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INFORMATION CAN BE OBTAINED ON THIS NEW AND EXCITING DEVELOPMENT DIRECT FROM THE SOLE SELLING AGENT HESKIN AUCTIONEERS ON 091-56 58 57.
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An Maolan is an attractive option for families, first time buyers and those looking for a spacious new home, built to the highest standards with all the features that make modern life easy.
Each property will achieve a BER Rating of A3 due to the excellent energy features such as Air to water heating throughout and highly efficient insulation to cavities and roof.
This development is ideally located in the much sought after idyllic village of Barna, 8 Km from Galway City.
Barna Village has a well-established thriving community which offers the homeowner numerous amenities such as supermarket, shops, bars, excellent restaurants including Donnellys and O Gradys on the Pier while the award winning Twelve Hotel is located in the heart of the village. The Barna area has many sporting facilities including a Golf Club, G.A.A. Club, Basketball Club and Soccer Club. The local primary school, Scoil Shéamais Naofa is located in the village.
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text:
Olli Jaakkola (* 17. Mai 1996) ist ein finnischer Biathlet.

Olli Jaakkola gab sein internationales Debüt bei den Juniorenrennen der Biathlon-Europameisterschaften 2014 in Nové Město na Moravě. Sprint und Verfolgung beendete er als 48., im Einzel wurde der Finne 31. Für das Staffelrennen wurde er in die Männerstaffel Finnlands berufen. Gemeinsam mit Antti Raatikainen, Sami Orpana und Tuomas Grönman erreichte er als Schlussläufer der Staffel den 17. Platz. Danach trat er erstmals in Presque Isle bei den Biathlon-Juniorenweltmeisterschaften 2014 an und wurde 18. des Sprints und 25. der Verfolgung. Mit Teemu Huhtala und Mikko Loukkaanhuhta gewann er zudem im Staffelrennen die Bronzemedaille hinter Russland und Kanada.

Weblinks

Biathlet (Finnland)
Finne
Geboren 1996
Mann

redpajama_set_name: RedPajamaWikipedia


Well, 43 people were fed, 2 turkeys were sacrificed, 5 pies and about 4 dozen cupcakes were consumed, mounds of rice, atole, coditos, salad, apple sauce, cranberry sauce, stuffing, arroz con leche, flan, casava bread, and empanadas were devoured. Am I missing anything? Thank YOU everyone for generously bringing food, drinks, and desserts to share in this truly wonderful meal. I hope everyone feels as blessed as I do to be surrounded by such caring and generous people each day.


In accordance with the Local Governance (Scotland) Act 2004 (Remuneration) Regulations 2007 as amended, Clackmannanshire Council hereby publish figures for Councillor's Remuneration, Allowances and Expenses.
These are determined by reference to the Local Governance (Scotland ) Act 2004 (Remuneration) Regulations 2007 (SS1 2007/183). A breakdown of the figures is available from Finance Services, Kilncraigs, Alloa, at all reasonable times, for inspection free of charge.
Clackmannanshire Council has a Protocol for Elected Members' Training and Development, which aims to ensure consistency in the Council's approach to training for Elected Members. The protocol is available to view or download from this page. However, if you would prefer us to send you a copy, please contact us using the details below.
Individual training plans are in place for each Member and the particulars are used to inform the annual training plan.
During 2017/18, a comprehensive range of training was attended by Elected Members following the Local Elections in May 2017. Training needs are reviewed on a regular basis.
Information on the specific training undertaken is also available on request.


text:
\section{Introduction}
	\subsection{Problem Statement}\label{sec::pbstatement}
	A major challenge in modern unconstrained convex optimization consists in building fast algorithms while maintaining low computational cost and memory footprint.
	This plays a central role in many key applications such as large-scale machine learning problems or data processing. The problems we are aiming to study are of the form
	\begin{equation*}
		\min_{x\in\mathbb{R}^n} f(x).
	\end{equation*}
	Large values of $n$ demand for algorithms at the interface of first- and second-order optimization.
	Limited computational capabilities explain why gradient-based (first-order) algorithms remain prominent in practice.
	Unfortunately, they often require many iterations which is true even for the provably best algorithms for certain classes of optimization problems; for example that of convex and strongly convex functions with Lipschitz continuous gradient \citep{polyak1964some,nesterov1983method,nesterov2003introductory}. On the other hand, algorithms using second-order information (the Hessian of $f$)---with Newton's method as prototype---adapt locally to the geometry of the objective, allowing them to progress much faster towards a solution. However, each iteration comes with high computational and memory costs, which highlights a challenging trade-off.
	It is therefore essential to develop algorithms that take the best of both worlds. They are commonly referred to as (limited-memory) quasi-Newton methods.
	Several quasi-Newton algorithms partly address this issue, for example BFGS methods \citep{broyden1970convergence,fletcher1970new,goldfarb1970family,shanno1970conditioning,liu1989limited}, yet, in very large-scale applications, first-order algorithms often remain the preferred choice.

	In order to reach a new level of efficiency, deep insights into the mechanism and relations between algorithms are required. To that aim, an insightful approach is to see optimization algorithms as discretization of ordinary differential equations (ODEs): for small-enough step-sizes, iterates can be modeled by a continuous-time trajectory \citep{ljung1977analysis,benaim2005stochastic}.
	Obtaining a fast algorithm following this strategy depends on two ingredients: choosing an ODE for which rapid convergence to a solution can be proved, and discretizing it with an appropriate scheme that preserves the favorable properties of the ODE.

	Both steps are highly challenging, our work focuses on the ODE matter. We study the following second-order dynamical system in a general setting:
	\begin{equation}\tag{VM-DIN-AVD}\label{eq::VMDINAVD}
		\varepsilon(t) \ddot{x}(t) + \alpha(t) \dot x(t) + \beta \nabla^2 f(x(t))\dot{x}(t)+\nabla f(x(t))=0,
		\quad t\geq 0,
	\end{equation}
	where $f\colon \mathbb{R}^n\to\mathbb{R}$ is a smooth convex twice continuously differentiable function, with gradient $\nabla f$ and Hessian $\nabla^2 f$ defined on $\mathbb{R}^n$ equipped with scalar product $\langle\cdot,\cdot\rangle$, and induced norm $\norm{\cdot}$. Additionally, $f$ is assumed to be coercive, and strongly convex on bounded subsets of $\mathbb{R}^P$.
	The functions $\varepsilon,\alpha\colon\mathbb{R}_+\to\mathbb{R}_+$ (where $\mathbb{R}_+=[0,+\infty[$) are differentiable, non-increasing, and $\varepsilon(t)>0$ for all $t\geq0$. Together with $\beta> 0$, they are control parameters that define the type of dynamics that drives the trajectory (or solution) $x\colon\mathbb{R}_+\to\mathbb{R}^n$, whose first- and second-order derivatives are denoted $\dot{x}$ and $\ddot{x}$ respectively.
	We call the above dynamics \eqref{eq::VMDINAVD}, which stands for ``Variable Mass Dynamical Inertial Newton-like system with Asymptotically Vanishing Damping'' since it generalizes a broad class of ODEs whose original member is DIN \citep{alvarez2002second}, where $\varepsilon$ and $\alpha$ were constant. DIN was then extended to the case of non-constant \emph{asymptotically vanishing dampings} (AVD) $\alpha$ \citep{attouch2016fast}. In this work we introduce the non-constant parameter $\varepsilon$ called \emph{variable mass} (VM) in front of the acceleration $\ddot{x}$, in the same way that $\alpha$ is called (viscous) \emph{damping} by analogy with classical mechanics.
	A key feature of these ODEs, that positions them at the interface of first- and second-order optimization, is that they possess equivalent forms involving only $\nabla f$ but not $\nabla^2f$, significantly reducing computational costs, hence enabling the design of practical algorithms, see e.g., \cite{castera2021inertial,attouch2020first,chen2019first}.
	The key idea behind this is the relation $\nabla^2 f(x(t))\dot{x}(t) = \frac{\diff}{\diff t}\nabla f(x(t))$, see Section~\ref{sec::prelim} for an equivalent formulation of \eqref{eq::VMDINAVD} exploiting this.

	This paper emphasizes the relation between \eqref{eq::VMDINAVD} and well-studied special cases.
	Indeed, taking $\varepsilon=\alpha=0$, one obtains\footnote{\ref{eq::CN} is usually considered with $\beta=1$, we put $\beta$ in the system to ease the discussions below.} the Continuous Newton (CN) method \citep{gavurin1958nonlinear}
	\begin{equation}\tag{CN}\label{eq::CN}
		\beta \nabla^2 f(x_N(t))\dot{x}_N(t)+\nabla f(x_N(t))=0, \quad t\geq 0,
	\end{equation}
	known notably for being invariant to affine transformations and yielding fast vanishing of the gradient (see Section~\ref{sec::Control}).
	In fact, this observation shows that \eqref{eq::VMDINAVD} is a singular perturbation of \eqref{eq::CN}, which also justifies the terminology ``Newton-like''  in DIN.
	When $\alpha\neq 0$ but $\varepsilon=0$, we recover the Levenberg--Marquardt (LM) method,
	\begin{equation}\tag{LM}\label{eq::LM}
		\alpha(t) \dot x_{LM}(t) + \beta \nabla^2 f(x_{LM}(t))\dot{x}_{LM}(t)+\nabla f(x_{LM}(t))=0, \quad t\geq 0,
	\end{equation}
	also known as regularized Newton method since it stabilizes \eqref{eq::CN}.
	In the rest of the paper, the solutions of \eqref{eq::CN} and \eqref{eq::LM} will always be denoted by $x_N$ and $x_{LM}$ respectively. \citet{alvarez2002second} showed that for $\alpha=0$, $\beta=1$, and $\varepsilon$ constant and small, \eqref{eq::VMDINAVD} is a ``perturbed'' Newton method since the distance between the solutions of \eqref{eq::VMDINAVD} and \eqref{eq::CN}  is at most proportional to $\sqrt{\varepsilon}$ at all time.
	Yet, despite the benefits of this class of ODEs, such as stabilization properties, see e.g., \cite{attouch2016fast,attouch2020first}, no improvement\footnote{DIN-like systems were thought to yield faster vanishing of the gradient compared to inertial first-order dynamics, until recently \citep{attouch2022ravine}.} of the rate of convergence (in values) has been shown compared to inertial first-order dynamics \citep{polyak1964some,su2014differential}. This raises the question:
	\begin{center}
		``\emph{are these ODEs really of Newton type?}'',
	\end{center}
	which is crucial in view of designing faster algorithms from them.

\begin{figure}[t]
\centering
	\begin{minipage}{0.53\linewidth}
		\centering
        \hspace{0.05\linewidth}\includegraphics[width=.8\linewidth]{Figures/horizontal_legend_labels}
		\includegraphics[width=\linewidth]{Figures/new_phase_diagram_alpha_eps}
	\end{minipage}
	\begin{minipage}{0.46\linewidth}
		\centering
		\includegraphics[width=\linewidth]{Figures/quad_illustration_noniso.pdf}
	\end{minipage}
	\caption{Left: phase diagram on distances from \eqref{eq::VMDINAVD} to \eqref{eq::CN} and \eqref{eq::LM} (see Section~\ref{sec::Control}). For each patch, the color indicates which of the distances $\norm{x_N(t)-x(t)}$ and $\norm{x_{LM}(t)-x(t)}$ is considered, the scaling of a corresponding upper-bound on this distance is written; in white for prior work and in black for our contributions.
	 The green line separates the cases $\varepsilon\geq \alpha$ (above) and $\varepsilon\leq \alpha$ (below).
	 Right: 2D illustration of the trajectories of \eqref{eq::VMDINAVD} for several choices of $\varepsilon$ on a quadratic function. Using fast-vanishing $\varepsilon(t)$ (dark-blue solid curves), one can bring the solution of \eqref{eq::VMDINAVD} close to that of \eqref{eq::CN}, making it, for example, more robust to bad conditioning compared to first-order dynamics (such as gradient descent).
	\label{fig::phasediagandquad}}
\end{figure}

	\subsection{Main Contributions} We show that the answer to this question is partially positive, and closely related to the choices of $\varepsilon$ and $\alpha$.
	We provide general results on the role played by these two control parameters and how they can be chosen to control \eqref{eq::VMDINAVD}, and make it close to \eqref{eq::CN} \emph{for all time}, as illustrated on the right-hand side of Figure~\ref{fig::phasediagandquad}, but also to obtain fast convergence. This represents a first step towards building new fast practical algorithms. Our main contributions are the following:
	\\
	--\ We provide a first-order equivalent formulation of \eqref{eq::VMDINAVD}, and address the questions of existence and uniqueness of the solutions of \eqref{eq::VMDINAVD} under mild assumptions.
	\\
	--\ We generalize the perturbed Newtonian property discussed above to non-constant and possibly vanishing variable masses $\varepsilon$, and ``not too large'' positive dampings $\alpha$, and derive bounds that (formally) take the form $\norm{x(t)-x_N(t)}=O(\sqrt{\varepsilon(t)})$. We then extend these results to larger dampings $\alpha$ and make the connection between \eqref{eq::VMDINAVD} and \eqref{eq::LM}. This contribution is summarized in the phase diagram of Figure~\ref{fig::phasediagandquad}.
	\\
	--\ Using quadratic functions as a model for strongly convex functions, we shed light on techniques to efficiently approximate solutions of \eqref{eq::VMDINAVD}.
	We then show how $\varepsilon$ and $\alpha$ affect the speed of convergence. Depending on their setting, the solutions of \eqref{eq::VMDINAVD} may either converge as fast as that of \eqref{eq::CN}, \emph{faster}, or rather have a \eqref{eq::LM} nature, as summarized in Table~\ref{tabl::speedsummary}.
	\\
	--\ We provide numerical experiments supporting our theoretical findings.

