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1. Discrete inverse problems with internal functionals ^*

   https://doi.org/10.1088/1361-6420/adbd69  

   Corbett, Marcus; Guevara_Vasquez, Fernando; Royzman, Alexander; Yang, Guang (March 2025, Inverse Problems)

   Abstract We study the problem of finding the resistors in a resistor network from measurements of the power dissipated by the resistors under different loads. We give sufficient conditions for local uniqueness, i.e. conditions that guarantee that the linearization of this non-linear inverse problem admits a unique solution. Our method is inspired by a method to study local uniqueness of inverse problems with internal functionals in the continuum, where the inverse problem is reformulated as a redundant system of differential equations. We use our method to derive local uniqueness conditions for other discrete inverse problems with internal functionals including a discrete analogue of the inverse Schrödinger problem and problems where the resistors are replaced by impedances and dissipated power at the zero and a positive frequency are available. Moreover, we show that the dissipated power measurements can be obtained from measurements of thermal noise induced currents. 

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2. A nonlocal prior in iterative CT reconstruction

   https://doi.org/10.1002/mp.17533  

   Shu, Ziyu; Entezari, Alireza (December 2024, Medical Physics)

   Abstract BackgroundComputed tomography (CT) reconstruction problems are always framed as inverse problems, where the attenuation map of an imaged object is reconstructed from the sinogram measurement. In practice, these inverse problems are often ill‐posed, especially under few‐view and limited‐angle conditions, which makes accurate reconstruction challenging. Existing solutions use regularizations such as total variation to steer reconstruction algorithms to the most plausible result. However, most prevalent regularizations rely on the same priors, such as piecewise constant prior, hindering their ability to collaborate effectively and further boost reconstruction precision. PurposeThis study aims to overcome the aforementioned challenge a prior previously limited to discrete tomography. This enables more accurate reconstructions when the proposed method is used in conjunction with most existing regularizations as they utilize different priors. The improvements will be demonstrated through experiments conducted under various conditions. MethodsInspired by the discrete algebraic reconstruction technique (DART) algorithm for discrete tomography, we find out that pixel grayscale values in CT images are not uniformly distributed and are actually highly clustered. Such discovery can be utilized as a powerful prior for CT reconstruction. In this paper, we leverage the collaborative filtering technique to enable the collaboration of the proposed prior and most existing regularizations, significantly enhancing the reconstruction accuracy. ResultsOur experiments show that the proposed method can work with most existing regularizations and significantly improve the reconstruction quality. Such improvement is most pronounced under limited‐angle and few‐view conditions. Furthermore, the proposed regularization also has the potential for further improvement and can be utilized in other image reconstruction areas. ConclusionsWe propose improving the performance of iterative CT reconstruction algorithms by applying the collaborative filtering technique along with a prior based on the densely clustered distribution of pixel grayscale values in CT images. Our experimental results indicate that the proposed methodology consistently enhances reconstruction accuracy when used in conjunction with most existing regularizations, particularly under few‐view and limited‐angle conditions. 

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3. A Galerkin Approach to the Generalized Karush–Kuhn–Tucker Conditions for the Solution of an Elliptic Distributed Optimal Control Problem with Pointwise State and Control Constraints

   https://doi.org/10.1515/cmam-2025-0007  

   Brenner, Susanne C; Sung, Li-Yeng (February 2025, Computational Methods in Applied Mathematics)

   Abstract We develop a convergence analysis for the simplest finite element method for a model linear-quadratic elliptic distributed optimal control problem with pointwise control and state constraints under minimal assumptions on the constraint functions.We then derive the generalized Karush–Kuhn–Tucker conditions for the solution of the optimal control problem from the convergence results of the finite element method and the Karush–Kuhn–Tucker conditions for the solutions of the discrete problems. 

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4. Topological superfluid defects with discrete point group symmetries

   https://doi.org/10.1038/s41467-022-32362-5  

   Xiao, Y.; Borgh, M. O.; Blinova, A.; Ollikainen, T.; Ruostekoski, J.; Hall, D. S. (August 2022, Nature Communications)

   Abstract Discrete symmetries are spatially ubiquitous but are often hidden in internal states of systems where they can have especially profound consequences. In this work we create and verify exotic magnetic phases of atomic spinor Bose–Einstein condensates that, despite their continuous character and intrinsic spatial isotropy, exhibit complex discrete polytope symmetries in their topological defects. Using carefully tailored spinor rotations and microwave transitions, we engineer singular line defects whose quantization conditions, exchange statistics, and dynamics are fundamentally determined by these underlying symmetries. We show how filling the vortex line singularities with atoms in a variety of different phases leads to core structures that possess magnetic interfaces with rich combinations of discrete and continuous symmetries. Such defects, with their non-commutative properties, could provide unconventional realizations of quantum information and interferometry. 

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5. A problem in control of elastodynamics with piezoelectric effects

   https://doi.org/10.1093/imanum/drz047  

   Antil, Harbir; Brown, Thomas S.; Sayas, Francisco-Javier (December 2019, IMA Journal of Numerical Analysis)

   Abstract We consider an optimal control problem where the state equations are a coupled hyperbolic–elliptic system. This system arises in elastodynamics with piezoelectric effects—the elastic stress tensor is a function of elastic displacement and electric potential. The electric flux acts as the control variable and bound constraints on the control are considered. We develop a complete analysis for the state equations and the control problem. The requisite regularity on the control, to show the well-posedness of the state equations, is enforced using the cost functional. We rigorously derive the first-order necessary and sufficient conditions using adjoint equations and further study their well-posedness. For spatially discrete (time-continuous) problems, we show the convergence of our numerical scheme. Three-dimensional numerical experiments are provided showing convergence properties of a fully discrete method and the practical applicability of our approach. 

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