tex-math><?CDATA $mathbb{E} [Vert widehat f_alpha - f^daggerVert^2]$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:mi mathvariant=\"double-struck\">E</mml:mi></mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>∥</mml:mo><mml:msub><mml:mover><mml:mi>f</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>f</mml:mi><mml:mo>†</mml:mo></mml:msup><mml:mrow><mml:msup><mml:mo>∥</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:math>\u0000<inline-graphic xlink:href=\"ipad12e0ieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> w.r.t. the true solution <inline-formula>\u0000<tex-math><?CDATA $f^dagger$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mi>f</mml:mi><mml:mo>†</mml:mo></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad12e0ieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> is minimized. In this paper, we introduce the Sharp Optimal Lepskiĭ-Inspired Tuning (SOLIT) method, which yields an <italic toggle=\"yes\">a posteriori</italic> parameter choice rule ensuring adaptive minimax rates of convergence. It depends only on <italic toggle=\"yes\">Y</italic> and the noise level <italic toggle=\"yes\">σ</italic> as well as the operator <italic toggle=\"yes\">T</italic> and th","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"43 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139051602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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