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1. Scanner: Simultaneously temporal trend and spatial cluster detection for spatial‐temporal data

   https://doi.org/10.1002/env.2849  

   Wang, Xin; Zhang, Xin (April 2024, Environmetrics)

   Abstract Identifying the underlying trajectory pattern in the spatial‐temporal data analysis is a fundamental but challenging task. In this paper, we study the problem of simultaneously identifying temporal trends and spatial clusters of spatial‐temporal trajectories. To achieve this goal, we propose a novel method named spatial clustered and sparse nonparametric regression (). Our method leverages the B‐spline model to fit the temporal data and penalty terms on spline coefficients to reveal the underlying spatial‐temporal patterns. In particular, our method estimates the model by solving a doubly‐penalized least square problem, in which we use a group sparse penalty for trend detection and a spanning tree‐based fusion penalty for spatial cluster recovery. We also develop an algorithm based on the alternating direction method of multipliers (ADMM) algorithm to efficiently minimize the penalized least square loss. The statistical consistency properties of estimator are established in our work. In the end, we conduct thorough numerical experiments to verify our theoretical findings and validate that our method outperforms the existing competitive approaches. 

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2. A convexity enforcing C0 interior penalty method for the Monge–Ampère equation on convex polygonal domains

   Brenner, Susanne C; Sung, Li-yeng; Tan, Zhiyu; Zhang, Hongchao (June 2021, Numerische Mathematik)

   We design and analyze a C0 interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions. 

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3. A convexity enforcing $${C}^{{0}}$$ interior penalty method for the Monge–Ampère equation on convex polygonal domains

   https://doi.org/10.1007/s00211-021-01210-x  

   Brenner, Susanne C.; Sung, Li-yeng; Tan, Zhiyu; Zhang, Hongchao (June 2021, Numerische Mathematik)

   null (Ed.)

   Abstract We design and analyze a $$C^0$$ C 0 interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions. 

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4. A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints

   https://doi.org/10.1515/cmam-2022-0148  

   Brenner, Susanne C; Gedicke, Joscha; Sung, Li-Yeng (December 2022, Computational Methods in Applied Mathematics)

   Abstract We construct a symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints on general polygonal domains.The resulting discrete problems are quadratic programs with simple box constraints that can be solved efficiently by a primal-dual active set algorithm.Both theoretical analysis and corroborating numerical results are presented. 

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5. An Energy-Based Summation-by-Parts Finite Difference Method For the Wave Equation in Second Order Form

   https://doi.org/10.1007/s10915-022-01829-4  

   Wang, Siyang; Appelö, Daniel; Kreiss, Gunilla (April 2022, Journal of Scientific Computing)

   Abstract We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the simultaneous-approximation-term (SAT) method, we derive energy estimates for the semi-discretization. In addition, error estimates are derived by the normal mode analysis. The proposed method is termed as energy-based because of its similarity with the energy-based discontinuous Galerkin method. When imposing the Dirichlet boundary condition and material interface conditions, the traditional SBP-SAT discretization uses a penalty term with a mesh-dependent parameter, which is not needed in our method. Furthermore, numerical dissipation can be added to the discretization through the boundary and interface conditions. We present numerical experiments that verify convergence and robustness of the proposed method. 

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