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   Hang, Nguyen T.; Mordukhovich, Boris S.; Sarabi, M. Ebrahim (January 2022, Journal of Global Optimization)
2. Second Order Adjoints in Optimization

   https://doi.org/10.1007/978-3-030-72040-7_10  

   Petra, Noemi; Sachs, Ekkehard W. (January 2022, Numerical Analysis and Optimization)

   Al-Baali, Mehiddin; Purnama, Anton; Grandinetti, Lucio (Ed.)

   Second order, Newton-like algorithms exhibit convergence properties superior to gradient-based or derivative-free optimization algorithms. However, deriving and computing second order derivatives--needed for the Hessian-vector product in a Krylov iteration for the Newton step--often is not trivial. Second order adjoints provide a systematic and efficient tool to derive second derivative infor- mation. In this paper, we consider equality constrained optimization problems in an infinite-dimensional setting. We phrase the optimization problem in a general Banach space framework and derive second order sensitivities and second order adjoints in a rigorous and general way. We apply the developed framework to a partial differential equation-constrained optimization problem. 

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3. Second order Zeeman interaction and ferroquadrupolar order in TmVO4

   https://doi.org/10.1038/s41535-022-00475-1  

   Vinograd, I.; Shirer, K. R.; Massat, P.; Wang, Z.; Kissikov, T.; Garcia, D.; Bachmann, M. D.; Horvatić, M.; Fisher, I. R.; Curro, N. J. (June 2022, npj Quantum Materials)

   Abstract TmVO₄exhibits ferroquadrupolar order of the Tm 4f electronic orbitals at low temperatures, and is a model system for Ising nematicity. A magnetic field oriented along thec-axis constitutes a transverse effective field for the quadrupolar order parameter, continuously tuning the system to a quantum phase transition as the field is increased from zero. In contrast, in-plane magnetic fields couple to the order parameter only at second order, such that orienting along the primary axes of the quadrupole order results in an effective longitudinal field, whereas orienting at 45 degrees results in a second effective transverse field. Not only do in-plane fields engender a marked in-plane anisotropy of the critical magnetic and quadrupole fluctuations above the ferroquadrupolar ordering temperature, but in-plane transverse fields initially enhance the ferroquadrupolar order, before eventually suppressing it, an effect that we attribute to admixing of the higher crystalline electric field levels. 

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4. Second-Order Fast-Slow Stochastic Systems

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5. Acoustic streaming in second-order fluids

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   Das, Pradipta Kr.; Snider, Arthur David; Bhethanabotla, Venkat R. (December 2020, Physics of Fluids)