	\begin{table}[t]
		\small
		\centering\setlength\extrarowheight{1.5pt}
		\caption{Informal summary of Section~\ref{sec::Quads}. Comparison of \eqref{eq::VMDINAVD} with other dynamics\label{tabl::speedsummary}}
		\begin{tabular}{@{\extracolsep{2pt}}*{2}{c@{\enspace}c@{\enspace}c}}
			\toprule
			\multicolumn{2}{c}{Parameters of \eqref{eq::VMDINAVD}} &\quad  & \multicolumn{2}{c}{Speed of convergence}\\
			\cmidrule{1-2} \cmidrule{4-5}
			{Dominant parameter}& {Integrability in $+\infty$} & & w.r.t.\ \eqref{eq::CN} & w.r.t.\ \eqref{eq::LM} \\
			\midrule
			\multirow{2}{*}{variable mass $\varepsilon$}
			& yes & & as fast & as fast   \\
			& no & & \color{Green3}faster & \color{Green3}faster \\
			\multirow{2}{*}{viscous damping $\alpha$}
			& yes & & as fast & \multirowcell{2}{\color{Seashell4}only depends\\ \color{Seashell4}on $\varepsilon$}  \\
			& no & & \color{IndianRed1}slower & \\
			\bottomrule
		\end{tabular}
	\end{table}

	\subsection{Related work}
	The system \eqref{eq::VMDINAVD} belongs to the class of inertial systems with viscous and geometric (``Hessian-driven'') dampings, initially introduced with constant $\varepsilon = 1$ and constant $\alpha$ in \cite{alvarez2002second} and called DIN (for Dynamical Inertial Newton-like system). Except in a few cases \citep{castera2021inertial,castera2021saddle}, most of the follow-up work then considered extensions of DIN with non-constant AVD $\alpha$, with in particular the DIN-AVD system with $\alpha(t)=\alpha_0/t$ as introduced in \cite{attouch2016fast}. The reason for this popular choice for $\alpha$ is its link with Nesterov's method \citep{su2014differential}. Non-constant choices for $\beta$ have been considered \citep{attouch2020first,alecsa2021extension,lin2022control,attouch2020fast}. We keep it constant here, and rather focus on non-constant $\varepsilon$, unlike prior work that used constant $\varepsilon=1$.
	The mass $\varepsilon$ was only considered in the original work \citep{alvarez2002second}, but only for fixed $\varepsilon$, $\beta=1$ and constant $\alpha=0$.
	\ref{eq::VMDINAVD} is however closely related to the IGS system considered in \cite{attouch2020fast} as it is actually equivalent to the latter after dividing both sides of \eqref{eq::VMDINAVD} by $\varepsilon(t)$. Our approach---which consists in studying the connections with other second-order dynamics as $\varepsilon$ vanishes asymptotically---is however different from the one followed in \citep{attouch2020fast}, which is of independent interest.
	The literature on DIN is rich, let us mention further connections with Nesterov's method \citep{shi2021understanding,alecsa2021extension}, extensions with Tikhonov regularization \citep{boct2021tikhonov} and closed-loop damping \citep{attouch2022fast,lin2022control}. The non-smooth and possibly non-convex cases have been considered in  \cite{attouch2020newton,attouch2021continuous,castera2021inertial}. Finally, avoidance of strict saddle points in smooth non-convex optimization has been shown in \cite{castera2021saddle}.

	The influence of the damping $\alpha$ on the \eqref{eq::LM} dynamics has been studied in \cite{attouch2011continuous,attouch2013global}. Interestingly, the conditions enforced on $\alpha$ in these papers (formally a sub-exponential decay) are very similar to those we make on $\varepsilon$ and $\alpha$ for \eqref{eq::VMDINAVD} (see Assumptions~\ref{ass:largeepsilon} and~\ref{ass:Onlysubexp}).

	Regarding the second part of our analysis, which deals with the case where $f$ is quadratic, \citet{attouch2020first} provided closed-form solutions to \eqref{eq::VMDINAVD} for $\varepsilon\equiv 1$ and special choices of $\alpha$. Our work rather deals with approximate solutions which allows considering a wide class of functions $\varepsilon$ and $\alpha$. We rely on the Liouville--Green (LG) method~\citep{Liouville1836,green1838} presented in Section~\ref{sec::Quads}. Generalizations of LG are also often referred to as WKB methods~\citep{wentzel1926verallgemeinerung,kramers1926wellenmechanik,brillouin1926remarques} and seem to be mostly used in physics so far. To the best of the authors' knowledge, the current work seem to be one of the first to use the LG method in optimization, and the first for DIN-like systems.

	\subsection{Organization}
	The paper is organized as follows. We discuss the existence of solutions in Section~\ref{sec::prelim}. Our main results, from a non-asymptotic control perspective, are then presented in Section~\ref{sec::Control}.
	An analysis of the role played asymptotically by $\varepsilon$ and $\alpha$ is then carried out on quadratic functions in Section~\ref{sec::Quads}. Finally, numerical experiments are presented in Section~\ref{sec::exp}, and some conclusions are then drawn.

	\section{Existence and Uniqueness of Solutions}\label{sec::prelim}
	In the sequel, we fix $x_0\in\mathbb{R}^n$ and $\dot{x}_0\in\mathbb{R}^n$, such that, unless stated otherwise, \eqref{eq::VMDINAVD} is always considered with initial condition $(x(0),\dot{x}(0))=(x_0,\dot{x}_0)$, and \eqref{eq::CN} and \eqref{eq::LM} with initial condition $x_{N}(0)=x_{LM}(0)=x_0$. We also fix initial values for the control parameters $\varepsilon(0)=\varepsilon_0>0$, $\varepsilon'(0)=\varepsilon'_0\leq0$ and $\alpha(0)=\alpha_0\geq 0$.
	In addition to the definitions of $\varepsilon$ and $\alpha$ in Section~\ref{sec::pbstatement}, we assume that $\varepsilon$ is twice differentiable, with bounded second derivative.
	In order to use the Cauchy--Lipschitz Theorem, we reformulate \eqref{eq::VMDINAVD} into a first-order (in time) system by introducing an auxiliary variable $y:\mathbb{R}_{+}\to\mathbb{R}^n$. Notably, our reformulation does not involve $\nabla^2 f$, in the same fashion as \cite{alvarez2002second,attouch2016fast}.
	For all $t$, defining $\nu(t) = \alpha(t)-\varepsilon'(t) - \frac 1 \beta \varepsilon(t)$, we show in Appendix~\ref{app:FOS} that \eqref{eq::VMDINAVD} is equivalent to
	\begin{equation}\label{eq::gVMDINAVD}\tag{gVM-DIN-AVD}
		\begin{cases}
			\varepsilon(t)\dot{x}(t) + \beta\nabla f(x(t)) + \nu(t)x(t) + y(t)&= 0
			\\\dot y(t)+ \nu'(t)x(t) + \frac{\nu(t)}{\beta} x(t) + \frac 1 \beta y(t) &= 0
		\end{cases}
	\end{equation}
	with initial conditions $(x(0),y(0)) = \left(x_0,-\varepsilon_0\dot{x}_0 - \beta\nabla f(x_0) - (\alpha_0-\varepsilon'_0-\frac{1}{\beta}\varepsilon_0)x_0\right)$.
	One can notice that in the special case where $\varepsilon$ is taken constant and equal to $1$ (that is when \eqref{eq::VMDINAVD} is simply the DIN-AVD system \citep{attouch2016fast}), we recover the same first-order formulation as that in \cite{attouch2016fast}.

	We can then apply the Cauchy--Lipschitz Theorem. For all $t\geq0$ and $(u,v)\in\mathbb{R}^n\times\mathbb{R}^n$, define the mapping
	$$ G\left(t,(u,v)\right) = \begin{pmatrix}
		\frac{1}{\varepsilon(t)}(-\beta\nabla f(u)-\nu(t)u-v)
		\\ -\nu'(t)u - \frac{\nu(t)}{\beta}u - \frac{1}{\beta}v
	\end{pmatrix}, $$
	so that \eqref{eq::gVMDINAVD} rewrites $(\dot{x}(t),\dot y(t)) = G\left(t,(x(t),y(t))\right)$ for all $t\geq 0$. Since $f$ is twice continuously differentiable, one can see that $G$ is continuously differentiable w.r.t.\ its second argument $(u,v)$. Consequently $G$ is locally Lipschitz continuous w.r.t.\ $(u,v)$ and by the Cauchy--Lipschitz Theorem, for each initial condition, there exists a unique local solution to \eqref{eq::gVMDINAVD} and thus to \eqref{eq::VMDINAVD}. We then show that this solution is actually global (in Appendix~\ref{app:FOS}) by proving the boundedness of $(x,y)$. We omit the existence and uniqueness of the solutions of \eqref{eq::CN} and \eqref{eq::LM} since these are standard results, obtained with similar arguments.

	\section{Non-asymptotic Control of \eqref{eq::VMDINAVD}}\label{sec::Control}
	The purpose of this section is to understand how close $x$ might be to $x_N$ and $x_{LM}$, as a function of $\alpha$ and $\varepsilon$.
	Since $f$ is coercive and strongly convex on bounded sets, it has a unique minimizer $x^\star\in \mathbb{R}^n$. Consequently, any two trajectories that converge to $x^\star$ will eventually be arbitrarily close to each other. Thus, asymptotic results of the form $\Vert x(t)-x_N(t)\Vert \xrightarrow[t\to+\infty]{} 0$ are not precise enough to claim, for example, that $x$ has a ``Newtonian behavior''.
	Instead, we will derive upper bounds on the distance between trajectories that hold \emph{for all time} $t\geq 0$, and which typically depend on $\varepsilon$ and/or $\alpha$. We first present the case where $\alpha$ is small relative to $\varepsilon$ and then generalize.

	\subsection{Comparison with \eqref{eq::CN} under Moderate Viscous Dampings}
	When the viscous damping $\alpha$ remains moderate w.r.t.\ the variable mass $\varepsilon$, one can expect the solutions of \eqref{eq::VMDINAVD} to be close to that of \eqref{eq::CN}. We make the following assumptions.
	\begin{assumption}\label{ass:largeepsilon}
		There exists $c_1,c_2\geq 0$ such that for all $t\geq 0$, $\vert\varepsilon'(t)\vert\leq c_1 \varepsilon(t)$ and $\alpha(t) \leq c_2\varepsilon(t)$.
	\end{assumption}
	The assumption states that $\alpha$ must decrease at least as fast as $\varepsilon$ (up to a constant).\footnote{Assumption~\ref{ass:largeepsilon} can actually hold only after some large-enough $t_0\geq 0$, we take $t_0=0$ for the sake of simplicity.}
	 The reason for assuming $\vert\varepsilon'(t)\vert \leq  c_1\varepsilon(t)$ is technical and will appear more clearly in the proofs below. It formally means that $\varepsilon$ can decrease at most exponentially fast.\footnote{This is a consequence of Gronwall's lemma, see e.g., \citep{emmrich1999discrete}.} This is a relatively mild assumption that holds, for example, for any polynomial decay $\varepsilon_0/(t+1)^a$, $a\in\mathbb{N}$.

