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1. Convergence of the boundary integral method for interfacial Stokes flow

   https://doi.org/10.1090/mcom/3787  

   Ambrose, David; Siegel, Michael; Zhang, Keyang (March 2023, Mathematics of Computation)

   Boundary integral numerical methods are among the most accurate methods for interfacial Stokes flow, and are widely applied. They have the advantage that only the boundary of the domain must be discretized, which reduces the number of discretization points and allows the treatment of complicated interfaces. Despite their popularity, there is no analysis of the convergence of these methods for interfacial Stokes flow. In practice, the stability of discretizations of the boundary integral formulation can depend sensitively on details of the discretization and on the application of numerical filters. We present a convergence analysis of the boundary integral method for Stokes flow, focusing on a rather general method for computing the evolution of an elastic capsule or viscous drop in 2D strain and shear flows. The analysis clarifies the role of numerical filters in practical computations. 

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2. Neural Network Method for Integral Fractional Laplace Equations

   https://doi.org/10.4208/eajam.010122.210722  

   Hao, Zhaopeng; Park, Moongyu; Cai, Zhiqiang (January 2023, East Asian Journal on Applied Mathematics)

   A neural network method for fractional order diffusion equations with integral fractional Laplacian is studied. We employ the Ritz formulation for the corresponding fractional equation and then derive an approximate solution of an optimization problem in the function class of neural network sets. Connecting the neural network sets with weighted Sobolev spaces, we prove the convergence and establish error estimates of the neural network method in the energy norm. To verify the theoretical results, we carry out numerical experiments and report their outcome. 

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3. Proportional Integral Derivative Control in Spark Plasma Sintering Simulations

   https://doi.org/10.3390/ma14071779  

   Jiang, Runjian; Torresani, Elisa; Cui, Guodong; Olevsky, Eugene A. (April 2021, Materials)

   null (Ed.)

   The prediction of microstructure evolution and densification behavior during the spark plasma sintering (SPS) process largely depends on accurate temperature regulation. A loop feedback control algorithm called proportional integral derivative (PID) control is a practical simulation tool, but its coefficients are often determined by an inefficient “trial and error” method. This paper is devoted to proposing a numerical method based on the principles of variable coefficients to construct an optimal linear PID controller in SPS electro-thermal simulations. Different types of temperature profiles were applied to evaluate the feasibility of the proposed method. Simulation results showed that, for temperature profiles conventionally used in SPS cycles, the PID output keeps pace with the desired profile. Characterized by an imperfect time delay and overshoot/undershoot, the constructed PID controller needs further advancement to provide a more satisfactory temperature regulation for non-continuous temperature profiles. The first step towards a numerical rule for the optimal PID controller design was undertaken in this work. It is expected to provide a valuable reference for the advanced electro-thermal modeling of SPS. 

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4. Integral equation methods for the Morse-Ingard equations

   https://doi.org/10.1016/j.jcp.2023.112416  

   Wei, Xiaoyu; Klöckner, Andreas; Kirby, Robert C. (November 2023, Journal of Computational Physics)

   We present two (a decoupled and a coupled) integral-equation-based methods for the Morse-Ingard equations subject to Neumann boundary conditions on the exterior domain. Both methods are based on second-kind integral equation (SKIE) formulations. The coupled method is well-conditioned and can achieve high accuracy. The decoupled method has lower computational cost and more flexibility in dealing with the boundary layer; however, it is prone to the ill-conditioning of the decoupling transform and cannot achieve as high accuracy as the coupled method. We show numerical examples using a Nyström method based on quadrature-by-expansion (QBX) with fast-multipole acceleration. We demonstrate the accuracy and efficiency of the solvers in both two and three dimensions with complex geometry. 

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5. DEEP NEURAL NETWORK SOLUTIONS FOR OSCILLATORY FREDHOLM INTEGRAL EQUATIONS

   https://doi.org/10.1216/jie.2024.36.23  

   Jiang, Jie; Xu, Yuesheng (April 2024, Journal of Integral Equations and Applications)

   We studied the use of deep neural networks (DNNs) in the numerical solution of the oscillatory Fredholm integral equation of the second kind. It is known that the solution of the equation exhibits certain oscillatory behaviors due to the oscillation of the kernel. It was pointed out recently that standard DNNs favor low frequency functions, and as a result, they often produce poor approximation for functions containing high frequency components. We addressed this issue in this study. We first developed a numerical method for solving the equation with DNNs as an approximate solution by designing a numerical quadrature that tailors to computing oscillatory integrals involving DNNs. We proved that the error of the DNN approximate solution of the equation is bounded by the training loss and the quadrature error. We then proposed a multigrade deep learning (MGDL) model to overcome the spectral bias issue of neural networks. Numerical experiments demonstrate that the MGDL model is effective in extracting multiscale information of the oscillatory solution and overcoming the spectral bias issue from which a standard DNN model suffers. 

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