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On the Convergence Rates of Iterative Regularization Algorithms for Composite Bi-Level Optimization
Shimrit Shtern, Adeolu Taiwo
https://arxiv.org/abs/2506.16382

On the Convergence Rates of Iterative Regularization Algorithms for Composite Bi-Level Optimization
This paper investigates iterative methods for solving bi-level optimization problems where both inner and outer functions have a composite structure. We provide novel theoretical results, including the first convergence rate analysis for the Iteratively REgularized Proximal Gradient (IRE-PG) method, a variant of Solodov's algorithm. Our results establish simultaneous convergence rates for the inner and outer functions, highlighting the inherent trade-offs between their respective convergence ra…

Shock formation in 1D conservation laws II: Vanishing viscosity
John Anderson, Sanchit Chaturvedi, Cole Graham
https://arxiv.org/abs/2506.17156 https://

Shock formation in 1D conservation laws II: Vanishing viscosity
We study the effects of weak viscosity on shock formation in 1D hyperbolic conservation laws. Given an inviscid solution that forms a nondegenerate shock, we add a small viscous regularization and study the limit as the viscosity vanishes. Using a matched asymptotic expansion, we determine the sharp rate of convergence in strong norms up to the time of inviscid shock formation, and we identify universal viscous behavior near the first singularity. To treat the complex interactions between multi…

$\ell_{1}^{2}-\eta\ell_{2}^{2}$ regularization for sparse recovery
Long Li, Liang Ding
https://arxiv.org/abs/2506.11372 https://arxiv…

$\ell_{1}^{2}-η\ell_{2}^{2}$ regularization for sparse recovery
This paper presents a regularization technique incorporating a non-convex and non-smooth term, $\ell_{1}^{2}-η\ell_{2}^{2}$, with parameters $0<η\leq 1$ designed to address ill-posed linear problems that yield sparse solutions. We explore the existence, stability, and convergence of the regularized solution, demonstrating that the $\ell_{1}^{2}-η\ell_{2}^{2}$ regularization is well-posed and results in sparse solutions. Under suitable source conditions, we establish a convergence rate of $\m…

Dynamic mode decomposition for detecting transient activity via sparsity and smoothness regularization
Yutaro Tanaka, Hiroya Nakao
https://arxiv.org/abs/2508.10266 https://

Dynamic mode decomposition for detecting transient activity via sparsity and smoothness regularization
Dynamic Mode Decomposition (DMD) is a data-driven modal decomposition technique that extracts coherent spatio-temporal structures from high-dimensional time-series data. By decomposing the dynamics into a set of modes, each associated with a single frequency and a growth rate, DMD enables a natural modal decomposition and dimensionality reduction of complex dynamical systems. However, when DMD is applied to transient dynamics, even if a large number of modes are used, it remains difficult to in…

Convergence Rate of Generalized Nash Equilibrium Learning in Strongly Monotone Games with Linear Constraints
Tatiana Tatarenko, Maryam Kamgarpour
https://arxiv.org/abs/2507.12112 …

Convergence Rate of Generalized Nash Equilibrium Learning in Strongly Monotone Games with Linear Constraints
We consider payoff-based learning of a generalized Nash equilibrium (GNE) in multi-agent systems. Our focus is on games with jointly convex constraints of a linear structure and strongly monotone pseudo-gradients. We present a convergent procedure based on a partial regularization technique and establish the convergence rate of its iterates under one- and two-point payoff-based feedback. To the best of our knowledge, this work is the first one characterizing the convergence speed of iterates to…

Direct numerical simulations of inhalation in a 23-generation lung model
Marco Atzori, Emanuele Gallorini, Ciro Cottini, Andrea Benassi, Maurizio Quadrio
https://arxiv.org/abs/2506.14729

Direct numerical simulations of inhalation in a 23-generation lung model
The air flows in the proximal and distal portions of the human lungs are interconnected: the lower Reynolds number in the deeper generations causes a progressive flow regularization, while mass conservation requires flow rate oscillations to propagate through the airway bifurcations. To explain how these two competing effects shape the flow state in the deeper generations, we have performed the first high-fidelity numerical simulations of the air flow in a lung model including 23 successive bif…

Viscoplastic flows in narrow channels: Herschel-Bulkley models versus its regularizations
Benedetta Calusi, Angiolo Farina, Lorenzo Fusi, Luigi Vergori
https://arxiv.org/abs/2506.12934

Viscoplastic flows in narrow channels: Herschel-Bulkley models versus its regularizations
We investigated the two-dimensional flows of a viscoplastic fluid in symmetric channels with impermeable walls under no-slip boundary conditions. As response functions for the Cauchy stress tensor of the viscoplastic fluid, we considered both the celebrated Herschel-Bulkley model and a very general class $\mathcal{C}$ of its regularizations that depend on a positive parameter with the same physical dimensions as the strain rate and known as the \emph{regularization parameter}. Within this class…