	We start with the main result of this section.
	\begin{theorem}\label{thm::mainEpsLarge}
		Let $x_N$ be the solution of \eqref{eq::CN}, and let $c_1,c_2\geq 0$. There exist $C_0,C_1,C_2\geq 0$, depending on $c_1$, $c_2$, such that for all $(\varepsilon,\alpha)$ for which Assumption~\ref{ass:largeepsilon} holds with constants $c_1$ and $c_2$, the corresponding solution $x$ of \eqref{eq::VMDINAVD} is such that for all $t\geq 0$,
		\begin{equation}\label{eq::ResLargeEps}
			\norm{x(t)-x_N(t)}\leq C_0 e^{-\frac{t}{\beta}}\varepsilon_0\norm{\dot{x}_0} + C_1\sqrt{\varepsilon(t)} +C_2 \int_{s=0}^{t} e^{\frac{1}{\beta}(s-t)} \sqrt{\varepsilon(s)}\diff s.
		\end{equation}
	\end{theorem}
	This extends a previous result from \cite[Proposition~3.1]{alvarez2002second} which states a similar bound for constant $\varepsilon$, $\alpha\equiv 0$ and $\beta=1$. Theorem~\ref{thm::mainEpsLarge} corresponds to the blue parts in the phase diagram of Figure~\ref{fig::phasediagandquad} (see also Corollary~\ref{cor::simpler} below).
	\begin{remark}
		The strength of the above result comes from the fact that the constants $C_0,C_1,C_2$ do not depend on $\varepsilon$ and $\alpha$, and that the result is \emph{non-asymptotic}. This allows in particular choosing $(\varepsilon,\alpha)$ to control the distance from $x$ to $x_N$, for all time $t\geq 0$.
	\end{remark}
    \begin{remark}
        Under Assumption~\ref{ass:largeepsilon}, the dynamics \eqref{eq::VMDINAVD} is dominated by the variable mass. The damping $\alpha$ does not appear in Theorem~\ref{thm::mainEpsLarge}.
    \end{remark}
	The above theorem and remarks emphasize the ``Newtonian nature'' of \eqref{eq::VMDINAVD}.
	We present two lemmas before proving Theorem~\ref{thm::mainEpsLarge}, and then state a simpler bound than \eqref{eq::ResLargeEps}, see Corollary~\ref{cor::simpler}.
	\begin{lemma}\label{lem::LyapLargeEps}
		Let $(\varepsilon,\alpha)$, and let $x$ be the corresponding solution of \eqref{eq::VMDINAVD}. For all $t\geq 0$, define the function,
		$$U(t) = \frac{\varepsilon(t)}{2} \norm{\dot{x}(t)}^2 + f(x(t))-f(x^\star).$$
		Then $U$ is differentiable and for all $t>0$,
		\begin{equation*
			\frac{\diff U}{\diff t} (t) = \frac{\varepsilon'(t)}{2} \norm{\dot{x}(t)}^2 - \alpha(t) \norm{\dot{x}(t)}^2 - \beta\langle \nabla^2 f(x(t))\dot{x}(t), \dot{x}(t)\rangle\leq 0.
		\end{equation*}
		Therefore, in particular, $U$ is non-increasing.
	\end{lemma}

	\begin{proof}
		Let $t\geq 0$, since $x$ is twice differentiable, $U$ is differentiable and,
		\begin{equation*}
			\frac{\diff U}{\diff t} (t) = \frac{\varepsilon'(t)}{2} \norm{\dot{x}(t)}^2 + \varepsilon(t)\langle\dot{x}(t),\ddot{x}(t)\rangle + \langle \dot{x}(t),\nabla f(x(t))\rangle.
		\end{equation*}
		We use the fact that $x$ is solution of \eqref{eq::VMDINAVD}, to substitute $\varepsilon(t)\ddot{x}(t)$ by its expression,
		\begin{equation*}
			\frac{\diff U}{\diff t} (t) = \frac{\varepsilon'(t)}{2} \norm{\dot{x}(t)}^2 - \alpha(t) \norm{\dot{x}(t)}^2 - \beta\langle \nabla^2 f(x(t))\dot{x}(t), \dot{x}(t)\rangle.
		\end{equation*}
		By assumption $\varepsilon$ is non-increasing so for all $t>0$, $\varepsilon'(t)\leq 0$. Furthermore $f$ is convex so $\langle \nabla^2 f(x(t))\dot{x}(t), \dot{x}(t)\rangle\geq 0$. Hence $U$ is non-increasing.
	\end{proof}
	We then state the following bound.
	\begin{lemma}\label{lem::boundeps}
		There exists $C\geq 0$ such that for any $(\varepsilon,\alpha)$ and the corresponding solution  $x$  of \eqref{eq::VMDINAVD}, for all $t\geq 0$ it holds that,
		\begin{equation*
			\varepsilon(t)\norm{\dot{x}(t)}\leq C\sqrt{\varepsilon(t)}.
		\end{equation*}
	\end{lemma}

	\begin{proof}
		Let $t\geq 0$, according to Lemma~\ref{lem::LyapLargeEps}, $U$ is non-increasing so $U(t)\leq U(0)$, or equivalently,
		\begin{equation*}
		 \frac{\varepsilon(t)}{2} \norm{\dot{x}(t)}^2 + f(x(t))-f(x^\star) \leq \frac{\varepsilon_0}{2} \norm{\dot{x}_0}^2 + f(x_0) - f(x^\star).
		\end{equation*}
		This implies in particular that,
		$$\varepsilon(t) \norm{\dot{x}(t)}^2 \leq \varepsilon_0 \norm{\dot{x}_0}^2 + 2(f(x_0) - f(x^\star)),$$ and hence by multiplying both sides by $\varepsilon(t)$ and composing with the square-root we obtain that,
		\begin{equation*}
			\varepsilon(t)\norm{\dot{x}(t)}\leq  C\sqrt{\varepsilon(t)},
		\end{equation*}
		where $C=\sqrt{\varepsilon_0 \norm{\dot{x}_0}^2 + 2(f(x_0) - f(x^\star))}$.
	\end{proof}
	\begin{proof}[Proof of Theorem~\ref{thm::mainEpsLarge}]
		 Let $(\varepsilon,\alpha)$ as defined in Sections~\ref{sec::pbstatement} and~\ref{sec::prelim}, and let $x$ be the corresponding solution of \eqref{eq::VMDINAVD}. Then, according to Lemma~\ref{lem::LyapLargeEps}, for all $t\geq 0$,  $U(t)\leq U(0)$, so  in particular
		 $$f(x(t)) \leq f(x_0) + \frac{\varepsilon_0}{2}\Vert\dot{x}_0\Vert^2.$$
		 Denoting $M_0= f(x_0) + \frac{\varepsilon_0}{2}\Vert\dot{x}_0\Vert^2$, the set $\mathsf{K}_0=\left\{y\in\mathbb{R}^n\mid f(y)\leq M_0\right\}$ is bounded, since $f$ is coercive ($\lim_{\norm{y}\to+\infty} f(y)=+\infty$). So for all $t\geq 0$, $x(t)\in\mathsf{K}_0$. Since $M_0$ (and hence $\mathsf{K}_0$) depends only on $\varepsilon_0$, $x_0$ and $\dot{x}_0$, we have proved that for any choice $(\varepsilon,\alpha)$, the corresponding solution $x$ of \eqref{eq::VMDINAVD} is inside $\mathsf{K}_0$ at all time.
		 Let $x_N$ be the solution of \eqref{eq::CN}. One can see similarly that for all $t\geq 0$, $f(x_N(t))\leq f(x_N(0))=f(x_0)\leq M_0$. So we also have $x_N(t)\in \mathsf{K}_0$ for all $t\geq 0$.

		 Now, fix $c_1,c_2>0$, and let $(\varepsilon,\alpha)$ such that Assumption~\ref{ass:largeepsilon} is satisfied with constants $c_1,c_2$. Let $x$ be the corresponding solution of \eqref{eq::VMDINAVD}.
		 Since $f$ is strongly convex on bounded sets, it is strongly convex on $\mathsf{K}_0$. We denote $\mu>0$ the strong-convexity parameter of $f$ on $\mathsf{K}_0$. Equivalently, we have that $\nabla f$ is strongly monotone on $\mathsf{K}_0$, that is, $\forall y_1,y_2\in\mathsf{K}_0$,
		 \begin{equation}\label{eq::gfMono}
		 	\langle\nabla f(y_1)-\nabla f(y_2),y_1-y_2\rangle\geq \mu \norm{y_1-y_2}^2.
		 \end{equation}
	 	Let $t\geq 0$, since $x(t)\in\mathsf{K}_0$ and $x_N(t)\in \mathsf{K}_0$, by
		combining \eqref{eq::gfMono} with the Cauchy--Schwarz inequality, we deduce that
		\begin{equation}\label{eq::boundwithgrad}
			\norm{x(t)-x_N(t)}\leq \frac 1 {\mu}\norm{\nabla f(x(t))-\nabla f(x_N(t))}.
		\end{equation}
		Therefore, it is sufficient to bound the difference of gradients in order to bound $\norm{x(t)-x_N(t)}$.
		First, remark that \eqref{eq::CN} can be rewritten as follows: $\frac{\diff}{\diff t} \nabla f(x_N(t)) + \frac 1 \beta\nabla f(x_N(t)) = 0$. So we can integrate, for all $t\geq0$,
		\begin{equation}\label{eq::gradN}
			\nabla f(x_N(t)) = e^{-\frac t \beta}\nabla f(x_0).
		\end{equation}

		We now turn our attention to $\nabla f(x(t))$, for which we cannot find a closed-form solution in general. We rewrite \eqref{eq::VMDINAVD} in the following equivalent form
		\begin{equation*
			\frac{\diff}{\diff t}\left[\varepsilon(t)\dot{x}(t)+\beta\nabla f(x(t))\right] + \frac{1}{\beta}\varepsilon(t)\dot{x}(t)+\nabla f(x(t)) = \left(\frac{1}{\beta}\varepsilon(t) + \varepsilon'(t)- \alpha(t)\right) \dot{x}(t).
		\end{equation*}
		Introducing the variable $\omega(t)=\varepsilon(t)\dot{x}(t)+\beta\nabla f(x(t))$, the latter is thus solution to
		\begin{equation*
			\begin{cases}
				\dot\omega(t)+ \frac 1 \beta \omega(t) = \left(\frac{1}{\beta}\varepsilon(t) + \varepsilon'(t)- \alpha(t)\right) \dot{x}(t),\quad t\geq 0,
				\\ \omega(0) = \varepsilon_0\dot{x}_0+\beta\nabla f(x_0).
			\end{cases}
		\end{equation*}
		This is a non-homogeneous first-order ODE in $\omega$, whose solution can be expressed using the integrating factor
		\begin{equation*}
			\omega(t)=e^{-\frac t \beta}(\varepsilon_0\dot{x}_0+\beta\nabla f(x_0))+ e^{-\frac t \beta}\int_{0}^{t} e^{\frac s \beta} \left(\frac 1 \beta \varepsilon(s) + \varepsilon'(s) - \alpha(s)\right)\dot{x}(s)\diff s.
		\end{equation*}
		We thus have the following expression for $\nabla f(x)$, for all $t\geq 0$,
		\begin{equation}\label{eq::gradDIN}
			\beta\nabla f(x(t))=\beta e^{-\frac t \beta}\nabla f(x_0) + e^{-\frac t \beta }\varepsilon_0\dot{x}_0 -\varepsilon(t)\dot{x}(t) + e^{-\frac t \beta}\int_{0}^{t} e^{ \frac s \beta} \left(\frac 1 \beta \varepsilon(s) + \varepsilon'(s) - \alpha(s)\right)\dot{x}(s)\diff s.
		\end{equation}

		We can now use \eqref{eq::gradN} and \eqref{eq::gradDIN} in \eqref{eq::boundwithgrad} to get
		\begin{equation*
			\norm{x(t)-x_N(t)}\leq\frac 1 {\beta\mu}\left\Vert e^{-\frac t \beta }\varepsilon_0\dot{x}_0 -\varepsilon(t)\dot{x}(t) + e^{-\frac t \beta}\int_{0}^{t} e^{ \frac s \beta} \left(\frac 1 \beta \varepsilon(s) + \varepsilon'(s) - \alpha(s)\right)\dot{x}(s)\diff s\right\Vert.
		\end{equation*}
		Using the triangle inequality, we obtain,
		\begin{equation}\label{eq::boundInt2}
			\norm{x(t)-x_N(t)}\leq \frac{\varepsilon_0\Vert\dot{x}_0\Vert}{\beta\mu} e^{-\frac t \beta } +  \frac{\varepsilon(t)\Vert\dot{x}(t)\Vert}{\beta\mu} + \frac{1}{\beta\mu} \int_{0}^{t} e^{ \frac 1 \beta (s-t)} \left\vert \frac 1 \beta \varepsilon(s) + \varepsilon'(s) - \alpha(s)\right\vert \Vert\dot{x}(s)\Vert\diff s.
		\end{equation}
		The first term in \eqref{eq::boundInt2} corresponds to the first one in \eqref{eq::ResLargeEps} with $C_0= 1/(\beta\mu)$. As for the second-one, by direct application of Lemma~\ref{lem::boundeps}, there exists $C>0$ such that for all $t\geq 0$, $\frac{\varepsilon(t)\Vert\dot{x}(t)\Vert}{\beta\mu}\leq C\sqrt{\varepsilon(t)}$, so we set $C_1 = C/(\beta\mu)$.
		Regarding the last term in \eqref{eq::boundInt2}, using Assumption~\ref{ass:largeepsilon} and again Lemma~\ref{lem::boundeps}, it holds that, for all $s\geq 0$,
		$$
		\left\vert \frac 1 \beta \varepsilon(s) + \varepsilon'(s) - \alpha(s)\right\vert \norm{\dot{x}(s)} \leq
		\left(\frac 1 \beta+c_1+c_2\right) \varepsilon(s)\norm{\dot{x}(s)}\leq \left(\frac 1 \beta+c_1+c_2\right)C \sqrt{\varepsilon(s)}.
		$$
		This proves the theorem with $C_2 = \frac{C}{\beta\mu}\left(\frac 1 \beta+c_1+c_2\right)$.
	\end{proof}

