Asymptotic Vanishing Research Articles

Emergent behaviors of two thermodynamically consistent inertial spin models

We propose two thermodynamically consistent inertial spin models generalizing the inertial spin model for the ensemble of mechanical particles with spin. For the collective dynamics modeling consistent with an entropy principle, we use two modeling ideas based on Hamiltonian formulation and flocking formalism for the coupling of the inertial spin model and temperatures introduced by Ha and Ruggeri [Arch. Ration. Mech. Anal. 223, 1397 (2017)]. The resulting models admit several constants of motions and are consistent with the entropy principle. We present several sufficient frameworks leading to the asymptotic vanishing of spins and velocity alignment. We also implement several numerical simulations and compare them with analytical results.

Accelerated optimization through time-scale analysis of inertial dynamics with asymptotic vanishing and Hessian-driven dampings

ABSTRACT Gradient optimization algorithms can be effectively studied from the perspective of ordinary differential equations (ODEs), where these algorithms are obtained through the temporal discretization of the continuous dynamics. This approach provides a powerful tool to comprehend acceleration phenomena in optimization, thanks to time-scale techniques and Lyapunov analysis. In the context of a Hilbert setting, our study focuses on the rapid optimization properties of inertial dynamics, which combine asymptotically vanishing viscous damping with Hessian-driven damping. These dynamics are derived as high-resolution ODEs from both the Nesterov accelerated gradient method and the Ravine method. For a general differentiable convex function f, choosing the viscous damping coefficient in the form of α / t with 3 $ ]]> α > 3 guarantees an inverse quadratic convergence rate of the values, i.e. o ( 1 / t 2 ) , the weak convergence of trajectories towards optimal solutions, and the fast convergence of gradients towards zero. By carefully scaling the dynamics temporally and based on the tuning of the damping coefficient in front of the Hessian term, we identify the limit dynamic as α becomes large. In particular, we examine the case where the limit dynamic corresponds to the Levenberg-Marquardt regularization of Newton's continuous method. This explanation accounts for the aforementioned fast convergence properties and provides fresh insights into the complexity of these methods, which play a central role in optimization for high-dimensional problems. The interplay between numerical algorithms and continuous-time ODEs plays a fundamental role in our analysis for the design and understanding of accelerated optimization algorithms. Numerical experiments are presented to illustrate and support the theoretical results.

Asymptotic vanishing of syzygies of algebraic varieties

The purpose of this paper is to prove Ein–Lazarsfeld’s conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein–Lazarsfeld’s asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows. Previously, Raicu reduced the problem to the case of products of three projective spaces, and we resolve this case here.

Improving MRAC Transient Response Using Generalized Dynamic Inversion and Nullspace Parametrization

We demonstrate how the transient response of Model Reference Adaptive Control (MRAC) is improved by utilizing Generalized Dynamic Inversion (GDI) and the corresponding nullspace parametrization in the powerful Greville formula. The proposed approach neither alters the reference model structure, nor it modifies it through feedback. Rather, the proposed approach utilizes the control redundancy in matching the reference model for the purpose of designing an MRAC control law that yields good transient closed loop adaptive control response. The adaptive GDI-MRAC control law is composed of two parts: a particular part that enforces the reference model dynamics, and an auxiliary part that is affine in a null control vector. The particular and the auxiliary parts of the adaptive control law are confined in the range space of the transposed control matrix and its orthogonal complement nullspace, respectively. The null control vector is projected onto the nullspace of the control matrix via a nullprojection matrix. Hence, the null control vector design does not interfere with the reference model following, yet it can be designed via state feedback to improve closed loop state transient response. An improved Lyapunov control function is used to design the adaptation gains such that asymptotic vanishing of the state error dynamics is guaranteed. Extensive simulations are conducted to demonstrate the efficacy of the proposed method.

Improved convergence rates and trajectory convergence for primal-dual dynamical systems with vanishing damping

In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with asymptotically vanishing damping term. The system is formulated in terms of the augmented Lagrangian associated to the minimization problem. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. In case the objective function has Lipschitz continuous gradient, we show that the primal-dual trajectory asymptotically weakly converges to a primal-dual optimal solution of the underlying minimization problem. To the best of our knowledge, this is the first result which guarantees the convergence of the trajectory generated by a primal-dual dynamical system with asymptotic vanishing damping. Moreover, we will rediscover in case of the unconstrained minimization of a convex differentiable function with Lipschitz continuous gradient all convergence statements obtained in the literature for Nesterov's accelerated gradient method.

