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Research Paper#Model Reduction, LTI Systems, Frequency Domain, Greedy Algorithms🔬 ResearchAnalyzed: Jan 3, 2026 18:28

Greedy Rational Approximation for Parametric LTI Systems

Published:Dec 29, 2025 19:18
1 min read
ArXiv

Analysis

This paper addresses the model reduction problem for parametric linear time-invariant (LTI) systems, a common challenge in engineering and control theory. The core contribution lies in proposing a greedy algorithm based on reduced basis methods (RBM) for approximating high-order rational functions with low-order ones in the frequency domain. This approach leverages the linearity of the frequency domain representation for efficient error estimation. The paper's significance lies in providing a principled and computationally efficient method for model reduction, particularly for parametric systems where multiple models need to be analyzed or simulated.
Reference

“The paper proposes to use a standard reduced basis method (RBM) to construct this low-order rational function. Algorithmically, this procedure is an iterative greedy approach, where the greedy objective is evaluated through an error estimator that exploits the linearity of the frequency domain representation.”

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Research Paper#Computational Fluid Dynamics (CFD)🔬 ResearchAnalyzed: Jan 3, 2026 18:32

High-Order Solver for Free Surface Flows

Published:Dec 29, 2025 17:59
1 min read
ArXiv

Analysis

This paper introduces a high-order spectral element solver for simulating steady-state free surface flows. The use of high-order methods, curvilinear elements, and the Firedrake framework suggests a focus on accuracy and efficiency. The application to benchmark cases, including those with free surfaces, validates the model and highlights its potential advantages over lower-order schemes. The paper's contribution lies in providing a more accurate and potentially faster method for simulating complex fluid dynamics problems involving free surfaces.
Reference

“The results confirm the high-order accuracy of the model through convergence studies and demonstrate a substantial speed-up over low-order numerical schemes.”

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Research Paper#Computational Fluid Dynamics, Reduced Order Modeling, Navier-Stokes Equations🔬 ResearchAnalyzed: Jan 3, 2026 19:54

ROM for Viscous, Incompressible Flow: Exponential Convergence

Published:Dec 27, 2025 11:50
1 min read
ArXiv

Analysis

This paper investigates the use of Reduced Order Models (ROMs) for approximating solutions to the Navier-Stokes equations, specifically focusing on viscous, incompressible flow within polygonal domains. The key contribution is demonstrating exponential convergence rates for these ROM approximations, which is a significant improvement over slower convergence rates often seen in numerical simulations. This is achieved by leveraging recent results on the regularity of solutions and applying them to the analysis of Kolmogorov n-widths and POD Galerkin methods. The paper's findings suggest that ROMs can provide highly accurate and efficient solutions for this class of problems.
Reference

“The paper demonstrates "exponential convergence rates of POD Galerkin methods that are based on truth solutions which are obtained offline from low-order, divergence stable mixed Finite Element discretizations."”

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Research Paper#Mathematics/Function Approximation🔬 ResearchAnalyzed: Jan 3, 2026 20:13

Continuous-Order Integral Operator for Function Reconstruction

Published:Dec 26, 2025 16:25
1 min read
ArXiv

Analysis

This paper introduces a novel continuous-order integral operator as an alternative to the Maclaurin expansion for reconstructing analytic functions. The core idea is to replace the discrete sum of derivatives with an integral over fractional derivative orders. The paper's significance lies in its potential to generalize the classical Taylor-Maclaurin expansion and provide a new perspective on function reconstruction. The use of fractional derivatives and the exploration of correction terms are key contributions.
Reference

“The operator reconstructs f accurately in the tested domains.”

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Research Paper#Computational Fluid Dynamics (CFD)🔬 ResearchAnalyzed: Jan 4, 2026 00:07

Semi-Implicit VMS for Navier-Stokes with Exact Adjoint Linearization

Published:Dec 25, 2025 19:46
1 min read
ArXiv

Analysis

This paper presents a novel semi-implicit variational multiscale (VMS) formulation for the incompressible Navier-Stokes equations. The key innovation is the use of an exact adjoint linearization of the convection term, which simplifies the VMS closure and avoids complex integrations by parts. This leads to a more efficient and robust numerical method, particularly in low-order FEM settings. The paper demonstrates significant speedups compared to fully implicit nonlinear formulations while maintaining accuracy, and validates the method on a range of benchmark problems.
Reference

“The method is linear by construction, each time step requires only one linear solve. Across the benchmark suite, this reduces wall-clock time by $2$--$4\times$ relative to fully implicit nonlinear formulations while maintaining comparable accuracy.”

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