	Let us analyze the bound in Theorem~\ref{thm::mainEpsLarge}. The first term in \eqref{eq::ResLargeEps} decays exponentially fast and can even be zero if the initial speed is $\dot{x}_0=0$, the second-one decays like $\sqrt{\varepsilon(t)}$, however, the rate at which the last term decreases is less obvious. The following corollary gives a less-tight but easier-to-understand estimate.
	\begin{corollary}\label{cor::simpler}
		Consider the same assumptions and variables as in Theorem~\ref{thm::mainEpsLarge}. If furthermore $c_1<\frac{2}{\beta}$, then there exists $C_3>0$ such that for all $t\geq 0$,
		\begin{equation*
			\norm{x(t)-x_N(t)}\leq C_0 e^{-\frac{t}{\beta}}\varepsilon_0\norm{\dot{x}_0} + C_3\sqrt{\varepsilon(t)}.
		\end{equation*}
	\end{corollary}

	\begin{proof}[Proof of Corollary~\ref{cor::simpler}]
		For all $t\geq 0$, define $J(t) = \int_{0}^{t} e^{\frac{s}{\beta}} \sqrt{\varepsilon(s)} \diff s$. An integration by parts yields
		\begin{equation}\label{eq::boundJ}
			J(t)  = \left[\beta e^{\frac{s}{\beta}}\sqrt{\varepsilon(s)}\right]_{s=0}^t - \int_{s=0}^t \beta e^{\frac{s}{\beta}} \frac{\varepsilon'(s)}{2\sqrt{\varepsilon(s)}}\diff s = \beta e^{\frac{t}{\beta}}\sqrt{\varepsilon(t)} - \beta \varepsilon_0 + \int_{s=0}^t \beta e^{\frac{s}{\beta}} \frac{-\varepsilon'(s)}{2\varepsilon(s)}\sqrt{\varepsilon(s)}\diff s.
		\end{equation}
		By assumption, $0\leq c_1<\frac{2}{\beta}$ such that for all $s>0$, $\vert\varepsilon'(s)\vert \leq c_1\varepsilon(s)$, which in our setting is equivalent to $\frac{-\varepsilon'(s)}{\varepsilon(s)}\leq c_1$.
		So we deduce from \eqref{eq::boundJ} that
		\begin{equation*}
			J(t)  \leq  \beta e^{\frac{t}{\beta}}\sqrt{\varepsilon(t)} + c_1\frac{\beta}{2}\int_{s=0}^t e^{\frac{s}{\beta}} \sqrt{\varepsilon(s)}\diff s = \beta e^{\frac{t}{\beta}}\sqrt{\varepsilon(t)} + c_1\frac{\beta}{2}J(t).
		\end{equation*}
		So, $\left(1-c_1\frac \beta 2\right)J(t)  \leq   \beta e^t\sqrt{\varepsilon(t)}$. By assumption $1-c_1\frac \beta 2>0$, therefore, $J(t)\leq 	\frac{2}{2-c_1\beta} e^{\frac t \beta}\sqrt{\varepsilon(t)}$.
		Finally, using this in \eqref{eq::ResLargeEps} and setting $C_3 = C_1 + C_2\frac{2}{2-c_1\beta}$, we obtain the result.
	\end{proof}

	\begin{remark}
		The above proofs suggest that an extension to the case where $f$ is non-smooth but strongly convex is possible using regularization techniques. This is left for future work.
	\end{remark}

	So far our results only cover the case where $\alpha$ is ``not too large'' w.r.t.\ $\varepsilon$, and do not study \eqref{eq::LM}.
	We now state a more general result that covers these cases.

	\subsection{Generalization to Arbitrary Viscous Dampings with Sub-exponential Decay}
	This time we do not assume a link between $\varepsilon$ and $\alpha$ but only sub-exponential decays.
	\begin{assumption}\label{ass:Onlysubexp}
		There exists $c_1,c_2\geq 0$ such that for all $t\geq 0$, $\vert\varepsilon'(t)\vert\leq c_1 \varepsilon(t)$ and $\vert\alpha'(t)\vert \leq c_2\alpha(t)$.
	\end{assumption}
	We are now in position to state the main result of this section.
	\begin{theorem}\label{thm::mainGenResult}
		Let $x_N$ and $x_{LM}$ be the solution of \eqref{eq::CN} and \eqref{eq::LM} respectively, and let $c_1,c_2\geq 0$.
		There exist constants $C,\tilde{C}\geq 0$, depending on $c_1$, $c_2$, such that for all $\varepsilon$ and $\alpha$ for which Assumption~\ref{ass:Onlysubexp} holds with $c_1$ and $c_2$, the corresponding solution $x$ of \eqref{eq::VMDINAVD} is such that for all $t\geq 0$,
		\begin{equation}\label{eq::BoundXN}
			\norm{x(t)-x_N(t)}\leq C\left[ e^{-\frac{t}{\beta}} + \sqrt{\varepsilon(t)} +
			\alpha(t)
			+ \int_{s=0}^{t} e^{\frac{1}{\beta}(s-t)} (\sqrt{\varepsilon(s)}+\alpha(s))\diff s\right],
		\end{equation}
		and,
		\begin{equation}\label{eq::BoundXLM}
			\norm{x(t)-x_{LM}(t)}\leq \tilde{C}\left[ e^{-\frac{t}{\beta}} + \sqrt{\varepsilon(t)} +
			\alpha(t)
			+ \int_{s=0}^{t} e^{\frac{1}{\beta}(s-t)} (\sqrt{\varepsilon(s)}+\alpha(s))\diff s\right].
		\end{equation}
	\end{theorem}
	The proof is postponed to Appendix~\ref{app::GenThm}. Although it follows a similar reasoning as that of Theorem~\ref{thm::mainEpsLarge}, more involved estimates are needed.

	Let us comment on these results. The bound \eqref{eq::BoundXN} generalizes Theorem~\ref{thm::mainEpsLarge}, although the constant involved will, in general, be larger than those in \eqref{eq::ResLargeEps} (see the proof of Theorem~\ref{thm::mainGenResult} in appendix). Theorem~\ref{thm::mainGenResult} allows for far more flexibility in the choice of $\varepsilon$ and $\alpha$ in order to control $x$ and make it possibly close to $x_N$.
	The bound in \eqref{eq::BoundXLM} is the same as that in  \eqref{eq::BoundXN} (up to a constant), but this time w.r.t.\ $x_{LM}$, thus connecting \eqref{eq::VMDINAVD} to \eqref{eq::LM}. We make the following two important remarks. First \eqref{eq::BoundXLM} involves  $\alpha$, suggesting that making $x$ close to $x_{LM}$  requires not only $\varepsilon$ but also $\alpha$ to vanish asymptotically.
    Additionally, Theorem~\ref{thm::mainGenResult} does not state to which of $x_{N}$ and $x_{LM}$ the solution of \eqref{eq::VMDINAVD} is the closest.
    It remains an open question to know whether one can make \eqref{eq::BoundXLM} independent of $\alpha$, and to state to which trajectory $x$ is the closest. Yet, the numerical experiments in Section~\ref{sec::exp} suggest that neither are possible. Indeed, we observe that for some functions $f$, $x$ is \emph{sometimes} closer to $x_{N}$ than to $x_{LM}$, even when $\varepsilon(t)\leq \alpha(t)$.

    Nevertheless, Theorem~\ref{thm::mainGenResult} answers the question asked in the introduction: yes, \eqref{eq::VMDINAVD} is really of second-order nature since it can be brought close to the second-order dynamics \eqref{eq::CN} and \eqref{eq::LM}. Doing so, it benefits from the good properties of these methods, such as the robustness to bad conditioning, as previously illustrated on the right of Figure~\ref{fig::phasediagandquad}.
	This concludes the analysis from a control perspective. We will now derive an approximation of the solution $x$ in order to study the impact that $\varepsilon$ and $\alpha$ have on the speed of convergence of $x$ to $x^\star$ compared to the speeds of convergence of $x_N$ and $x_{LM}$.

	\section{Approximate Solutions and Asymptotic Analysis on Quadratic Functions}\label{sec::Quads}
	We consider the particular case where $f$ is a strongly-convex quadratic function in order to study the asymptotic behavior of \eqref{eq::VMDINAVD} w.r.t.\ \eqref{eq::CN} and \eqref{eq::LM}. Quadratic functions are the prototypical example of strongly-convex functions. In particular, any strongly-convex function can be locally approximated by a quadratic one around its minimizer, making the latter a good model for understanding the local behavior of dynamics.
	In this section, $f$ is quadratic: $f(y)=\frac{1}{2}\Vert Ay-b\Vert_2^2$ for all $y\in \mathbb{R}^n$, where $A\in\mathbb{R}^{n\times n}$ is symmetric positive definite and $b\in\mathbb{R}^n$. Without loss of generality, we take $b=0$, so that the unique minimum is $x^\star=0$.
	\subsection{Setting: the Special Case of Quadratic Functions}
	    Quadratic functions are particularly interesting in our setting since DIN-like ODEs take a simpler form (as observed in \cite{attouch2020first,shi2021understanding}). Indeed, $\forall y\in\mathbb{R}^n$, $\nabla f(y) = A^TAy$ and $\nabla^2 f(y)=A^T A$. Since $\nabla^2 f(y)$ is independent of $y$ we can rewrite \eqref{eq::VMDINAVD} in an eigenspace\footnote{This can be generalized to the case where $A^TA$ is only semi-definite by considering orthogonal projections on an eigenspace spanned by the positive eigenvalues of $A^TA$.} of  $A^TA$. That is, we can study the ODE coordinate-wise by looking at one-dimensional problems of the form
	\begin{equation}\label{eq::QuadeDINAVD}\tag{Q1-VM-DIN-AVD}
		\varepsilon(t)\ddot{x}(t)+(\alpha(t)+\beta\lambda)\dot{x}(t)+\lambda x(t)=0, \quad t\geq 0.
	\end{equation}
	Here (and throughout what follows) $\lambda>0$ denotes any eigenvalue of $A^TA$ and $x\colon\mathbb{R}_+\to\mathbb{R}$ now denotes the corresponding  coordinate (function) of the solution of \eqref{eq::VMDINAVD} in an eigenspace of $A^TA$. The dynamics \eqref{eq::QuadeDINAVD} is a \emph{linear} second-order ODE in $x$ with non-constant coefficients. Similarly, \eqref{eq::LM} can be rewritten coordinate-wise as
	\begin{equation}\label{eq::quadLM}\tag{Q1-LM}
		(\alpha(t)+\beta\lambda)\dot{x}_{LM}(t)+\lambda x_{LM}(t)=0, \quad t\geq 0,
	\end{equation}
	where $x_{LM}\colon\mathbb{R}_+\to\mathbb{R}$, and \eqref{eq::CN} becomes
	\begin{equation}\label{eq::quadCN}\tag{Q1-CN}
		\beta\dot{x}_N(t)+ x_N(t)=0, \quad t\geq 0,
	\end{equation}
	where again, $x_N\colon\mathbb{R}_+\to\mathbb{R}$ is one-dimensional.
	Observe in particular that \eqref{eq::CN} and \eqref{eq::LM} are now first-order \emph{linear} ODEs, whose solutions have the closed forms, for all $t\geq 0$,
	\begin{equation}\label{eq::closeformLMN}
		x_N(t) = x_0e^{-\frac{t}{\beta}}
		\quad \text{and}\quad
		x_{LM}(t) = x_0\exp\left(-\int_0^t \frac{\lambda}{\alpha(s)+\beta\lambda}\diff s \right).
	\end{equation}
	Since the minimizer is $x^\star=0$, we directly see that $x_N$ converges exponentially fast to $x^\star$, with a rate independent of $\lambda$. The rate of $x_{LM}$ depends however on $\lambda$ and how fast $\alpha$ vanishes.

	Unfortunately, except for some special choices of $\varepsilon$ and $\alpha$ (see \cite{attouch2020first}), one cannot solve the second-order linear ODE \eqref{eq::QuadeDINAVD} in closed form in general.
	Additionally, it is hopeless to circumvent the difficulty by finding a closed form for $\nabla f(x)$, accordingly to what we did in Section~\ref{sec::Control}, since here $\nabla f(x)=\lambda x$.
	In order to study the speed of convergence of $x$ despite not having access to a closed form, we will approximate it with a controlled error, via a method that we now present.

