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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. 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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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. Partially polaron-transformed quantum master equation for exciton and charge transport dynamics

   https://doi.org/10.1063/5.0106546  

   Jang, Seogjoo Julian (January 2022, The Journal of Chemical Physics)

   Polaron-transformed quantum master equation (PQME) offers a unified framework to describe the dynamics of quantum systems in both limits of weak and strong couplings to environmental degrees of freedom. Thus, PQME serves as an efficient method to describe charge and exciton transfer/transport dynamics for a broad range of parameters in condensed or complex environments. However, in some cases, the polaron transformation (PT) being employed in the formulation invokes an over-relaxation of slow modes and results in premature suppression of important coherence terms. A formal framework to address this issue is developed in the present work by employing a partial PT that has smaller weights for low frequency bath modes. It is shown here that a closed form expression of a 2nd order time-local PQME including all the inhomogeneous terms can be derived for a general form of partial PT, although more complicated than that for the full PT. All the expressions needed for numerical calculation are derived in detail. Applications to a model of two-level system coupled to a bath of harmonic oscillators, with test calculations focused on those due to homogeneous relaxation terms, demonstrate the feasibility and the utility of the present approach. 

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5. Characterizing THz Scattering Loss in Nano-Scale SOI Waveguides Exhibiting Stochastic Surface Roughness with Exponential Autocorrelation

   https://doi.org/10.3390/electronics11030307  

   Guiana, Brian; Zadehgol, Ata (February 2022, Electronics)

   Electromagnetic (EM) scattering may be a significant source of degradation in signal and power integrity of high-contrast silicon-on-insulator (SOI) nano-scale interconnects, such as opto-electronic or optical interconnects operating at 100 s of THz where two-dimensional (2D) analytical models of dielectric slab waveguides are often used to approximate scattering loss. In this work, a formulation is presented to relate the scattering (propagation) loss to the scattering parameters (S-parameters) for the smooth waveguide; the results are correlated with results from the finite-difference time-domain (FDTD) method in 2D space. We propose a normalization factor to the previous 2D analytical formulation for the stochastic scattering loss based on physical parameters of waveguides exhibiting random surface roughness under the exponential autocorrelation function (ACF), and validate the results by comparing against numerical experiments via the 2D FDTD method, through simulation of hundreds of rough waveguides; additionally, results are compared to other 2D analytical and previous 3D experimental results. The FDTD environment is described and validated by comparing results of the smooth waveguide against analytical solutions for wave impedance, propagation constant, and S-parameters. Results show that the FDTD model is in agreement with the analytical solution for the smooth waveguide and is a reasonable approximation of the stochastic scattering loss for the rough waveguide. 

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