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   https://doi.org/doi:10.24033/asens.2466  

   Diaconis, Persi; Hough, Robert (May 2021, Annales scientifiques de lÉcole normale supérieure)

   null (Ed.)

   We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure. As a second illustration, the method is used to study walks on the nn uni-upper triangular group with entries taken modulo p. The method allows sharp answers to the behavior of individual coordinates: coordinates immediately above the diagonal require order p^2 steps for randomness, coordinates on the second diagonal require order p steps; coordinates on the kth diagonal require order p^{2/k} steps. 

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2. Tunable High‐Resolution Synthetic Aperture Radar Imaging

   https://doi.org/10.1029/2022RS007572  

   Kim, Arnold D.; Tsogka, Chrysoula (October 2022, Radio Science)

   Abstract We have recently introduced a modification of the multiple signal classification method for synthetic aperture radar. This method incorporates a user‐defined parameter,ϵ, that allows for tunable quantitative high‐resolution imaging. However, this method requires relatively large single‐to‐noise ratios (SNR) to work effectively. Here, we first identify the fundamental mechanism in that method that produces high‐resolution images. Then we introduce a modification to Kirchhoff Migration (KM) that uses the same mechanism to produce tunable, high‐resolution images. This modified KM method can be applied to low SNR measurements. We show simulation results that demonstrate the features of this method. 

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3. Towards a spacelike characterization of the null singularity inside a black hole

   https://doi.org/10.1088/1361-6382/ae0387  

   Garfinkle, David (September 2025, Classical and Quantum Gravity)

   Abstract We propose a method for recognizing null singularities in a computer simulation that uses a foliation by spacelike surfaces. The method involves harmonic time slicing as well as rescaled tetrad variables. As a ‘proof of concept’ we show that the method works in Reissner–Nordstrom spacetime. 

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4. A Riemann–Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals

   https://doi.org/10.1111/sapm.12630  

   Ballew, Cade; Trogdon, Thomas (August 2023, Studies in Applied Mathematics)

   Abstract We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas–Its–Kitaev Riemann–Hilbert representation of the orthogonal polynomials to produce an method to compute the firstNrecurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory. 

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5. A stochastic collocation method based on sparse grids for a stochastic Stokes-Darcy model

   https://doi.org/10.3934/dcdss.2021104  

   Yang, Zhipeng; Li, Xuejian; He, Xiaoming; Ming, Ju (January 2021, Discrete and Continuous Dynamical Systems - Series S)

   In this paper, we develop a sparse grid stochastic collocation method to improve the computational efficiency in handling the steady Stokes-Darcy model with random hydraulic conductivity. To represent the random hydraulic conductivity, the truncated Karhunen-Loève expansion is used. For the discrete form in probability space, we adopt the stochastic collocation method and then use the Smolyak sparse grid method to improve the efficiency. For the uncoupled deterministic subproblems at collocation nodes, we apply the general coupled finite element method. Numerical experiment results are presented to illustrate the features of this method, such as the sample size, convergence, and randomness transmission through the interface. 

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