	\subsection{The Liouville--Green Method}
	In what follows, we rely on the Liouville--Green method~\citep{Liouville1836,green1838}, a technique for obtaining \emph{non-asymptotic} approximations to solutions of linear second-order ODEs with non-constant coefficients. First, we give the intuition behind the method, following the presentation of \citep{olver1997asymptotics}.
	Consider the differential equation
	\begin{equation}\label{eq::GenericODE}
		\ddot{z}(t)-r(t)z(t)=0,\quad t\geq 0,
	\end{equation}
	where $r$ is real-valued, positive, and twice continuously differentiable. Any linear second-order ODE can be reformulated in the form \eqref{eq::GenericODE}, see Lemma~\ref{lem::canonForm} below. Since for all $t\geq 0$, $r(t)\neq 0$, we can use the changes of variables $\tau = \int_0^t \sqrt{r(s)}\diff s$ and $w=r^{1/4}z$ and show that $w$ is solution to
	\begin{equation}\label{eq::GenericODE2}
		\ddot{w}(\tau)-(1+\psi(\tau))w(\tau)=0,\quad t\geq 0,
	\end{equation}
	where\footnote{We express $\psi(\tau)$ using $t$ for the sake of readability, using the one-to-one correspondence between $\tau$ and $t$.} $\psi(\tau)=\frac{4r(t)r''(t)-5r'(t)^2}{16r(t)^3}$. The LG method consists in neglecting the term $\psi(\tau)$ in \eqref{eq::GenericODE2}, which simply yields two approximate solutions $\hat{w}_1(\tau) = e^{\tau}$ and $\hat w_2(\tau)=e^{-\tau}$. Expressing this in terms of $z$ and $t$, we obtain
	\begin{equation}\label{eq::LG}
		\hat z_1(t) = r(t)^{-1/4}\exp\left(\int_0^t \sqrt{r(s)}\diff s\right) \quad \text{and}\quad \hat z_2(t) = r(t)^{-1/4}\exp\left(\int_0^t -\sqrt{r(s)}\diff s\right).
	\end{equation}
	Those are the LG approximations of the solutions of \eqref{eq::GenericODE}. They are formally valid on any interval $[0,T]$, $T> 0$ when $\psi$ is ``not too large'', provided that $\sqrt{r}$ is integrable on $[0,T]$.

	\begin{remark}
		There exists other (but less intuitive) ways to derive the approximations above, which allow for generalization to higher-order linear ODEs, see e.g., \citep[Chapter~10]{bender1999asymptotic}.
	\end{remark}
	The advantage of this approach is the possibility to estimate the error made using \eqref{eq::LG} w.r.t.\ the true solutions of \eqref{eq::GenericODE}. This is expressed in the following theorem which gathers results from \cite{blumenthal1899uber,olver1961error,taylor1982improved}.
	\begin{theorem}[\citet{olver1997asymptotics}]\label{thm::olver}
		Let $r\colon \mathbb{R}_{+}\to \mathbb{R}$ be a real, positive, twice continuously differentiable function, and define $\varphi(t) = \frac{4r(t)r''(t)-5r'(t)^2}{16r(t)^{5/2}}$ for all $t\geq 0$. Then for any $T>0$, the differential equation,
		\begin{equation}\label{eq::GenericODEthm}
			\ddot{z}(t)-r(t)z(t)=0,\quad t\in [0,T],
		\end{equation}
		has two real and twice continuously differentiable solutions defined for all $t\in[0,T]$ by,
		\begin{equation*}
			z_1(t) = \frac{1+\delta_1(t)}{r(t)^{1/4}}\exp\left(\int_0^t \sqrt{r(s)}\diff s\right) \quad\text{and} \quad
			z_2(t) = \frac{1+\delta_2(t)}{r(t)^{1/4}}\exp\left(-\int_0^t \sqrt{r(s)}\diff s\right),
		\end{equation*}
		where $\displaystyle\vert\delta_1(t)\vert\leq \exp\left(\frac{1}{2} \int_0^t \vert \varphi(s)\vert\diff s\right)-1$ and $\displaystyle\vert\delta_2(t)\vert\leq \exp\left(-\frac{1}{2} \int_t^{T} \vert \varphi(s)\vert\diff s\right)-1$.
		If in addition $\displaystyle\int_0^{+\infty} \vert\varphi(s)\vert\diff s <+\infty$, then the results above also hold for $T=+\infty$.
	\end{theorem}
	\begin{remark}\label{rem::olver} We make the following remarks regarding the above result.\\
		--\ Note that $z_1$ and $z_2$ in Theorem~\ref{thm::olver} are \emph{exact} solutions to \eqref{eq::GenericODEthm}. The LG approximations $\hat z_1$ and $\hat z_2$ are obtained by neglecting the unknown functions $\delta_1$ and $\delta_2$ in $z_1$ and $z_2$. The theorem gives a \emph{non-asymptotic} bound for the errors $\vert z_1(t)-\hat{z}_1(t)\vert$ and $\vert z_2(t)-\hat{z}_2(t)\vert$, $t\geq 0$.
	   \\
        --\ Since we assumed $r$ to be twice continuously differentiable and positive, $\varphi$ is continuous, so it is integrable except maybe for $t\to+\infty$.
		\\
		--\ For the sake of simplicity, the formulation of Theorem~\ref{thm::olver} slightly differs from that in \cite{olver1997asymptotics}, the original formulation can be recovered by a change of variable.
	\end{remark}

	%
	\subsection{Liouville--Green Approximation of \eqref{eq::VMDINAVD}}
	We now proceed to make use of the LG method for approximating the solutions of \eqref{eq::QuadeDINAVD}. The reader only interested in the result can jump directly to the  Section~\ref{sec::mainasymptoticres}. We first make the following assumption.
	\begin{assumption}\label{ass:smooth}
		The functions $\alpha$ and $\varepsilon$ are three times continuously differentiable, and $\varepsilon_0$ is such that $\forall t\geq 0$, $\varepsilon_0<\frac{(\beta\lambda)^2}{2\vert\alpha'(t)\vert+4\lambda}$.
	\end{assumption}
	\begin{remark}
		The condition on $\varepsilon_0$ in Assumption~\ref{ass:smooth} is only technical, so that $r$ defined below is positive.
		It can be easily satisfied since $\vert\alpha'(t)\vert$ is uniformly bounded. Indeed,  $\alpha$ is non-increasing and non-negative (see Section~\ref{sec::pbstatement}), from which one can deduce that $\int_{0}^{+\infty} \vert\alpha'(s)\vert\diff s\leq \alpha_0$.
	\end{remark}
	We now rewrite \eqref{eq::QuadeDINAVD} in the form \eqref{eq::GenericODEthm}.
	\begin{lemma}\label{lem::canonForm}
		Suppose that Assumption~\ref{ass:smooth} holds, and let $x$ be the solution of \eqref{eq::QuadeDINAVD}. For all $t\geq 0$, define
		\begin{equation}\label{eq::pr}
			p(t) = \frac{\alpha(t)+\beta\lambda}{\varepsilon(t)} \quad \text{and}  \quad r(t) = \frac{p(t)^2}{4} + \frac{p'(t)}{2} - \frac{\lambda}{\varepsilon(t)}.
		\end{equation}
		Then, $p$ and $r$ are twice continuously differentiable, $r$ is positive and the function $y$ defined for all $t\geq 0$ by $y(t) = x(t)\exp\left(\int_0^t \frac{p(s)}{2}\diff s\right)$ is a solution to
		\begin{equation}\label{eq::ODErewritten}
			\ddot y(t) - r(t)y(t) = 0,\quad t\geq 0
		\end{equation}
		with initial condition $(y(0),\dot{y}(0) )= (x_0,\dot{x}_0 + \frac{p(0)}{2}x_0)$.
	\end{lemma}
	\begin{proof}
		We first check that for all $t\geq 0$, $r(t)$ is positive. Let $t>0$,
		\begin{equation}\label{eq::rpositive}
			r(t)> 0 \iff \frac{(\alpha(t)+\beta\lambda)^2}{4\varepsilon(t)^2} + \frac{\alpha'(t)}{2\varepsilon(t)} -  \frac{(\alpha(t)+\beta\lambda)\varepsilon'(t)}{\varepsilon(t)^2} - \frac{\lambda}{\varepsilon(t)} >0.
		\end{equation}
		Since $\varepsilon'(t)\leq 0$ and $\alpha'(t)\leq 0$, one can check that a sufficient condition for \eqref{eq::rpositive} to hold is,
		\begin{equation*}
			r(t)> 0 \impliedby \frac{(\alpha(t)+\beta\lambda)^2}{4} > \left(\frac{\vert\alpha'(t)\vert}{2}+\lambda\right)\varepsilon(t) \impliedby \frac{(\beta\lambda)^2}{2\vert\alpha'(t)\vert+4\lambda}>\varepsilon_0.
		\end{equation*}
		So under Assumption~\ref{ass:smooth}, for all $t\geq 0$, $r(t)>0$. We then check that $y$ is indeed solution to \eqref{eq::ODErewritten}. Let $t>0$,
		\begin{equation*}
			\dot{y}(t) = \frac{p(t)}{2}x(t)\exp\left(\int_0^t \frac{p(s)}{2}\diff s\right) + \dot{x}(t)\exp\left(\int_0^t \frac{p(s)}{2}\diff s\right),
		\end{equation*}
		and
		\begin{equation*}
			\ddot{y}(t) = \exp\left(\int_0^t \frac{p(s)}{2}\diff s\right)\left[\left(\frac{p(t)^2}{4}+\frac{p'(t)}{2}\right)x(t) + p(t)\dot{x}(t) + \ddot{x}(t)\right].
		\end{equation*}
		Since $x$ solves \eqref{eq::QuadeDINAVD}, it holds that $\ddot{x}(t) = -p(t)\dot{x}(t) - \frac{\lambda}{\varepsilon(t)}x(t)$, so,
		\begin{multline*}
			\ddot{y}(t) = \exp\left(\int_0^t \frac{p(s)}{2}\diff s\right)\left(\frac{p(t)^2}{4}+\frac{p'(t)}{2}- \frac{\lambda}{\varepsilon(t)}\right)x(t) \\= \left(\frac{p(t)^2}{4}+\frac{p'(t)}{2}- \frac{\lambda}{\varepsilon(t)}\right)y(t)
			= r(t)y(t).
		\end{multline*}
	\end{proof}
	Lemma~\ref{lem::canonForm} gives a reformulation of \eqref{eq::QuadeDINAVD} suited to apply Theorem~\ref{thm::olver}.
	To use the theorem for all $t\geq 0$, we need to ensure that $\varphi(t)=\frac{4r(t)r''(t)-5r'(t)^2}{16r(t)^{5/2}}$ is integrable. To this aim we make the following assumption.
	\begin{assumption}\label{ass:subexp}
		The functions $\varepsilon$ and $\alpha$ have first, second and third-order derivatives that are integrable on $[0,+\infty[$. In addition, $\lim\limits_{t\to\infty}{\varepsilon(t)}=0$ and $\varepsilon'(t)^2/\varepsilon(t)$ is integrable on $[0,+\infty[$.
	\end{assumption}
	\begin{remark}
		Assumption~\ref{ass:subexp} holds for most decays used in practice, with in particular any polynomial decay of the form $\frac{\varepsilon_0}{(t+1)^a}$ and $\frac{\alpha_0}{(t+1)^b}$, $a\in\mathbb{N}\setminus \{0\}$ and $b\in\mathbb{N}$. Note that $\varepsilon$ and $\alpha$ need not be integrable and $\alpha$ can even be constant.
	\end{remark}
	The next lemma states the integrability of $\varphi$ on $[0,+\infty[$.
	\begin{lemma}\label{lem::phiintegrable}
		Under Assumption~\ref{ass:smooth} and~\ref{ass:subexp}, $\int_0^{+\infty} \vert \varphi(s)\vert\diff s<+\infty$.
	\end{lemma}
	The proof of this lemma, relies on relatively simple arguments but involves long computations and is thus postponed to Appendix~\ref{app::integrability}.
	We can now use Theorem~\ref{thm::olver} to obtain an exact form for the solution of \eqref{eq::QuadeDINAVD} based on the LG approximations.
	\begin{theorem}\label{thm::MainResLG}
		Suppose that Assumptions~\ref{ass:smooth} and~\ref{ass:subexp} hold. There exists $A,B\in\mathbb{R}$ such that $x(0)=x_0$, $\dot{x}(0)=\dot{x}_0$ and for all $t\geq 0$, the solution of \eqref{eq::QuadeDINAVD} is
		\begin{align}\label{eq::formofx}
			\begin{split}
				&x(t) = A\frac{1+\delta_1(t)}{r(t)^{1/4}}\frac{\sqrt{\alpha(t)+\beta\lambda}}{\sqrt{\varepsilon(t)}}\exp\left(\int_0^t -\frac{\lambda}{\alpha(s)+\beta\lambda} - \frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))\diff s\right)
				\\+ &B\frac{1+\delta_2(t)}{r(t)^{1/4}}\frac{\sqrt{\varepsilon(t)}}{\sqrt{\alpha(t)+\beta\lambda}}\exp\left(\int_0^t -\frac{\alpha(s)+\beta\lambda}{\varepsilon(s)}+\frac{\lambda}{\alpha(s)+\beta\lambda}+\frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))\diff s\right),
			\end{split}
		\end{align}
		where for all  $t\geq 0$,
		\begin{equation}\label{eq::deltabounds}
			\vert\delta_1(t)\vert \leq \exp\left(\frac{1}{2} \int_0^t \vert \varphi(s)\vert\diff s\right)-1
			\quad \text{and}\quad
			\vert\delta_2(t)\vert \leq \exp\left(-\frac{1}{2} \int_t^{+\infty} \vert \varphi(s)\vert\diff s\right)-1.
		\end{equation}
	\end{theorem}
	Thanks to the bounds \eqref{eq::deltabounds}, we now have an approximation of $x$.  We will use it in particular to compare $x$ asymptotically to the solutions of \eqref{eq::quadLM} and \eqref{eq::quadCN}.
	Before this, we prove Theorem~\ref{thm::MainResLG}.

	\begin{proof}[Proof of Theorem~\ref{thm::MainResLG}]
		Let $x$ be the solution of \eqref{eq::QuadeDINAVD} define $p,r$ as in \eqref{eq::pr}. Let us also define $y(t)\stackrel{\textrm{def}}{=}x(t)\exp\left(\int_0^t \frac{p(s)}{2}\diff s\right)$. According to Lemma~\ref{lem::canonForm}, $r$ is positive and $y$ is solution to \eqref{eq::ODErewritten}. Then, from Lemma~\ref{lem::phiintegrable}, $\int_t^{T} \vert \varphi(s)\vert\diff s<+\infty$, so we can apply Theorem~\ref{thm::olver} to $y$ on $[0,+\infty[$. Therefore, there exists $A,B\in\mathbb{R}$, such that $\forall t\geq 0$,
		$$
		y(t)= A\frac{1+\delta_1(t)}{r(t)^{1/4}}\exp\left(\int_0^t \sqrt{r(s)}\diff s\right) + B\frac{1+\delta_2(t)}{r(t)^{1/4}}\exp\left(\int_0^t -\sqrt{r(s)}\diff s\right),
		$$
		where $A$ and $B$ are determined by the initial conditions, and $\delta_1$, $\delta_2$ are such that \eqref{eq::deltabounds} holds.