First-order inertial algorithms involving dry friction damping

In a Hilbert space $$ {\mathcal H}$$ , we introduce a new class of first-order algorithms which naturally occur as discrete temporal versions of an inertial differential inclusion jointly involving viscous friction and dry friction. The function $$f:{\mathcal H}\rightarrow {\mathbb {R}}$$ to be minimized is supposed to be differentiable (not necessarily convex), and enters the algorithm via its gradient. The dry friction damping function $$\phi :{\mathcal H}\rightarrow {\mathbb {R}}_+$$ is convex with a sharp minimum at the origin, (typically $$\phi (x) = r \Vert x\Vert $$ with $$r >0$$ ). It enters the algorithm via its proximal mapping, which acts as a soft threshold operator on the velocities. As a result, we obtain a new class of splitting algorithms involving separately the proximal and gradient steps. The sequence of iterates has a finite length, and therefore strongly converges towards an approximate critical point $$x_{\infty }$$ of f (typically $$\Vert \nabla f(x_{\infty })\Vert \le r$$ ). Under a geometric property satisfied by the limit point $$x_{\infty }$$ , we obtain geometric and finite rates of convergence. The convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence towards zero. By replacing the function f by its Moreau envelope, we extend the results to the case of nonsmooth convex functions. In this case, the algorithm involves the proximal operators of f and $$\phi $$ separately. Several variants of this algorithm are considered, including the case of the Nesterov accelerated gradient method. We then consider the extension in the case of additive composite optimization, thus leading to new splitting methods. Numerical experiments are given for Lasso-type problems. The performance profiles, as a comparison tool, demonstrate the efficiency of the Nesterov accelerated method with asymptotic vanishing damping combined with dry friction.

An asymptotic vanishing theorem for the cohomology of thickenings

Let X be a closed equidimensional local complete intersection subscheme of a smooth projective scheme Y over a field, and let \(X_t\) denote the t-th thickening of X in Y. Fix an ample line bundle \(\mathcal {O}_Y(1)\) on Y. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer c, such that for all integers \(t \geqslant 1\), the cohomology group \(H^k(X_t,\mathcal {O}_{X_t}(j))\) vanishes for \(k < \dim X\) and \(j < -ct\). Note that there are no restrictions on the characteristic of the field, or on the singular locus of X. We also construct examples illustrating that a linear bound is indeed the best possible, and that the constant c is unbounded, even in a fixed dimension.

Vanishing for Frobenius twists of ample vector bundles

We prove several asymptotic vanishing theorems for Frobenius twists of ample vector bundles in positive characteristic. As an application, we improve the Bott-Danilov-Steenbrink vanishing theorem for ample vector bundles on toric varieties.

Characterizing finite length local cohomology in terms of bounds on Koszul cohomology

Let (R,m,κ) be a local ring. We give a characterization of R-modules M whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module M is asymptotically Cohen-Macaulay if and only if M is Cohen-Macaulay on the punctured spectrum if and only if sup⁡{ℓ(Hi(f1,…,fd;M))|f1,…,fd=m, i<d}<∞ for d=dim⁡(M)=dim⁡(R).

On asymptotic vanishing behavior of local cohomology

Let R be a standard graded algebra over a field k, with irrelevant maximal ideal $$\mathfrak {m}$$, and I a homogeneous R-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules $$\{{\text {H}}^{i}_{\mathfrak {m}}(R/I^n)\}_{n\in {\mathbb {N}}}$$ for $$i<\dim R/I$$. We show that, when $${{\,\mathrm{{{char}}}\,}}k= 0$$, R / I is Cohen–Macaulay, and I is a complete intersection locally on $${{\,\mathrm{{Spec}}\,}}R {\setminus }\{\mathfrak {m}\}$$, the lowest degrees of the modules $$\{{\text {H}}^{i}_{\mathfrak {m}}(R/I^n)\}_{n\in {\mathbb {N}}}$$ are bounded by a linear function whose slope is controlled by the generating degrees of the dual of $$I/I^2$$. Our result is a direct consequence of a related bound for symmetric powers of locally free modules. If no assumptions are made on the ideal or the field k, we show that the complexity of the sequence of lowest degrees is at most polynomial, provided they are finite. Our methods also provide a result on stabilization of maps between local cohomology of consecutive powers of ideals.

