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1. A semi-adaptive preservative scheme for a fractional quenching convective-diffusion problem

   https://doi.org/10.1016/j.camwa.2023.09.043  

   Liu, Nabing; Zhu, Lin; Sheng, Qin (December 2023, Computers & Mathematics with Applications)

   A preservative scheme is presented and analyzed for the solution of a quenching type convective-diffusion problem modeled through one-sided Riemann-Liouville space-fractional derivatives. Properly weighted Grünwald formulas are employed for the discretization of the fractional derivative. A forward difference approximation is considered in the approximation of the convective term of the nonlinear equation. Temporal steps are optimized via an asymptotic arc-length monitoring mechanism till the quenching point. Under suitable constraints on spatial-temporal discretization steps, the monotonicity, positivity preservations of the numerical solution and numerical stability of the scheme are proved. Three numerical experiments are designed to demonstrate and simulate key characteristics of the semi-adaptive scheme constructed, including critical length, quenching time and quenching location of the fractional quenching phenomena formulated through the one-sided space-fractional convective-diffusion initial-boundary value problem. Effects of the convective function to quenching are discussed. Numerical estimates of the order of convergence are obtained. Computational results obtained are carefully compared with those acquired from conventional integer order quenching convection-diffusion problems for validating anticipated accuracy. The experiments have demonstrated expected accuracy and feasibility of the new method. 

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2. 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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3. A simulation expressivity of the quenching phenomenon in a two-sided space-fractional diffusion equation

   https://doi.org/10.1016/j.amc.2022.127523  

   Zhu, Lin; Liu, Nabing; Sheng, Qin (January 2023, Applied Mathematics and Computation)

   The aims of this paper are to investigate and propose a numerical approximation for a quenching type diffusion problem associated with a two-sided Riemann-Liouville space- fractional derivative. The approach adopts weighted Grünwald formulas for suitable spatial discretization. An implicit Crank-Nicolson scheme combined with adaptive technology is then implemented for a temporal integration. Monotonicity, positivity preservation and linearized stability are proved under suitable constraints on spatial and temporal discretization parameters. Two specially designed simulation experiments are presented for illustrating and outreaching properties of the numerical method constructed. Connections between the two-sided fractional differential operator and critical values including quenching time, critical length and quenching location are investigated. 

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4. On the convergence of Fourier spectral methods involving noncompact operators

   https://doi.org/10.1093/imanum/drae044  

   Trogdon, Thomas (July 2024, IMA Journal of Numerical Analysis)

   Abstract Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $$\mathcal L u = f$$. The framework posits the existence of a left-Fredholm regulator for $$\mathcal L$$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann–Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients. 

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5. Variational quasi-harmonic maps for computing diffeomorphisms

   https://doi.org/10.1145/3592105  

   Wang, Yu; Guo, Minghao; Solomon, Justin (July 2023, ACM Transactions on Graphics)

   Computation of injective (or inversion-free) maps is a key task in geometry processing, physical simulation, and shape optimization. Despite being a longstanding problem, it remains challenging due to its highly nonconvex and combinatoric nature. We propose computation ofvariational quasi-harmonic mapsto obtain smooth inversion-free maps. Our work is built on a key observation about inversion-free maps: A planar map is a diffeomorphism if and only if it is quasi-harmonic and satisfies a special Cauchy boundary condition. We hence equate the inversion-free mapping problem to an optimal control problem derived from our theoretical result, in which we search in the space of parameters that define an elliptic PDE. We show that this problem can be solved by minimizing within a family of functionals. Similarly, our discretized functionals admit exactly injective maps as the minimizers, empirically producing inversion-free discrete maps of triangle meshes. We design efficient numerical procedures for our problem that prioritize robust convergence paths. Experiments show that on challenging examples our methods can achieve up to orders of magnitude improvement over state-of-the-art, in terms of speed or quality. Moreover, we demonstrate how to optimize a generic energy in our framework while restricting to quasi-harmonic maps. 

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