		Going back to $x(t) = y(t)\exp\left(\int_0^t -\frac{p(s)}{2}\diff s\right)$, we obtain 	that for all $t\geq 0$,
		\begin{equation}\label{eq::formofx0}
			x(t) = A\frac{1+\delta_1(t)}{r(t)^{1/4}}\exp\left(\int_0^t -\frac{p(s)}{2} +\sqrt{r(s)}\diff s\right) + B\frac{1+\delta_2(t)}{r(t)^{1/4}}\exp\left(\int_0^t -\frac{p(s)}{2} -\sqrt{r(s)}\diff s\right).
		\end{equation}
		It now remains to expand the terms in the two exponentials in \eqref{eq::formofx0} in order to obtain \eqref{eq::formofx}. To this aim, we approximate $\sqrt{r(s)}$, let $s\geq 0$,
		\begin{align}\label{eq::sqrtr}
			\begin{split}
				\sqrt{r(s)}&=\frac{p(s)}{2}\sqrt{1+ \frac{2p'(s)}{p(s)^2} - \frac{4\lambda}{\varepsilon(s)p(s)^2}}
				\\&= \frac{p(s)}{2}\left(1 + \frac{p'(s)}{p(s)^2} - \frac{2\lambda}{\varepsilon(s)p(s)^2} - \frac{1}{8}\left(\frac{2p'(s)}{p(s)^2} - \frac{4\lambda}{\varepsilon(s)p(s)^2}\right)^2 + o(\varepsilon(s)^2) \right)
				\\ & = \frac{p(s)}{2} +\frac{ p'(s)}{2p(s)} - \frac{\lambda}{\varepsilon(s)p(s)} - \frac{1}{16}\left(\frac{2p'(s)}{p(s)^{3/2}} - \frac{4\lambda}{\varepsilon(s)p(s)^{3/2}}\right)^2 + o(\varepsilon(s))
				\\ & = \frac{p(s)}{2} + \frac{p'(s)\varepsilon(s)}{2(\alpha(s)+\beta\lambda)} - \frac{\lambda}{\alpha(s)+\beta\lambda} - \frac{1}{16}\left(\frac{2p'(s)}{p(s)^{3/2}} - \frac{4\lambda\sqrt{\varepsilon(s)}}{(\alpha(s)+\beta\lambda)^{3/2}}\right)^2 + o(\varepsilon(s))
				\\  = \frac{p(s)}{2} &+ \frac{\alpha'(s)}{2(\alpha(s)+\beta\lambda)} -\frac{\varepsilon'(s)}{2\varepsilon(s)} -\frac{\lambda}{\alpha(s)+\beta\lambda} - \frac{1}{16}\left(\frac{2p'(s)}{p(s)^{3/2}} - \frac{4\lambda\sqrt{\varepsilon(s)}}{(\alpha(s)+\beta\lambda)^{3/2}}\right)^2 + o(\varepsilon(s))
			\end{split}
		\end{align}
		Focusing on the first exponential term in \eqref{eq::formofx0}, we deduce from \eqref{eq::sqrtr} that for all $t\geq 0$,
		\begin{align*
			\begin{split}
				&\exp\left(\int_0^t -\frac{p(s)}{2} +\sqrt{r(s)}\diff s\right)
				\\&= \exp\left(\int_0^t \frac{\alpha'(s)/2}{\alpha(s)+\beta\lambda} -\frac{\varepsilon'(s)}{2\varepsilon(s)} -\frac{\lambda}{\alpha(s)+\beta\lambda} - \frac{1}{16}\left(\frac{2p'(s)}{p(s)^{3/2}} - \frac{4\lambda\sqrt{\varepsilon(s)}}{(\alpha(s)+\beta\lambda)^{3/2}}\right)^2 + o(\varepsilon(s))\diff s\right)
				\\&=  \frac{\sqrt{\alpha(t)+\beta\lambda})}{\sqrt{\alpha_0+\beta\lambda}} \frac{\sqrt{\varepsilon_0}}{\sqrt{\varepsilon(t)}} \exp\left(
				\int_0^t \frac{-\lambda}{\alpha(s)+\beta\lambda} -
				\frac{1}{16}\left(\frac{2p'(s)}{p(s)^{3/2}} - \frac{4\lambda\sqrt{\varepsilon(s)}}{(\alpha(s)+\beta\lambda)^{3/2}}\right)^2 + o(\varepsilon(s))\diff s\right)
				\\&=  \frac{\sqrt{\alpha(t)+\beta\lambda})}{\sqrt{\alpha_0+\beta\lambda}} \frac{\sqrt{\varepsilon_0}}{\sqrt{\varepsilon(t)}} \exp\left(
				\int_0^t \frac{-\lambda}{\alpha(s)+\beta\lambda} -
				\frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}} + o(\varepsilon(s))\diff s\right),
			\end{split}
		\end{align*}
		where the last line relies on further computations postponed to Lemma~\ref{lem::remainderterm} in Appendix~\ref{app::integrability}. Performing the exact same type of computations on $\exp\left(\int_0^t -\frac{p(s)}{2} -\sqrt{r(s)}\diff s\right)$, and up to redefining $A$ and $B$ so as to encompass all the constants, we obtain \eqref{eq::formofx} and the result is proved.
	\end{proof}

	\subsection{Comparison of \texorpdfstring{$x$}{x} with \texorpdfstring{$x_{LM}$}{xLM} and \texorpdfstring{$x_N$}{xN}}\label{sec::mainasymptoticres}

	We now have an expression for $x$ which is almost explicit: we do not know $\delta_1$ and $\delta_2$ in closed form, but they are uniformly bounded (by Lemma~\ref{lem::phiintegrable}). We will now  compare the asymptotic behavior of \eqref{eq::formofx} with those of the solutions of \eqref{eq::quadLM} and \eqref{eq::quadCN} that we denoted $x_{LM}$ and $x_N$ respectively.
    Our main result of Section~\ref{sec::Quads} is the following, where $\sim_{+\infty}$ denotes the asymptotic equivalence\footnote{Two real-valued functions $g_1$ and $g_2$ are asymptotically equivalent in $+\infty$ if and only if $\lim_{t\to\infty}\frac{g_1(t)}{g_2(t)}=1$.} between two functions as $t\to\infty$.
	\begin{theorem}\label{thm::MainResAsymptotic}
		Let $x$ be the solution of \eqref{eq::QuadeDINAVD}, given in \eqref{eq::formofx}, and $x_{LM}$ and $x_N$ whose closed forms are stated in \eqref{eq::closeformLMN}. Under Assumptions~\ref{ass:smooth} and~\ref{ass:subexp}, there exists $C>0$ such that the following asymptotic equivalences hold:
		\begin{align}\label{eq::equivx}
        \begin{split}
			x(t) &\sim_{+\infty}
			x_{LM}(t)C\exp\left(\int_0^t - \frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))\diff s\right),\quad \text{and}
			\\x(t)&\sim_{+\infty}
			x_{N}(t)C\exp\left(\int_0^t \frac{\alpha(s)}{\beta(\alpha(s)+\beta\lambda)}- \frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))\diff s\right).
        \end{split}
		\end{align}
		As a consequence, the convergence of $x$ to $x^\star$ is:
		\vspace{-0.2cm
		\begin{enumerate}[label=(\roman*)]
			\itemsep0em
			\item \emph{Faster} than that of $x_{LM}$
			if $\varepsilon$ is non-integrable and as fast otherwise.
			\item Slower than that of $x_{N}$ if $\alpha$ is non-integrable and as fast if $\alpha$ is integrable, in the case where $\forall t\geq 0$, $\alpha(t)>\varepsilon(t)$.
			\item \emph{Faster} than that of $x_{N}$ if $\varepsilon$ is non-integrable and as fast if $\varepsilon$ is integrable, in the case where $\forall t\geq 0$, $\alpha(t)< \varepsilon(t)$.
		\end{enumerate}
	\end{theorem}

	While the results of Section~\ref{sec::Control} were related to the closeness of \eqref{eq::VMDINAVD} w.r.t.\ \eqref{eq::CN} and \eqref{eq::LM} from a control perspective, Theorem~\ref{thm::MainResAsymptotic} provides a different type of insight. First, the results are asymptotic, so they only allow to control \eqref{eq::VMDINAVD} for large $t$. They provide however a clear understanding of the nature of the solutions of \eqref{eq::VMDINAVD} and their convergence. The conclusions (summarized in Table~\ref{tabl::speedsummary}) are in accordance with what we would expect: when the viscous damping is larger than the variable mass,
	\eqref{eq::VMDINAVD} behaves more like the Levenberg--Marquardt method than the Newton one, but it actually becomes an accelerated Levenberg--Marquardt dynamics when $\varepsilon$ is non-integrable but vanishing. However, when the variable mass $\varepsilon$ is larger than $\alpha$, the dynamics is closer to the one of the Newton method, and can actually be an accelerated Newton dynamics, again for non-integrable $\varepsilon$. This is analogous to the necessary condition that $\alpha$ must be non-integrable in order to accelerate first-order methods in convex optimization (see \cite{attouch2017asymptotic}). We conclude this section by proving Theorem~\ref{thm::MainResAsymptotic}.

	\begin{proof}[Proof of Theorem~\ref{thm::MainResAsymptotic}]
		Thanks to Assumptions~\ref{ass:smooth} and~\ref{ass:subexp}, Theorem~\ref{thm::MainResLG} tells us that $x$ has the form \eqref{eq::formofx}. We now analyze the two terms in \eqref{eq::formofx}.

		First, we know from Theorem~\ref{thm::MainResLG} that $\delta_1(0)=0$ and $\lim_{t\to+\infty}\delta_2(t)=0$. In addition, by Lemma~\ref{lem::phiintegrable}, $\delta_1$ and $\delta_2$ are uniformly bounded by some positive constant.
		Then $r(t)^{-1/4}$ decays asymptotically like $\sqrt{\varepsilon(t)}$ and $\alpha$ is bounded. So
		$A\frac{1+\delta_1(t)}{r(t)^{1/4}}\frac{\sqrt{\alpha(t)+\beta\lambda}}{\sqrt{\varepsilon(t)}}$ is asymptotically equivalent to some constant $c_1\in\mathbb{R}$ as $t\to +\infty$. Similarly, $B\frac{1+\delta_2(t)}{r(t)^{1/4}}\frac{\sqrt{\varepsilon(t)}}{\sqrt{\alpha(t)+\beta\lambda}}$ is equivalent to $c_2\varepsilon(t)$, with $c_2\in\mathbb{R}$.

		We now analyze the ``exponential factors'' in \eqref{eq::formofx}.
		On the one hand, $\frac{\lambda}{\alpha(s)+\beta\lambda}+\frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))$ converges to $\frac{1}{\beta}$ as $s\to \infty$, while $\frac{\alpha(s)+\beta\lambda}{\varepsilon(s)}$ diverges to $+\infty$. Therefore, we deduce that,
		\begin{multline*}
			\exp\left(\int_0^t -\frac{\alpha(s)+\beta\lambda}{\varepsilon(s)}+\frac{\lambda}{\alpha(s)+\beta\lambda}+\frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))\diff s\right)
			\\= o\left(\exp\left(\int_0^t -\frac{\lambda}{\alpha(s)+\beta\lambda} - \frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))\diff s\right)\right).
		\end{multline*}
		As a consequence, the second term in \eqref{eq::formofx} will decrease to $0$ faster than the first-one (let alone the additional $\varepsilon(t)$ decay that we have just discussed). The asymptotic behavior of $x$ will thus be governed by the first term in \eqref{eq::formofx}.

