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1. Direct/Iterative Hybrid Solver for Scattering by Inhomogeneous Media

   https://doi.org/10.1137/22M1521547  

   Bruno, Oscar P; Pandey, Ambuj (April 2024, SIAM Journal on Scientific Computing)

   Bruno, Oscar; Pandey, Ambuj (Ed.)

   This paper presents a fast high-order method for the solution of two-dimensional problems of scattering by penetrable inhomogeneous media, with application to high-frequency configurations containing (possibly) discontinuous refractivities. The method relies on a hybrid direct/iterative combination of 1)~A differential volumetric formulation (which is based on the use of appropriate Chebyshev differentiation matrices enacting the Laplace operator) and, 2)~A second-kind boundary integral formulation (which, once again, utilizes Chebyshev discretization, but, in this case, in the boundary-integral context). The approach enjoys low dispersion and high-order accuracy for smooth refractivities, as well as second-order accuracy (while maintaining low dispersion) in the discontinuous refractivity case. The solution approach proceeds by application of Impedance-to-Impedance (ItI) maps to couple the volumetric and boundary discretizations. The volumetric linear algebra solutions are obtained by means of a multifrontal solver, and the coupling with the boundary integral formulation is achieved via an application of the iterative linear-algebra solver GMRES. In particular, the existence and uniqueness theory presented in the present paper provides an affirmative answer to an open question concerning the existence of a uniquely solvable second-kind ItI-based formulation for the overall scattering problem under consideration. Relying on a modestly-demanding scatterer-dependent precomputation stage (requiring in practice a computing cost of the order of $$O(N^{\alpha})$$ operations, with $$\alpha \approx 1.07$$, for an $$N$$-point discretization \textcolor{black}{and for the relevant Chebyshev accuracy orders $$q$$ used)}, together with fast ($O(N)$-cost) single-core runs for each incident field considered, the proposed algorithm can effectively solve scattering problems for large and complex objects possibly containing discontinuities and strong refractivity contrasts. 

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2. On analyticity of scattered fields in layered structures with interfacial graphene

   https://doi.org/10.1111/sapm.12389  

   Nicholls, David P. (May 2021, Studies in Applied Mathematics)

   Abstract Two‐dimensional materials such as graphene have become crucial components of most state‐of‐the‐art plasmonic devices. The possibility of not only generating plasmons in the terahertz regime, but also tuning them in real time via chemical doping or electrical gating make them compelling materials for engineers seeking to build accurate sensors. Thus, the faithful modeling of the propagation of linear waves in a layered, periodic structure with such materials at the interfaces is of paramount importance in many branches of the applied sciences. In this paper, we present a novel formulation of the problem featuring surface currents to model the two‐dimensional materials which not only is free of the artificial singularities present in related approaches, but also can be used to deliver a proof of existence, uniqueness, and analytic dependence of solutions. We advocate for a surface integral formulation which is phrased in terms of well‐chosen Impedance–Impedance Operators that are immune to the Dirichlet eigenvalues which plague the Dirichlet–Neumann Operators that appear in classical formulations. With a High‐Order Perturbation of Surfaces approach we are able to give a straightforward demonstration of this new well‐posedness result which only requires the verification that a finite collection of explicitly stated transcendental expressions be nonzero. We further illustrate the utility of this formulation by displaying results of a High‐Order Spectral numerical implementation which is flexible, rapid, and robust. 

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3. Robust formulation of Wick’s theorem for computing matrix elements between Hartree–Fock–Bogoliubov wavefunctions

   https://doi.org/10.1063/5.0156124  

   Chen, Guo P.; Scuseria, Gustavo E. (June 2023, The Journal of Chemical Physics)

   Numerical difficulties associated with computing matrix elements of operators between Hartree–Fock–Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick’s theorem in the limit of vanishing HFB overlap. In this Communication, we present a robust formulation of Wick’s theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree–Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan–Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick’s theorem is a most promising development for methods using quasiparticle vacuum states. 

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4. Tightened Formulation and Resolution of Energy-Efficient Job-Shop Scheduling

   Bing Yan, Mikhail A. (August 2020, IEEE Conference on Automation Science and Engineering)

   null (Ed.)

   Job shops are an important production environment for low-volume high-variety manufacturing. When there are urgent orders, the speeds of certain machines can be adjusted with a high energy and wear and tear cost. Scheduling in such an environment is to achieve on-time deliveries and low energy costs. The problem is, however, complicated because part processing time depends on machine speeds, and machines need to be modeled individually to capture energy costs. This paper is to obtain near-optimal solutions efficiently. The problem is formulated as a Mixed-Integer Linear Programming (MILP) form to make effective use of available MILP methods. This is done by modeling machines in groups for simplicity while approximating energy costs, and by linking part processing status and machine speed variables. Nevertheless, the resulting problem is still complicated. The formulation is therefore transformed by extending our previous tightening approach for machines with constant speeds. The idea is that if constraints can be transformed to directly delineate the convex hull, then the problem can be solved by linear programming methods. To solve the problem efficiently, our advanced decomposition and coordination method is used. Numerical results show that near-optimal solutions are obtained, demonstrating significant benefits of our approach on on-time deliveries and energy costs. 

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5. An Eulerian Vortex Method on Flow Maps

   https://doi.org/10.1145/3687996  

   Wang, Sinan; Deng, Yitong; Deng, Molin; Yu, Hong-Xing; Zhou, Junwei; Chen, Duowen; Komura, Taku; Wu, Jiajun; Zhu, Bo (November 2024, ACM Transactions on Graphics)

   We present an Eulerian vortex method based on the theory of flow maps to simulate the complex vortical motions of incompressible fluids. Central to our method is the novel incorporation of the flow-map transport equations forline elements, which, in combination with a bi-directional marching scheme for flow maps, enables the high-fidelity Eulerian advection of vorticity variables. The fundamental motivation is that, compared to impulsem, which has been recently bridged with flow maps to encouraging results, vorticityωpromises to be preferable for its numerical stability and physical interpretability. To realize the full potential of this novel formulation, we develop a new Poisson solving scheme for vorticity-to-velocity reconstruction that is both efficient and able to accurately handle the coupling near solid boundaries. We demonstrate the efficacy of our approach with a range of vortex simulation examples, including leapfrog vortices, vortex collisions, cavity flow, and the formation of complex vortical structures due to solid-fluid interactions. 

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