Development and implementation of an effective constitutive model for architected cellular iron-based shape memory alloys: Pressure dependency and transformation-plasticity interaction

Using architected cellular iron-based shape memory alloys (AC Fe-SMAs) can help compensate the relatively higher density of the base material compared to NiTi-based and Cu-based shape memory alloys, while providing good shape recovery, lower production cost, and greater energy dissipation. This article is dedicated to the development of an effective and pressure-dependent constitutive model that predicts the thermomechanical response of AC Fe-SMAs. We first simulate the behavior of the cellular material using the dense model that was previously developed by the authors, along with different unit cells (UCs) subject to periodic boundary. The shape memory effect is simulated by compressing the UCs by 2% of their height, followed by mechanical unloading, and heating above the austenite finish temperature. The results highlight stress concentration, maximum phase transformation, and maximum plastic deformation at the geometry discontinuities or strands necks. Post-processing the ABAQUS ODB files with Python scripts shows that the bending-dominated unit cell with the highest maximum local values of state variables exhibits the lowest volume-averaged outputs. Comparison of the unit cell to cubic multi-cell structures points out an asymptotic vanishing of the effects of the free boundaries as the number of cells in the multi-cell structure increases. The results of the unit cells are used to calibrate the parameters of the pressure-dependent effective model. The ratios of the inelastic hydrostatic strains to the equivalent total inelastic strains indicate higher pressure effects in the bending-dominated cellular lattice than in the stretch-dominated structures. The force-displacement and “dissipated energy”-temperature curves of the cellular beam and its equivalent bulk structure obtained by simulations of 4-point bending tests are found to be close enough that the effective model can be considered as an efficient design tool for architected cellular iron-based shape memory alloy structures.

Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3

In a Hilbert space setting ℋ, given Φ : ℋ → ℝ a convex continuously differentiable function, and α a positive parameter, we consider the inertial dynamic system with Asymptotic Vanishing Damping (AVD)α ẍ(t) + α/tẋ(t) + ∇Φ(x(t)) = 0. Depending on the value of α with respect to 3, we give a complete picture of the convergence properties as t → +∞ of the trajectories generated by (AVD)α, as well as iterations of the corresponding algorithms. Indeed, as shown by Su-Boyd-Candès, the case α = 3 corresponds to a continuous version of the accelerated gradient method of Nesterov, with the rate of convergence Φ(x(t)) − min Φ = O(t−2) for α ≥ 3. Our main result concerns the subcritical case α ≤ 3, where we show that Φ(x(t)) − min Φ = O(t−⅔α). This overall picture shows a continuous variation of the rate of convergence of the values Φ(x(t)) − minℋ Φ = O(t−p(α)) with respect to α &gt; 0: the coefficient p(α) increases linearly up to 2 when α goes from 0 to 3, then displays a plateau. Then we examine the convergence of trajectories to optimal solutions. As a new result, in the one-dimensional framework, for the critical value α = 3, we prove the convergence of the trajectories. In the second part of this paper, we study the convergence properties of the associated forward-backward inertial algorithms. They aim to solve structured convex minimization problems of the form min {Θ := Φ + Ψ}, with Φ smooth and Ψ nonsmooth. The continuous dynamics serves as a guideline for this study. We obtain a similar rate of convergence for the sequence of iterates (xk): for α ≤ 3 we have Θ(xk) − min Θ = O(k−p) for all p &lt; 2α/3, and for α &gt; 3 Θ(xk) − min Θ = o(k−2). Finally, we show that the results are robust with respect to external perturbations.

Asymptotic vanishing of homogeneous components of multigraded modules and its applications

In this article, we give a condition on the vanishing of finitely many homogeneous components which must imply the asymptotic vanishing for multigraded modules. We apply our result to multi-Rees algebras of ideals. As a consequence, we obtain a result on normality of monomial ideals, which extends and improves the results of Reid, Roberts and Vitulli [14], Singla [16], and Sarkar and Verma [15].