		Let us now focus on the first term in \eqref{eq::formofx}. Observe that
		$\exp\left(\int_0^t -\frac{\lambda}{\alpha(s)+\beta\lambda}\diff s\right)$ is exactly the decay of $x_{LM}$ in \eqref{eq::closeformLMN}. Thus, we have proved that there exists $C>0$, such that the following asymptotic equivalence holds,
		\begin{multline*
			A\frac{1+\delta_1(t)}{r(t)^{1/4}}\frac{\sqrt{\alpha(t)+\beta\lambda}}{\sqrt{\varepsilon(t)}}\exp\left(\int_0^t -\frac{\lambda}{\alpha(s)+\beta\lambda} - \frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))\diff s\right)
			\\ \sim_{+\infty}
			x_{LM}(t)C\exp\left(\int_0^t - \frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))\diff s\right),
		\end{multline*}
		which proves the first part of \eqref{eq::equivx}. The second equivalence in \eqref{eq::equivx} is obtained using the following identity,
		\begin{equation}\label{eq::rewritteLM}
			\int_0^t -\frac{\lambda}{\alpha(s)+\beta\lambda}\diff s = \int_0^t -\frac{1}{\beta} + \frac{\alpha(s)}{\beta(\alpha(s)+\beta\lambda)}\diff s = -\frac{t}{\beta} + \int_0^t  \frac{\alpha(s)}{\beta(\alpha(s)+\beta\lambda)}\diff s
		\end{equation}
		and $e^{-t/\beta}$ is precisely the rate at which $x_N$ decreases. So \eqref{eq::equivx} holds.

		It finally remains to deduce the conclusions of the theorem from \eqref{eq::equivx}.
		\\--\ Regarding the comparison with $x_{LM}$, the integral $\int_0^t - \frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))\diff s$ converges if and only if $\varepsilon$ is integrable on $[0,+\infty[$, and diverges to $-\infty$ when $\varepsilon$ is not. So $x$ converges to $0$ at least as fast as $x_{LM}$ and faster when $\varepsilon$ is not integrable.
		\\-- As for the comparison with $x_N$, if $\alpha(s)>\varepsilon(s)\geq 0$ for all $s\geq 0$, then the integral $\int_0^t \frac{\alpha(s)}{\beta(\alpha(s)+\beta\lambda)}- \frac{\lambda^2\varepsilon(s)}{(\alpha(s)+\beta\lambda)^{3}}+o(\varepsilon(s))\diff s$ is convergent in $+\infty$ if and only if $\alpha$ is integrable and diverges to $+\infty$ when $\alpha$ is non-integrable. So when $\alpha$ is integrable, the speed of convergence of $x$ is the same as that of $x_N$. When $\alpha$ is not integrable, the convergence to $0$ is slower but still holds. Indeed, for all $s\geq 0$
		$\frac{\alpha(s)}{\beta(\alpha(s)+\frac{1}{\beta})}< \frac{\alpha(s)}{\beta\alpha(s)}= \frac{1}{\beta}$. Thus for all $t>0$, $-\frac{t}{\beta} + \int_0^t \frac{\alpha(s)}{\beta(\alpha(s)+\beta\lambda)}\diff s<0$.
		\\-- Finally, the comparison with $x_N$ in the case $\varepsilon(s)> \alpha(s)$ is exactly the same as the comparison with $x_{LM}$ using \eqref{eq::rewritteLM}.
	\end{proof}

	\section{Numerical Experiments}\label{sec::exp}
	\begin{figure}[htb]
		\begin{minipage}{\linewidth}
			\centering
			\includegraphics[width=0.9\linewidth]{Figures/fig1_brent_alphanoninteg}
			\\
			\includegraphics[width=0.9\linewidth]{Figures/fig1_brent_alphainteg}
		\end{minipage}
		\caption{\label{fig::expmt}
			Comparison of the solutions $x_N$, $x_{LM}$ and $x$ of \eqref{eq::CN}, \eqref{eq::LM} and \eqref{eq::VMDINAVD} respectively, for a strongly-convex function of the form $f(x)=e^{-\Vert x\Vert^2} + \frac 1 2 \norm{Ax}^2 $.
			Left figures: distance $\norm{x(t)-x_N(t)}$ versus time $t$, each curve corresponds to a different choice of $\varepsilon$; middle figures: distance $\norm{x(t)-x_{LM}(t)}$, again for several $\varepsilon$. Right figures: distance to the optimum $x^\star$ for reference, $x_N$ and $x_{LM}$ are in dotted and dashed lines, other curves correspond to \eqref{eq::VMDINAVD} for several choices of $\varepsilon$. The brown curve is often hidden behind the purple (and sometimes the pink) curve.
			Top and bottom rows show results respectively for non-integrable and integrable viscous dampings $\alpha$.
			The theoretical bounds from Theorem~\ref{thm::mainGenResult} are only displayed on Figure~\ref{fig::LP} below, for the sake of readability.}
	\end{figure}

	\begin{figure}[htb]
		\begin{minipage}{\linewidth}
			\centering
			\includegraphics[width=0.9\linewidth]{Figures/fig1_logsumexpdouble_alphanoninteg}
			\includegraphics[width=0.9\linewidth]{Figures/fig1_logsumexpdouble_alphainteg}
		\end{minipage}
		\caption{\label{fig::logsumexpexp} Similar experiment and figures as those described in Figure~\ref{fig::expmt}, but for the function $f(x)=\log\left(\sum_{i=1}^n e^{x_i} + e^{-x_i}\right) + \frac 1 2 \norm{Ax}^2 $.}
	\end{figure}

	\begin{figure}[htb]
		\begin{minipage}{\linewidth}
			\centering
			\includegraphics[width=0.9\linewidth]{Figures/fig1_LP_alphanoninteg}
			\\
			\includegraphics[width=0.9\linewidth]{Figures/fig1_LP_alphainteg}
		\end{minipage}
		\caption{\label{fig::LP} Similar experiment and figures as those described in Figure~\ref{fig::expmt}, but for the function $f(x)=\norm{x}^{50} + \frac 1 2 \norm{Ax}^2 $.
		The thin ``dash dotted'' curves represent the theoretical bounds from Theorem~\ref{thm::mainGenResult} for each choice of $(\varepsilon,\alpha)$ considered.}
	\end{figure}

	\begin{figure}[htb]
		\centering
		\includegraphics[width=0.45\linewidth]{Figures/speed_vrt_Newton} \hspace{0.1cm}\includegraphics[width=0.45\linewidth]{Figures/speed_vrt_LM}
		\caption{\label{fig::quadspeed} Numerical validation of Theorem~\ref{thm::MainResAsymptotic}: distance to the optimum $x^\star$ as a function of time. on a quadratic function $f(x)=\frac{1}{2}\norm{Ax}^2$. Left: speed comparison w.r.t.\ \eqref{eq::CN} for several choices of $\varepsilon$ and $\alpha$. Right: Comparison with LM in for $\alpha$ integrable or not and several choices of $\varepsilon$. Shades of blue represent cases where $\varepsilon(t)>\alpha(t)$ while shades of red represent the opposite setting.}
	\end{figure}
	We present two set of experiments that illustrate our main results from Sections~\ref{sec::Control} and~\ref{sec::Quads}. We first detail the general methodology.
	\subsection{Methodology}
	We compare the solutions of \eqref{eq::CN}, \eqref{eq::LM} and \eqref{eq::VMDINAVD} obtained for strongly-convex functions in dimension $n=100$. Since closed-form solutions are not available, they are estimated via discretization schemes with small step-sizes $\gamma=10^{-1}$. We used Euler semi-explicit schemes, where a linear system is solved at each iteration, for the sake of stability.
	The resulting algorithms are detailed in Appendix~\ref{app:Exp}.
	\subsection{First Experiment: Distance between Trajectories}
	We begin with an empirical validation of the results of Section~\ref{sec::Control} on the distance between $x$, $x_{LM}$ and $x_N$.
	Each of Figures~\ref{fig::expmt}, \ref{fig::logsumexpexp} and~\ref{fig::LP} (as well as Figures~\ref{fig::quadANDlogsumexp} in Appendix~\ref{app:Exp}) corresponds to a different strongly-convex function, specified below its corresponding figure. In order to ensure strong convexity, each function contains a quadratic term of the form $\Vert Ax\Vert^2$, where $A$ is symmetric positive definite.

	Several observations can be made from the numerical results, but we first note on the right plots of Figures~\ref{fig::expmt} to~\ref{fig::LP} that $x_N$ always converges asymptotically linearly (i.e., exponentially fast). This is also the case for $x$ and $x_{LM}$ in some (but not all) cases. This is important because $\norm{x(t)-x_N(t)}\leq \norm{x(t)-x^\star} + \norm{x_N(t)-x^\star}$, so if both $x$ and $x_N$ converge linearly, then the bounds of Theorems~\ref{thm::mainEpsLarge} and~\ref{thm::mainGenResult} need not be tight asymptotically. That being said, the strength of these results is to be non-asymptotic and this is highlighted by the experiments as we now explain.

	Looking at the left and middle plots of Figures~\ref{fig::expmt}, \ref{fig::logsumexpexp} and~\ref{fig::LP}, we observe that Theorems~\ref{thm::mainEpsLarge} and~\ref{thm::mainGenResult} seem empirically validated, since the distances $\norm{x(t)-x_{N}(t)}$ and $\norm{x(t)-x_{LM}(t)}$ decrease relatively fast to zero. Again, when $x$ converges rapidly to $x^\star$ this is not very insightful, however, the main interest of our theorems appears on the left of Figures~\ref{fig::logsumexpexp} and~\ref{fig::LP}: the blue and green curves, corresponding to slowly decaying choices of $\varepsilon$, converge more slowly than other trajectories. However, when taking faster decays, we recover fast convergence and closeness to $x_N$ (this is particularly true for the purple curve).
	Very similar observations are made w.r.t.\ $x_{LM}$ on the middle plots. Despite not being stated in the theorems of Section~\ref{sec::Control}, the experiments match the intuition that when $\varepsilon> \alpha$, $x$ may be closer to $x_N$ and when $\varepsilon\leq\alpha$, $x$ would rather be closer to $x_{LM}$. This is more noticeable on the top rows of the figures, where $\alpha$ is not integrable.

	Figure~\ref{fig::LP} suggests that the bounds in Theorem~\ref{thm::mainGenResult} are rather tight for small $t$, since, for example, the blue and green curves on the left show a relatively slow vanishing of $\norm{x(t)-x_N(t)}$ for slowly decaying $\varepsilon$. The experiments show that the bounds seem however often too pessimistic for large $t$, for which the second part of our study provides better insights (see Section~\ref{sec::Quads} and below).
	Interestingly, slow decays of $\varepsilon$ may sometimes result in faster convergence for $x$ than fast decays (and also faster convergence than $x_{LM}$), notably on Figure~\ref{fig::expmt}. We also note that $\varepsilon(t)=\varepsilon_0/t$ combined either with $\alpha(t)=\alpha_0/t$ or $\alpha(t)=\alpha_0/t^2$ seems to always yield fast convergence on these experiments (and sometimes the fastest of all dynamics).

	\subsection{Second Experiment: Empirical Validation of Theorem~\ref{thm::MainResAsymptotic}}
	We now turn our attention to the solutions $x$, $x_N$ and $x_{LM}$ for a quadratic function of the form $f(y)=\frac{1}{2}\Vert Ay\Vert^2$, $y\in\mathbb{R}^n$, and for several choices of $\varepsilon$ and $\alpha$. The results in Figure~\ref{fig::quadspeed} exactly match  the expected behavior summarized in Table~\ref{tabl::speedsummary}. Indeed, looking first at the right-hand side of Figure~\ref{fig::quadspeed}, $x$ is as fast as the corresponding\footnote{That is, the solution of \eqref{eq::LM} for the same $\alpha$ as that considered for \eqref{eq::VMDINAVD}.} $x_{LM}$ when $\varepsilon$ is not integrable and regardless of $\alpha$, and $x$ is faster when $\varepsilon$ is non-integrable. Then on the left-hand side, when comparing to $x_N$, $x$ is slower in settings where $\alpha$ is larger than $\varepsilon$ and non-integrable (red curves), or almost as fast when $\alpha$ is integrable (pink curve). However, acceleration w.r.t.\ to $x_N$ is indeed achieved for non-integrable $\varepsilon$ regardless of $\alpha$ (first-two blue curves), and the rate is the same as that of $x_N$ when $\varepsilon$ is integrable (third blue curve).

	\section{Conclusions and Perspectives}
	We introduced a general ODE \eqref{eq::VMDINAVD} featuring variable mass, and provided a deep understanding on the behavior of its solutions w.r.t.\ time dependent control parameters $\varepsilon$ and $\alpha$, both, asymptotically and non-asymptotically. We can conclude that \eqref{eq::VMDINAVD} is indeed of  (regularized) Newton type, since it can be controlled to be  close to both \eqref{eq::CN} and \eqref{eq::LM}. Yet we also showed that \eqref{eq::VMDINAVD} fundamentally differs from the other two dynamics in its nature. In particular, Theorem~\ref{thm::MainResAsymptotic} and the numerical experiments emphasized that $\varepsilon$ and $\alpha$ can accelerate (or slow down) \eqref{eq::VMDINAVD} w.r.t.\ \eqref{eq::CN} and \eqref{eq::LM}. We also note that our bounds in Theorems~\ref{thm::mainEpsLarge} and~\ref{thm::mainGenResult} seem relatively tight, in particular for functions with large gradients (see Figure~\ref{fig::LP}).
    Our contribution yields a complete and satisfying picture on the relation between the three systems, which was only partially understood. We believe that our results build a strong foundation for the development of algorithms that combine the best properties of first- and second-order optimization methods.

