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1. Optimal Control, Numerics, and Applications of Fractional PDEs

   https://doi.org/10.1016/bs.hna.2021.12.003  

   Antil, H. ; Brown T. ; Khatri, R. ; Onwunta, A. ; Verma, D. ; Warma, M. ( February 2022 , Handbook of numerical analysis)

   Trélat, E. ; Zuazua, E. (Ed.)

   This chapter provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps, and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems. 

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2. Fast implicit difference schemes for time‐space fractional diffusion equations with the integral fractional Laplacian

   https://doi.org/10.1002/mma.6746  

   Gu, Xian‐Ming ; Sun, Hai‐Wei ; Zhang, Yanzhi ; Zhao, Yong‐Liang ( July 2020 , Mathematical Methods in the Applied Sciences)

   In this paper, we develop two fast implicit difference schemes for solving a class of variable‐coefficient time–space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the gradedL1formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via theM‐matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use the fast sum‐of‐exponential approximation and Toeplitz matrix algorithms to reduce the computational cost for the nonlocal property of time and space fractional derivatives, respectively. The fast schemes greatly reduce the computational work of solving the discretized linear systems from by a direct solver to per preconditioned Krylov subspace iteration and a memory requirement from𝒪(MN²)to𝒪(NN_exp), whereNand(N_exp ≪)Mare the number of spatial and temporal grid nodes, respectively. The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners. Finally, numerical results are presented to show the utility of the proposed methods.

    

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3. Hydraulic resistance of three-dimensional pial perivascular spaces in the brain

   https://doi.org/10.1186/s12987-023-00505-5  

   Boster, Kimberly A. S. ; Sun, Jiatong ; Shang, Jessica K. ; Kelley, Douglas H. ; Thomas, John H. ( January 2024 , Fluids and Barriers of the CNS)

   Perivascular spaces (PVSs) carry cerebrospinal fluid (CSF) around the brain, facilitating healthy waste clearance. Measuring those flows in vivo is difficult, and often impossible, because PVSs are small, so accurate modeling is essential for understanding brain clearance. The most important parameter for modeling flow in a PVS is its hydraulic resistance, defined as the ratio of pressure drop to volume flow rate, which depends on its size and shape. In particular, the local resistance per unit length varies along a PVS and depends on variations in the local cross section.

   Using segmented, three-dimensional images of pial PVSs in mice, we performed fluid dynamical simulations to calculate the resistance per unit length. We applied extended lubrication theory to elucidate the difference between the calculated resistance and the expected resistance assuming a uniform flow. We tested four different approximation methods, and a novel correction factor to determine how to accurately estimate resistance per unit length with low computational cost. To assess the impact of assuming unidirectional flow, we also considered a circular duct whose cross-sectional area varied sinusoidally along its length.

   We found that modeling a PVS as a series of short ducts with uniform flow, and numerically solving for the flow in each, yields good resistance estimates at low cost. If the second derivative of area with respect to axial location is less than 2, error is typically less than 15%, and can be reduced further with our correction factor. To make estimates with even lower cost, we found that instead of solving for the resistance numerically, the well-known resistance of a circular duct could be scaled by a shape factor. As long as the aspect ratio of the cross section was less than 0.7, the additional error was less than 10%.

   Neglecting off-axis velocity components underestimates the average resistance, but the error can be reduced with a simple correction factor. These results could increase the accuracy of future models of brain-wide and local CSF flow, enabling better prediction of clearance, for example, as it varies with age, brain state, and pathological conditions.

    

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4. Bilinear Systems With Initial Gaps Involving Inelastic Collision: Forced Response Experiments and Simulations

   https://doi.org/10.1115/1.4051493  

   Noguchi, Kohei ; Saito, Akira ; Tien, Meng-Hsuan ; D’Souza, Kiran ( April 2022 , Journal of Vibration and Acoustics)

   Abstract In this paper, the forced response of a two degrees-of-freedom (DOF) bilinear oscillator with initial gaps involving inelastic collision is discussed. In particular, a focus is placed upon the experimental verification of the generalized bilinear amplitude approximation (BAA) method, which can be used for the accurate estimation of forced responses for bilinear systems with initial gaps. Both experimental and numerical investigations on the system have been carried out. An experimental setup that is capable of representing the dynamics of a 2DOF oscillator has been developed, and forced response tests have been conducted under swept-sine base excitation for different initial gap sizes. The steady-state response of the system under base excitation was computed by both traditional time integration and BAA. It is shown that the results of experiments and numerical predictions are in good agreement especially at resonance. However, slight differences in the responses obtained from both numerical methods are observed. It was found that the time duration where the DOFs are in contact with each other predicted by BAA is longer than that predicted by time integration. Spectral analyses have also been conducted on both experimental and numerical results. It was observed that in a frequency range where intermittent contact between the masses occurs, super-harmonic components of the excitation frequency are present in the spectra. Moreover, as the initial gap size increases, the frequency band where the super-harmonic components are observed decreases. 

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5. Analysis of an approximation to a fractional extension problem

   https://doi.org/10.1007/s10543-019-00787-y  

   Padgett, Joshua L. ( November 2019 , BIT Numerical Mathematics)

   The purpose of this article is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework herein. The proposed method differs from many standard techniques, as we approximate the true solution to the abstract problem, rather than solve an associated discrete problem. The numerical method is shown to be consistent, stable, and convergent in an appropriate Banach space. These results are built upon well understood results from semigroup theory. Numerical experiments are provided to demonstrate the theoretical results. 

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