Spin squeezing and pairwise entanglement under non-Markovian environments with dynamical decoupling pulses

We study the effect of non-Markovian environments on spin squeezing and pairwise entanglement of a qubit ensemble in the presence of dynamical decoupling pulses. Each qubit of the ensemble interacts with its own environment, and the environments are independent and identical. It is shown that spin squeezing and entanglement may display sudden death, asymptotic vanishing and revival behaviors. Moreover, it is indicated that spin squeezing and entanglement can be protected effectively by employing appropriate pulse separate time and pulse number as well as non-Markovian properties.

Fault-tolerant sensorless control of wind turbines achieving efficiency maximization in the presence of electrical faults

This paper proposes a sensorless fault-tolerant control strategy solving the tracking problem of the maximum delivered power characteristic for a wind energy conversion system equipped with a permanent magnet synchronous generator. A previously published control scheme ensuring the maximum power efficiency of the wind turbine, not requiring feedback information about rotor speed and position, and about wind velocity, is here extended to make the control scheme fault-tolerant with respect to possible electrical faults affecting the equations of the permanent magnet synchronous generator (PMSG) in the original (α, β) frame. The control law is based on a number of interconnected nonlinear observers. Closed loop asymptotic vanishing of the observation errors is proved. The proposed control solution has been validated on the National Renewable Energy Laboratory (NREL) 5-MW three-blade wind turbine model.

Regularity, positivity and asymptotic vanishing of solutions of a φ-Laplacian

AbstractIn this note we prove that solutions of a φ-Laplacian operator on the entire space ℝNare locally regular (Hölder continuous), positive and vanish at infinity. Mild restrictions are imposed on the right-hand side of the equation. For example, we assume a Lieberman-like condition but the hypothesis of differentiability is dropped. This is in striking contrast with the classical case.

The exponent in the orthogonality catastrophe for Fermi gases

We quantify the asymptotic vanishing of the ground-state overlap of two non-interacting Fermi gases in d -dimensional Euclidean space in the thermodynamic limit. Given two one-particle Schrödinger operators in finite-volume which differ by a compactly supported bounded potential, we prove a power-law upper bound on the ground-state overlap of the corresponding non-interacting N -Fermion systems. We interpret the decay exponent \gamma in terms of scattering theory and find \gamma = \pi^{-2}\|\mathrm {arcsin}|T_E/2|\|_{\mathrm {HS}}^2 , where T_E is the transition matrix at the Fermi energy E . This exponent reduces to the one predicted by Anderson [Phys. Rev. 164, 352–359 (1967)] for the exact asymptotics in the special case of a repulsive point-like perturbation.

Representation stability for syzygies of line bundles on Segre–Veronese varieties

The rational homology groups of packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre–Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology groups of packing complexes. This allows us to deduce stability properties for the syzygies of line bundles on Segre–Veronese varieties. We provide bounds for when stabilization occurs and show that these bounds are sometimes sharp by describing the linear syzygies for a family of line bundles on Segre varieties. As a motivation for our investigation, we show in an appendix that Ein and Lazarsfeld’s conjecture on the asymptotic vanishing of syzygies of coherent sheaves on arbitrary projective varieties reduces to the case of line bundles on a product of (at most three) projective spaces.

Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity

In a Hilbert space setting $${{\mathcal {H}}}$$H, we study the fast convergence properties as $$t \rightarrow + \infty $$tź+ź of the trajectories of the second-order differential equation $$\begin{...

Sensorless efficient fault-tolerant control of wind turbines with geared generator

This paper proposes a sensorless (i.e. not requiring feedback information about rotor speed and position, and about wind velocity) control technique solving the tracking problem of the maximum delivered power characteristic for a wind energy conversion system equipped with a permanent magnet synchronous generator. A previously published sensorless control scheme ensuring the achievement of maximum power efficiency of the wind turbine is here extended to consider the drive-train dynamics. A novel observer of the electrical speed, based on the electrical variables only, is proposed, and asymptotic vanishing of the observation error is proved, as well as boundedness of the observer variables. Moreover, it is proved that the derived control algorithm can be easily modified also to ensure fault tolerance with respect to a class of system faults, sensor faults occurring in mechanical sensors being absent because of the sensorless approach here pursued. The proposed control solution has been validated on a 5−MW three-blade wind turbine using the National Renewable Energy Laboratory (NREL) wind turbine simulator FAST (Fatigue, Aerodynamics, Structures, and Turbulence) code.