	As for future work, we showed that \eqref{eq::VMDINAVD} is promising from an optimization perspective. So far we approximated solutions of \eqref{eq::VMDINAVD} via schemes that required solving a linear system at each iteration (this is also true for \eqref{eq::CN} and \eqref{eq::LM}). Our new understanding on $(\varepsilon,\alpha)$ paves the way towards designing new Newton-like algorithms with a significantly reduced computational cost, which is crucial for large-scale optimization. Another open question is whether it is possible to preserve the properties evidenced in this work when $\varepsilon$ is defined in a closed-loop manner (formally depending on $x$ rather than on $t$). Finally, it would be worth investigating how the current work can be extended to general convex and/or non-smooth functions.

	\FloatBarrier

	\section*{Acknowledgment}
	C. Castera, J. Fadili and P. Ochs are supported by the ANR-DFG joint project TRINOM-DS under number ANR-20-CE92-0037-01. The numerical experiments were made thanks to the development teams of the following libraries: Python \citep{rossum1995python}, Numpy \citep{walt2011numpy} and
	Matplotlib \citep{hunter2007matplotlib}.

	\newpage

redpajama_set_name: RedPajamaArXiv


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A. Sannino, A. Maffezzoli, L. Nicolais. "Introduction of molecular spacers between the crosslinks of a cellulose based superabsorbent hydrogel: effects on the equilibrium sorption properties." J Appl Polym Sci 90: 168-174, 2003.
F. Lionetto, A. Sannino, G. Mensitieri, A. Maffezzoli. "Evaluation of the degree of crosslinking of cellulose based superabsorbent hydrogels: a comparison between different techniques." Macromol Symp 200: 199-207, 2003.
A. Sannino, A. Esposito, A. De Rosa, A. Cozzolino, L. Ambrosio and L. Nicolais. "Biomedical application of a superabsorbent hydrogel for body water elimination in the treatment of oedemas." J Biomed Mat Res 67A: 1016-1024, 2003.
A. Sannino, P.A. Netti, G. Mensitieri and L. Nicolais. "Designing Microporous Macromolecular Hydrogels for Biomedical Applications: a Comparison between Two Techniques." Comp Sci Technol 63: 2411-2416, 2003.
A. Sannino, M. Madaghiele, F. Conversano, A. Maffezzoli, P.A. Netti, L. Ambrosio and L. Nicolais. "oupling cellulose derivatives to hyaluronic acid based microporous hydrogel to modulating equilibrium sorption capacity and network stability." C Biomacromolecules 5(1): 92-96, 2004 – TOP TEN CITED JOURNAL ARTICLES YEAR 2004.
A. Sannino, G. Mensitieri and L. Nicolais."Water and Synthetic Urine Sorption Capacity of Cellulose Based Hydrogels Under Compressive Stress Field." J Appl Polym Sci 91(6): 3791-3796, 2004.
A. Sannino, X. Xuejun, L. Ambrosio, P.A. Netti, L. Nicolais. "Preparation and Characteristics of Low Molecular Weight HA Hydrogel." J Bioactive Compatible Polym 19: 5-15, 2004.
A. Esposito, A. Sannino, A. Cozzolino, S. Nappo Quintiliano, A. De Rosa, L. Ambrosio and L. Nicolais. "A Cellulose-PEG based polymer as a new biomaterial for biomedical application in the treatment of intractable oedemas." Biomaterials 26/19: 4101-4110, 2005.
F. Lionetto, A. Sannino and A. Maffezzoli. "Ultrasonic monitoring of the network formation in superabsorbent cellulose based hydrogels." Polymer 46: 1796-1803, 2005.
A. Sannino and L. Nicolais. "Concurrent effect of microporosity and chemical structure on the equilibrium sorption properties of cellulose-based hydrogels." Polymer 46(13): 4676-4685, 2005.
A. Sannino, S. Pappadà, M. Madaghiele, A. Maffezzoli, L. Ambrosio and L. Nicolais. "Crosslinking of cellulose derivatives and hyaluronic acid with water soluble carbodiimide." Polymer 46: 11206-11212, 2005.
A. Sannino, P. Pulito, G. Mele, G. Marcì, L. Palmisano and A. Maffezzoli. "Photocatalytic TiO2 based processes used for the purification of washing waters deriving from the preparation of superabsorbent hydrogels." Green Chem 8: 439 – 444, 2006 – 'BEST COVER' YEAR 2006.
A. Sannino, P.A. Netti, M. Madaghiele, V. Coccoli, A. Luciani, A. Maffezzoli and L. Nicolais. "Synthesis and characterization of macroporous poly(ethylene glycol)-based hydrogels for tissue engineering application." J Biomed Mater Res 79A: 229-236, 2006.
A. Sannino, M. Madaghiele, G. Lionetto, A. Maffezzoli and L. Nicolais. "A cellulose-based hydrogel as potential bulking agent for hypocaloric diets: an in vitro biocompatibility study on rat intestine." J Appl Polym Sci 102: 1524-1530, 2006.
A. Sannino, S. Pappadà, L. Valli and A. Maffezzoli. "Spin coating of cellulose derivatives on QCM to obtain hydrogel based sensors and actuators." J Appl Polym Sci 106: 3040-3050, 2007.
C. Demitri, L. Mercurio, F. Montagna, A. Sannino, A. Maffezzoli. "Acrylic-based hydrogel phantom for in vitro ultrasound contrast agent characterization." Virtual and Physical Prototyping 2(4): 191 – 196, 2007.
C. Demitri, A. Sannino, F. Conversano, S. Casciaro, A. Distante, A. Maffezzoli. "Hydrogel based tissue mimicking phantom for in vitro ultrasound contrast agents studies." J Biomed Mater Res B Appl Biomater 87(2): 338-345, 2008.
C. Demitri, R. Del Sole, F. Scalera, A. Sannino, G. Vasapollo, A. Maffezzoli. "Novel superabsorbent cellulose-based hydrogels crosslinked with citric acid." J Appl Polym Sci 110: 2453-2460, 2008.
M. Pollini, A. Sannino, A. Maffezzoli, A. Licciulli, and L. Nicolais. "Non-supercritical synthesis of microporous gels." J Appl Polym Sci 110: 2563-2568, 2008.
S. Casciaro, F. Conversano, S. Musio, E. Casciaro, C. Demitri, A. Sannino. "Full experimental modelling of a liver tissue mimicking phantom for medical ultrasound studies employing different hydrogels." J Mater Sci Mater Med 20(4): 983-989, 2009.
A. Sannino, C. Demitri, M. Madaghiele. "Biodegradable cellulose-based hydrogels: design and applications." Materials 2(2): 353-373, 2009.
M. Madaghiele, A. Piccinno, M. Saponaro, A. Maffezzoli, A. Sannino. "Collagen- and gelatine-based films sealing vascular prostheses: evaluation of the degree of crosslinking for optimal blood impermeability." J Mater Sci Mater Med 20: 1979-1989, 2009.
A. Sannino, M. Madaghiele, C. Demitri, F. Scalera, A. Esposito, V. Esposito, A. Maffezzoli. "Development and characterization of cellulose-based hydrogels for use as dietary bulking agents." J Appl Polym Sci. Published online October 7, 2009.


text:
Pete the Ramp Guy
Community December 12, 2019 Amanda Creasey
Local handyman builds wheelchair ramps to improve neighbors' lives In the summer of 2005, Pete Adler attended a meeting of the Chester Rotary Club....
Local handyman builds wheelchair ramps to improve neighbors' lives
In the summer of 2005, Pete Adler attended a meeting of the Chester Rotary Club. A guest speaker from Project:HOMES attended to explain the group's program and to recruit volunteers to build wheelchair ramps. The mission of Project:HOMES, as Adler explains it, is to "help keep people in their home as long as possible by providing accessibility through wheelchair ramps and other home improvements." The Rotary Club in Chester "embraced the idea as an avenue of community support," Adler said. In the beginning, the group volunteered "at the ElderHomes facility to prefabricate modules used by other groups to build ramps." The first ramp Adler's group built itself wasn't for a house at all, but "on a trailer that we used as a float in the Chester Christmas Parade."
The following spring, the group constructed its first ramp on a home in Colonial Heights. Since then, Adler has helped build 186 ramps in the area, mostly for local residents' personal homes, but also for "the community centers at the Greenleigh and Shady Hill communities, the Chesterfield Food Bank, and most recently, the Faith Gospel Family Worship Center, a church in South Richmond."
Adler emphasizes that what he does wouldn't be achievable without his "team of dedicated volunteers [who] come when called." Adler said he never worries about having enough helping hands; he's "learned that God will provide what we need." His volunteer pool includes friends at the Chester Rotary Club, at his church, and at other local churches, as well as people he knows from the Tri-Cities Workcamps, a group with which he volunteers as a handyman.
Typically, a team of volunteers can complete a ramp within a day. According to Adler, a project usually begins at 8:00 in the morning, and is finished by mid-afternoon. Some builds, though, are more challenging than others. "One of the earliest ramps we built was at a country home for a little boy with cerebral palsy," Adler recalled. "We knew it was going to be a long ramp because of the way the ground sloped away from the house." Upon completion, the ramp totaled 63 feet in length and included multiple landings and turns. "It reminded me of a queuing line at Disney [Park]," Adler said. When he returned to the home after construction to take photographs, the boy's grandmother told him how much her grandson enjoyed "'rides' up and down the ramp."
Adler and his volunteers provide joy, but also improve community members' lives by increasing accessibility and mobility. "The most rewarding aspect of what we do is the immediate impact we can make on the quality of life for someone who has been struggling to get in and out of [his or her] house," Adler said. "The gratefulness of the residents is all the fuel we need to keep moving forward to the next project."
Adler is no stranger to home improvement projects. He was raised "in a home where we were always fixing things, working with Dad." From his do-it-yourself upbringing, Adler learned to "enjoy the sense of accomplishment of solving problems." This desire to problem-solve means that over the years, Adler has been able to help make the building process more efficient.
In addition to a more streamlined process, a dependable team makes all the difference to Adler. "Everyone has something to contribute," he said. "Many hands make a burden light. And together we can make a significant positive impact on someone's life."

redpajama_set_name: RedPajamaCommonCrawl


Big night for the Canadians behind @StudioMDHR at @thegameawards last night – congratulations on taking home Best Debut Indie Game and Best Art Direction for Cuphead!
A big congratulations to all the nominations, and winners. While Warframe was nominated for Best Ongoing Game, last years Game of the Year Overwatch took home the prize for that.


text:
Q: Get all contacts at least one phone number in ngCordova Contacts plugin I am using the ngcordova contacts plugin to retrieve the contacts in an app. I'd like to know if it is possible to get only the contacts that have at least one phone Number.
I have use the following code , It returns the my google contacts which contains email but not phone numbers. But I want only available phone number not emails.
Is this possible ? or any other option available to get this result.
$scope.getContactList = function() {
         $ionicLoading.show({
            template: 'Loading...'
        });
        var options = {};
        options.multiple = true;
        options.hasPhoneNumber = true;
        options.fields = ['name.formatted', 'phoneNumbers'];
        $cordovaContacts.find(options).then(function(result) {
            $scope.contacts = result;
            $ionicLoading.hide();

        }, function(error) {
            console.log("ERROR: " + error);
        });
    }

A: I'd suggest using http://underscorejs.org/ in order to filter the contacts result. Something like this should suit your needs:
$scope.getContactList = function() {
     $ionicLoading.show({
        template: 'Loading...'
    });
    var options = {};
    options.multiple = true;
    options.hasPhoneNumber = true;
    options.fields = ['name.formatted', 'phoneNumbers'];
    $cordovaContacts.find(options).then(function(result) {
        $scope.contacts = result;

        var contactsWithAtLeastOnePhoneNumber = _.filter(result, function(contact){
            return contact.phoneNumbers.length > 0
        });

        //
        // Contacts with at least one phone number...
        console.log(contactsWithAtLeastOnePhoneNumber);

        $ionicLoading.hide();

    }, function(error) {
        console.log("ERROR: " + error);
    });
}

Because the phoneNumbers array can be returned and be blank, this quick method ensures that at least one entry is present.

A: I got solution without using  any external js , code shown as follows :
$scope.getContactList = function() {
      $scope.contacts = [];
             $ionicLoading.show({
                template: 'Loading...'
            });
            var options = {};
            options.multiple = true;
            $cordovaContacts.find(options).then(function(result) {
                 for (var i = 0; i < result.length; i++) {
                    var contact = result[i];
                     if(contact.phoneNumbers != null)
                       $scope.contacts.push(contact);
                  }
                $ionicLoading.hide();
            }, function(error) {
                console.log("ERROR: " + error);
            });
    }

redpajama_set_name: RedPajamaStackExchange