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1. Towards Data-Driven Model Reduction of the Navier-Stokes Equations using the Loewner Framework

   https://doi.org/10.1007/978-3-030-90727-3_14  

   Diaz, Alejandro N.; Heinkenschloss, Matthias (April 2022, Active Flow and Combustion Control 2021)

   King, R.; Peitsch, D. (Ed.)

   The Loewner framework is extended to compute reduced order models (ROMs) for systems governed by the incompressible Navier-Stokes (NS) equations. For quadratic ordinary differential equations (ODEs) it constructs a ROM directly from measurements of transfer function components derived from an expansion of the system’s input-to-output map. Given measurements, no explicit access to the system is required to construct the ROM. To extend the Loewner framework, the NS equations are transformed into ODEs by projecting onto the subspace defined by the incompressibility condition. This projection is used theoretically, but avoided computationally. This paper presents the overall approach. Currently, transfer function measurements are obtained via computational simulations; obtaining them from experiments is an open issue. Numerical results show the potential of the Loewner framework, but also reveal possible lack of stability of the ROM. A possible approach, which currently requires access to the NS system, to deal with these instabilities is outlined. 

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2. Reduced Order Model Hessian Approximations in Newton Methods for Optimal Control

   https://doi.org/10.1007/978-3-030-95157-3_18  

   Heinkenschloss, Matthias; Magruder, Caleb (July 2022, Realization and Model Reduction of Dynamical Systems - A Festschrift in Honor of the 70th Birthday of Thanos Antoulas)

   Beattie, C.A.; Benner, P.; Embree, M.; Gugercin, S.; Lefteriu, S. (Ed.)

   This paper introduces reduced order model (ROM) based Hessian approximations for use in inexact Newton methods for the solution of optimization problems implicitly constrained by a large-scale system, typically a discretization of a partial differential equation (PDE). The direct application of an inexact Newton method to this problem requires the solution of many PDEs per optimization iteration. To reduce the computational complexity, a ROM Hessian approximation is proposed. Since only the Hessian is approximated, but the original objective function and its gradient is used, the resulting inexact Newton method maintains the first-order global convergence property, under suitable assumptions. Thus even computationally inexpensive lower fidelity ROMs can be used, which is different from ROM approaches that replace the original optimization problem by a sequence of ROM optimization problem and typically need to accurately approximate function and gradient information of the original problem. In the proposed approach, the quality of the ROM Hessian approximation determines the rate of convergence, but not whether the method converges. The projection based ROM is constructed from state and adjoint snapshots, and is relatively inexpensive to compute. Numerical examples on semilinear parabolic optimal control problems demonstrate that the proposed approach can lead to substantial savings in terms of overall PDE solves required. 

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3. BooLigero: Improved Sublinear Zero Knowledge Proofs for Boolean Circuits

   Gvili, Yaron; Scheffler, Sarah; Varia, Mayank (March 2021, Financial Cryptography and Data Security)

   null (Ed.)

   We provide a modified version of the Ligero sublinear zero knowledge proof system for arithmetic circuits provided by Ames et. al. (CCS ‘17). Our modification "BooLigero" tailors Ligero for use in Boolean circuits to achieve a significant improvement in proof size. Although the original Ligero system could be used for Boolean circuits, Ligero generally requires allocating an entire field element to represent a single bit on a wire in a Boolean circuit. In contrast, our system performs operations over words of bits, allowing a proof size savings of between O(log(|F|)^1/4) and O(log(|F|)^1/2) compared to Ligero, where F is the field that leads to the optimal proof size in original Ligero. We achieve improvements in proof size of approximately 1.1-1.6x for SHA-2 and 1.7-2.8x for SHA-3. In addition to checking constraints of standard Boolean operations such as AND, XOR, and NOT over words, BooLigero also supports several other constraints such as multiplication in GF(2^w), bit masking, bit rearrangement within and across words, and bitwise outer product. Like Ligero, construction requires no trusted setup and no computational assumptions, which is ideal for blockchain applications. It is plausibly post-quantum secure in the standard model. Furthermore, it is public-coin, perfect honest-verifier zero knowledge, and can be made non-interactive in the random oracle model using the Fiat-Shamir transform. 

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4. SECOND ORDER, LINEAR, AND UNCONDITIONALLY ENERGY STABLE SCHEMES FOR A HYDRODYNAMIC MODEL OF SMECTIC-A LIQUID CRYSTALS

   Chen, R; Yang, X; Zhang, H. (November 2017, SIAM journal on scientific computing)

   In this paper, we consider the numerical approximations for a hydrodynamical model of smectic-A liquid crystals. The model, derived from the variational approach of the modified Oseen– Frank energy, is a highly nonlinear system that couples the incompressible Navier–Stokes equations and a constitutive equation for the layer variable. We develop two linear, second order time marching schemes based on the Invariant Energy Quadratization method for nonlinear terms in the constitutive equation, the projection method for the Navier–Stokes equations, and some subtle implicit-explicit treatments for the convective and stress terms. Moreover, we prove the well-posedness of the linear system and their unconditionally energy stabilities rigorously. Various numerical experiments are presented to demonstrate the stability and the accuracy of the numerical schemes in simulating the dynamics under shear flow and the magnetic field. 

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5. A semi-adaptive finite difference method for simulating two-sided fractional convection–diffusion quenching problems

   https://doi.org/10.1016/j.cam.2025.116796  

   Dong, Rumin; Zhu, Lin; Sheng, Qin; Zhao, Bingxin (July 2025, Journal of Computational and Applied Mathematics)

   na (Ed.)

   This paper investigates quenching solutions of an one-dimensional, two-sided Riemann–Liouville fractional order convection–diffusion problem. Fractional order spatial derivatives are discretized using weighted averaging approximations in conjunction with standard and shifted Grünwald formulas. The advective term is handled utilizing a straightforward Euler formula, resulting in a semi-discretized system of nonlinear ordinary differential equations. The conservativeness of the proposed scheme is rigorously proved and validated through simulation experiments. The study is further advanced to a fully discretized, semi-adaptive finite difference method. Detailed analysis is implemented for the monotonicity, positivity and stability of the scheme. Investigations are carried out to assess the potential impacts of the fractional order on quenching location, quenching time, and critical length. The computational results are thoroughly discussed and analyzed, providing a more comprehensive understanding of the quenching phenomena modeled through two-sided fractional order convection-diffusion problems. 

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