From 4d132f781f576645367e7adf3d75efcf84a3301a Mon Sep 17 00:00:00 2001 From: Urbain Vaes Date: Tue, 26 Sep 2023 01:20:33 +0200 Subject: Shorten presentation --- main.tex | 655 ++++++++++++++++++++++++++++++++++----------------------------- 1 file changed, 354 insertions(+), 301 deletions(-) diff --git a/main.tex b/main.tex index ceb383b..1c84dbc 100755 --- a/main.tex +++ b/main.tex @@ -240,7 +240,7 @@ \mathcal L_{\rm ovd} \Big\vert_{\red \eta = 0} = - \grad V \cdot \grad + \laplacian = - \grad^* \grad, \qquad - \mu(\d q) = \frac{1}{Z} \e^{-V(q)} \, \d q. + \mu(\d q) = \frac{1}{Z} \e^{- V(q)} \, \d q. \] where $\grad^* := (\grad V - \grad) \cdot $. For any $f, g \in C^{\infty}_{\rm c}(\mathcal E)$, we have @@ -249,14 +249,13 @@ = - \int_{\mathcal E} \nabla f \cdot \nabla g \, \d \mu = \int_{\mathcal E} (\mathcal L_{\rm ovd} g ) f \, \d \mu. \] - \item For Langevin dynamics + \item For Langevin dynamics, $\mu(\d q \, \d p) = \frac{1}{Z} \exp \left( - V(q) - \frac{\abs{p}^2}{2} \right) \, \d q \, \d p$. \begin{align*} \mathcal L\Big\vert_{\red \eta = 0} = p \cdot \grad_q - \grad V \cdot \grad_p + \gamma \left( - p \cdot \grad_p + \laplacian_p \right) - = \grad_p^* \grad_q - \grad_q^* \grad_p - \gamma \grad_p^* \grad_p^*, + = \grad_p^* \grad_q - \grad_q^* \grad_p - \gamma \grad_p^* \grad_p, \end{align*} - where $\grad_q^* := (\grad V - \grad_q) \cdot $ and $\grad_p^* = (p -\grad_p) \cdot$ are the formal adjoints. - We have + where $\grad_q^* := (\grad V - \grad_q) \cdot $ and $\grad_p^* = (p -\grad_p) \cdot$ are the formal~$L^2(\mu)$ adjoints. \begin{align*} \int_{\mathcal E} (\mathcal Lf ) g \, \d \mu &= \int_{\mathcal E} g \left(\grad_p^* \grad_q - \grad_q^* \grad_p\right) f - \gamma \grad_p f \cdot \grad_p g \, \d \mu \\ @@ -273,8 +272,8 @@ \[ \left\{ \begin{aligned} - \d q_t & = M^{-1} p_t \, \d t, \\* - \d p_t & = -\nabla V(q_t) \, \d t - \gamma M^{-1} p_t \, \d t + \d q_t & = p_t \, \d t, \\* + \d p_t & = -\nabla V(q_t) \, \d t - \gamma p_t \, \d t + \sqrt{2\gamma {\red T_\eta(q)}} \, \d W_t, \end{aligned} \right. @@ -370,8 +369,8 @@ \mathcal P^{\dagger} \mu \geq \alpha \pi, \qquad \text{ (Minorization condition) } \] - then there exists $\mu_*$ such that $\mathcal P^{\dagger} \mu_* = \mu_*$. - Furthermore $d(\mathcal P^{\dagger^n} \mu, \mu_*) \leq \alpha^n d(\mu, \mu_*)$. + then there exists $\mu_*$ such that $\mathcal P^{\dagger} \mu_* = \mu_*$, + and $d(\mathcal P^{\dagger^n} \mu, \mu_*) \leq (1-\alpha)^n d(\mu, \mu_*)$. \end{theorem} \emph{Sketch of proof.} Define the Markov transition kernel @@ -437,12 +436,11 @@ \begin{align*} \mathcal P^{\dagger}\mu (A) &= \expect \left[ q_t \in A \, \middle| \, q_0 \sim \mu \right] - = \int_{\mathcal E} \int_{A} p_t(x, y) \, \mu(\d x) \, + = \int_{\mathcal E} \int_{A} p_t(x, y) \, \mu(\d x) \, && p_t = \text{transition pdf} \\ &\geq \left( \inf_{(x,y) \in \mathcal E^2} p_t(x, y) \right) \lambda(A) && \lambda := \text{Lebesgue measure}. \end{align*} - The infimum is achieved by parabolic regularity, - and achieved by {\blue Harnack's inequality}. + The infimum is $> 0$ by parabolic regularity and Harnack's inequality. \item \textbf{Decay of the semigroup}: For $t \in [0, \infty)$ and $\varphi \in L^{\infty}_*$, it holds that @@ -467,7 +465,7 @@ {Existence of an invariant measure for perturbed Langevin dynamics} Consider the paradigmatic dynamics \begin{align*} - \d q_t &= M^{-1} p_t \, \d t, \\ + \d q_t &= p_t \, \d t, \\ \d p_t &= - \grad V(q_t) \, \d t + {\red \eta F \, \d t} - \gamma p_t \, \d t + \sqrt{2 \gamma} \, \d W_t, \end{align*} where $(q_t, p_t) = \torus^d \times \real^d$ and $F \in \real^d$ with $\abs{F} = 1$ is a given direction. @@ -483,13 +481,14 @@ \end{frame} \begin{frame} - {Harris' theorem} + {Harris' theorem \footnote{M. Hairer and J. Mattingly, \emph{Progr. Probab.} (2011)}} Let $p(x, A)$ denote a Markov transition kernel and let \[ (\mathcal P \phi)(x) := \int_{\mathcal E} \phi(y) \, p(x, \d y), \qquad (\mathcal P^{\dagger} \mu)(A) := \int_{A} p(x, A) \, \mu(\d x). \] + \vspace{-.2cm} \begin{theorem} [Harris's theorem] Suppose that the following conditions are satisfied: @@ -516,6 +515,7 @@ \qquad \overline \phi := \int_{\mathcal E} \phi \, \d \mu_*. \] \end{theorem} + \end{frame} \begin{frame} @@ -549,7 +549,7 @@ \end{frame} \begin{frame} -{Perturbation expansion for {\yellow $\eta$ sufficiently small} (1/2)} +{Perturbation expansion for {\yellow $\eta$ sufficiently small} (1/3)} Consider the perturbed Langevin dynamics and write \[ \mathcal L_{\eta} = \mathcal L_0 + {\red \eta \widetilde {\mathcal L}}, @@ -570,7 +570,7 @@ \] Observe that $\mathcal L_{\eta}^* = \mathcal L_0^* + \widetilde {\mathcal L}^*$ with \[ - \mathcal L_0^* = - \grad_p^* \grad_q + \grad_q^* \grad_p - \gamma \grad_p^* \grad_p^*, + \mathcal L_0^* = - \grad_p^* \grad_q + \grad_q^* \grad_p - \gamma \grad_p^* \grad_p, \qquad \widetilde {\mathcal L}^* \placeholder = \grad_p^* (F \placeholder) \] \end{block} @@ -611,7 +611,7 @@ \vspace{-.3cm} \begin{itemize} \itemsep.2cm - \item + \item The operator $\mathcal L_0^{-1}$ is a well defined bounded operator on $L_0^2(\psi_0)$ \\ ({\red Hypocoercivity} + {\red hypoelliptic regularization}) @@ -677,13 +677,13 @@ % % &\leq \frac{1}{\gamma} \norm{(\mathcal L_0^*)^{-1}}_{\mathcal B\bigl(L^2(\psi_0)\bigr)}^2 % % \norm{\varphi}_{L^2(\psi_0)} % % \end{align*} -% +% % \item Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L_0^* + \eta \wcL^*$ % \vspace{-0.2cm} % \[ % \mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0. % \] -% +% % \item {\red Prove that $f_\eta \geq 0$}. % \end{itemize} % \end{frame} @@ -735,7 +735,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. \begin{itemize} \itemsep.2cm \item - Non-equilibrium techniques. + Non-equilibrium steady state techniques. \begin{itemize} \item Calculations from the steady state of a system out of equilibrium. \item Comprises bulk-driven and boundary-driven approaches. @@ -761,9 +761,9 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. \begin{frame} {Linear response of nonequilibrium dynamics} - Consider the nonequilibirium dynamics + Consider the nonequilibirium dynamics with $V$ periodic: \begin{align*} - \d q_t &= M^{-1} p_t \, \d t, \\ + \d q_t &= p_t \, \d t, \\ \d p_t &= - \grad V(q_t) \, \d t + {\red \eta F \, \d t} - \gamma p_t \, \d t + \sqrt{2 \gamma} \, \d W_t, \end{align*} @@ -784,7 +784,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. We proved that $\psi_\eta = f_\eta\psi_0$ with $\psi_0(q,p) = Z^{-1} \e^{-\beta H(q,p)}$ and \[ - f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathrm{O}(\eta^2), \qquad \mathfrak f_1 = - (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \mathbf 1. + f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathcal O(\eta^2), \qquad \mathfrak f_1 = - (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \mathbf 1. \] Therefore \[ @@ -811,7 +811,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. \end{frame} \begin{frame} - {Reformulation as integrated correlation functions} + {Reformulation as integrated correlation function} Define the conjugate response \[ S @@ -892,7 +892,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. \begin{align*} \d \phi(q_s, p_s) % &= \frac{1}{\varepsilon^2} \mathcal L_{L} \phi (q_t, p_t) + \frac{1}{\varepsilon} \, \sqrt{2 \gamma \beta^{-1}} \, \derivative{1}[\phi]{p}(q_t, p_t) \, \d W_t, \\ - &= - p_s \, \d s + \sqrt{2} \, \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s + &= - p_s \, \d s + \sqrt{2 \gamma} \, \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s \end{align*} and then rearrange: \begin{align*} @@ -962,7 +962,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. \end{frame} \begin{frame} - {Thermal transport in one-dimensional chain (1)} + {Thermal transport in one-dimensional chain (1/3)} Consider a chain of $N$ atoms with nearest-neighbor interactions \begin{tikzpicture} \coordinate (origin) at (0,0); @@ -991,21 +991,21 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. \begin{equation*} \left\{ \begin{aligned} \d r_n &= (p_{n+1} - p_n) \, \d t, \\ - \d p_1 &= v'(r_1) \, \d t - \gamma p_1 \dt + \sqrt{2 \gamma {\color{red} (T+\Delta T)}} \, \d W_t^L, \\ + \d p_1 &= v'(r_1) \, \d t - \gamma p_1 \d t + \sqrt{2 \gamma {\color{red} (T+\Delta T)}} \, \d W_t^L, \\ \d p_n &= \bigl(v'(r_n) - v'(r_{n-1})\bigr) \, \d t, \\ - \d p_N &= -v'(r_{N-1}) \, \d t - \gamma p_N \dt + \sqrt{2 \gamma {\color{blue} (T-\Delta T)}} \, \d W_t^R, + \d p_N &= -v'(r_{N-1}) \, \d t - \gamma p_N \d t + \sqrt{2 \gamma {\color{blue} (T-\Delta T)}} \, \d W_t^R, \end{aligned} \right. \end{equation*} The Hamiltonian of the system is the sum of the potential and kinetic energies: \begin{equation*} H(r,p) = V(r) + \sum_{n=1}^N \frac {p_n^2}{2}, - \quad V(r) = \sum_{n=1}^{N-1} v(r_n). + \qquad V(r) = \sum_{n=1}^{N-1} v(r_n). \end{equation*} \end{frame} \begin{frame} - {Thermal transport in one-dimensional chains (2)} + {Thermal transport in one-dimensional chains (2/3)} \begin{itemize} \item @@ -1038,7 +1038,7 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies: \begin{frame} - {Thermal transport in one-dimensional chains (3)} + {Thermal transport in one-dimensional chains (3/3)} \bu Response function: {\blue total energy current} \begin{block} @@ -1067,9 +1067,9 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies: \end{frame} \begin{frame} - {Shear viscosity in fluids (1)} + {Shear viscosity in fluids (1/4)} - Consider a fluid $\mathcal{D} = \left( L_x\mathbb{T} \times L_y\mathbb{T} \right)^N$ subjected to a sinusoidal forcing + Consider a fluid in $\mathcal{D} = \left( L_x\mathbb{T} \times L_y\mathbb{T} \right)^N$ subjected to a sinusoidal forcing \begin{figure} \centering \includegraphics[height=.5\textwidth]{figures/osc_shear.eps} @@ -1079,76 +1079,64 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies: each subjected to a force $F$. \end{frame} -\begin{frame} - {Shear viscosity in fluids (2)} - Macroscopic description by Navier--Stokes equation - \[ - \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \nu \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x} - \] - Substitution of steady state ansatz $\mathbf{u} = U_x(y) \, \mathbf e_x$ gives - \[ - - \nu U_x''(y) = \overline{\rho} F(y), \qquad \overline \rho := \frac{\rho}{m} = \frac{N}{|\mathcal D|} - \] - Therefore, for the test function~$g(y) = \e^{2i\pi \frac{y}{L_y}}$ - \[ - \nu \int_0^{L_y} U_x(y) g''(y) \, \d y = \overline{\rho} \int_{0}^{L_y} F(y) g(y) \, \d y - \] - - $\rightarrow$ Suggests estimating the shear viscosity from molecular dynamics as - \[ - \nu = \frac{\dps \frac{\overline{\rho}}{L_y}\int_{0}^{L_y} F(y) g(y) \, \d y} - {\dps \expect_{F} \left[ \frac{1}{N}\sum_{n=1}^{N} \frac{p_{xi}}{m} g''(q_{yi}) \right]}. - \] -\end{frame} - +% \begin{frame} +% {Shear viscosity in fluids (2/4)} +% Macroscopic description by Navier--Stokes equation +% \[ +% \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \nu \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x} +% \] +% Substitution of steady state ansatz $\mathbf{u} = U_x(y) \, \mathbf e_x$ gives +% \[ +% - \nu U_x''(y) = \overline{\rho} F(y), \qquad \overline \rho := \frac{\rho}{m} = \frac{N}{|\mathcal D|} +% \] +% \end{frame} \begin{frame} - {Shear viscosity in fluids (3)} + {Shear viscosity in fluids (2/4)} Assume pairwise interactions \[ - V(q) = \sum_{1 \leq i < j \leq N} \mathcal V(\abs{q_i - q_j}). + V(q) = \sum_{1 \leq \ell < n \leq N} \mathcal V(\abs{q_\ell - q_n}). \] -\bu Add a smooth {\blue nongradient force} in the $x$ direction, depending on~$y$ -\begin{block}{Langevin dynamics under flow} -\centerequation{\left \{ \begin{aligned} - \d q_{i,t} &= \frac{p_{i,t}}{m} \, \d t,\\ - \d p_{xi,t} &= -\nabla_{q_{xi}} V(q_t) \, \d t + {\red \eta F(q_{yi,t}) \, \d t} - - \gamma \frac{p_{xi,t}}{m} \, \d t + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{xi}_t, \\ - \d p_{yi,t} &= -\nabla_{q_{yi}} V(q_t) \, \d t - \gamma \frac{p_{yi,t}}{m} \, \d t - + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{yi}_t. -\end{aligned} \right. -} -\end{block} + \bu Add a smooth {\blue nongradient force} in the $x$ direction, depending on~$y$ + \begin{block}{Langevin dynamics under flow} + \centerequation{\left \{ \begin{aligned} + \d q_{n} &= \frac{p_{n}}{m} \, \d t,\\ + \d p_{n,x} &= - \partial_{q_{n,x}} V(q_t) \, \d t + {\red \eta F(q_{n,y}) \, \d t} + - \gamma \frac{p_{n,x}}{m} \, \d t + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{n,x}_t, \\ + \d p_{n,y} &= - \partial_{q_{n,y}} V(q_t) \, \d t - \gamma \frac{p_{n,y}}{m} \, \d t + + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{n,y}_t. + \end{aligned} \right. + } + \end{block} -\smallskip + \smallskip -\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma>0$ + \bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma>0$ -\smallskip + \smallskip -\bu The perturbation $\dps \wcL = \sum_{i=1}^N \! F(q_{y,i}) \partial_{p_{x,i}}$ is $\mathcal{L}_0$-bounded + \bu The perturbation $\dps \wcL = \sum_{n=1}^N \! F(q_{n,y}) \partial_{p_{n,x}}$ is $\mathcal{L}_0$-bounded \end{frame} \begin{frame} - {Shear viscosity in fluids (4)} - -\bu {\blue Linear response}: -\[ - \lim_{\eta \rightarrow 0} \frac{\expect_{\eta} [\mathcal L_0 h]}{\eta} - = - \frac{\beta}{m} \! - \left\langle \!h, \sum_{i=1}^N p_{xi} F(q_{yi}) \!\right\rangle_{L^2(\psi_0)}. -\] + {Shear viscosity in fluids (3/4)} + \bu {\blue Linear response}: + \[ + \lim_{\eta \rightarrow 0} \frac{\expect_{\eta} [\mathcal L_0 h]}{\eta} + = - \frac{\beta}{m} \! + \left\langle \!h, \sum_{n=1}^N p_{n,x} F(q_{n,y}) \!\right\rangle_{L^2(\psi_0)}. + \] \bu Average {\red longitudinal velocity} $u_x(Y) = \dps \lim_{\varepsilon \to 0} - \lim_{\eta \to 0} \frac{\expect_{\eta} \left[ U_x^\varepsilon(Y,\cdot) \right]}{\eta}$ + \lim_{\eta \to 0} \frac{\expect_{\eta} \left[ U_x^\varepsilon(Y,\placeholder) \right]}{\eta}$ where \vspace{-0.3cm} \[ - U_x^\varepsilon(Y,q,p) = \frac{L_y}{Nm}\sum_{i=1}^N p_{xi} - \chi_{\varepsilon}\left(q_{yi}-Y\right) + U_x^\varepsilon(Y,q,p) = \frac{L_y}{Nm}\sum_{n=1}^N p_{n,x} + \, \chi_{\varepsilon}(q_{n,y}-Y) \] \vspace{-0.5cm} @@ -1160,36 +1148,65 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies: \[ \hspace{-0.1cm} ... = - \frac{1}{L_x} \left( \sum_{i=1}^N \frac{p_{xi} p_{yi}}{m}\chi_{\varepsilon}\left(q_{yi}-Y\right) + \frac{1}{L_x} \left( \sum_{n=1}^N \frac{p_{n,x} p_{n,y}}{m}\chi_{\varepsilon}(q_{n,y}-Y) - \! \! \! \! \! \! \! \! - \sum_{1 \leq i < j \leq N} \! \! \! \! - v'(|q_i-q_j|)\frac{ q_{xi}-q_{xj}}{|q_i-q_j|} - \!\int_{q_{yj}}^{q_{yi}} \!\chi_{\varepsilon}(s-Y) \, ds \right) + \sum_{1 \leq n < \ell \leq N} \! \! \! \! + \mathcal V'(|q_n-q_\ell|)\frac{ q_{n,x}-q_{\ell,x}}{|q_n-q_\ell|} + \!\int_{q_{\ell,y}}^{q_{n,y}} \!\chi_{\varepsilon}(s-Y) \, ds \right) \] -\bu {\blue Local conservation} of momentum\footnote{Irving and Kirkwood, {\it J. Chem. Phys.} {\bf 18} (1950)}: replace $h$ by $U_x^\varepsilon$ (with $\overline{\rho} = N/|\mathcal{D}|$) +\bu {\blue Local conservation} of momentum\footnote{Irving and Kirkwood, {\it J. Chem. Phys.} {\bf 18} (1950)}: replace $h$ by $U_x^\varepsilon$ \[ - \frac{d\sigma_{xy}(Y)}{dY} + \gamma_{x} \overline{\rho} u_x(Y) = \overline{\rho} F(Y) + \frac{\d\sigma_{xy}(Y)}{\d Y} + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y), + \qquad \overline{\rho} = \frac{N}{|\mathcal{D}|}. \] \end{frame} \begin{frame} - {Shear viscosity in fluids (4)} + {Shear viscosity in fluids (4/4)} -\bu {\blue Definition} $\sigma_{xy}(Y) := -\eta(Y)\dfrac{du_x(Y)}{dY}$, {\red closure} assumption $\eta(Y) = \eta > 0$ +\bu {\blue Definition} $\sigma_{xy}(Y) := -\nu(Y) u_x'(Y)$, +{\red closure} assumption $\nu(Y) = \nu > 0$. \begin{block}{Velocity profile in Langevin dynamics under flow} -\centerequation{-\eta u_x''(Y) + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y)} +\centerequation{-\nu u_x''(Y) + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y)} \end{block} -\begin{figure}[ht] - \centering - \includegraphics[width=\linewidth]{figures/shear1.png} -\end{figure} +Therefore, integrating against the test function~$\e^{2i\pi \frac{y}{L_y}}$ and rearranging, +we have +\[ + \nu = \overline \rho \left( \frac{F_1}{U_1} - \gamma \right) \left(\frac{L_y}{2\pi}\right)^2, +\] +where +\[ + U_1 = \frac{1}{L_y} \int_{0}^{L_y} u_x(x) \e^{2i\pi \frac{y}{L_y}} \, \d y, + \qquad + F_1 = \frac{1}{L_y} \int_{0}^{L_y} F(y) \e^{2i\pi \frac{y}{L_y}} \, \d y. +\] +The coefficient $U_1$ can be rewritten as +\[ + U_1 = \lim_{\eta \to 0} \frac{1}{\eta} {\dps \expect_{\eta} \left[ \frac{1}{N}\sum_{n=1}^{N} \frac{p_{n,x}}{m} \exp \left( 2i\pi \frac{q_{n,y}}{L_y} \right) \right]}. +\] + \end{frame} -% \begin{frame} -% \end{frame} +\begin{frame} + {Numerical illustration} + \begin{figure}[ht] + \centering + \includegraphics[width=\linewidth]{figures/shear1.png} + \caption{Numerical results from~\footnote{See R.~Joubaud and G.~Stoltz, \emph{Multiscale Model. Simul.} (2012)}} + \end{figure} +\end{frame} + +\begin{frame} + {Numerical illustration} + \begin{figure}[ht] + \centering + \includegraphics[width=\linewidth]{figures/shear2.png} + \caption{Numerical results from~\footnote{See R.~Joubaud and G.~Stoltz, \emph{Multiscale Model. Simul.} (2012)}} + \end{figure} +\end{frame} \begin{frame} \begin{center} @@ -1203,7 +1220,7 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies: \begin{itemize} \item Reminders: strong order, weak order \item Error analysis for the linear response method - \item Error analysis for Green--Kubo method + \item Error analysis for the Green--Kubo method \end{itemize} \end{minipage} \end{frame} @@ -1235,7 +1252,7 @@ x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \stackrel{\ \begin{block}{Error estimates for {\red finite} trajectory averages} \[ -\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) +\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) = \expect_\pi(A) + \underbrace{\frac{C}{N_{\rm iter} \dt}}_{\rm bias} + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error} \] \end{block} @@ -1265,7 +1282,7 @@ where $(x^n)$ is an approximation of $(x_{n \dt})$ \[ \dps \sup_{0 \leq n \leq T/\dt} \expect | x^n - x_{n\dt} | \leq C \dt^p \] -\item {\blue Weak error}: for any $\varphi$, +\item {\blue Weak error}: for any $\varphi$, \[ \dps \!\!\!\! \sup_{0 \leq n \leq T/\dt} \Big| \expect\left[\varphi\left(x^n\right)\right] - \expect\left[\varphi\left(x_{n\dt}\right)\right] \Big| \leq C \dt^p \] @@ -1287,6 +1304,15 @@ where $(x^n)$ is an approximation of $(x_{n \dt})$ \begin{frame} {Elements of proof} \begin{itemize} + \item + Rewrite the weak error as a telescopic sum + \begin{align*} + \expect \bigl[\varphi(x^N)\bigr] - \expect \bigl[\varphi(x_{N \dt})\bigr] + &= P_{\dt}^N \varphi (x_0) - \e^{N \dt \mathcal L} \varphi(x_0) \\ + &= \sum_{n=0}^{N-1} \left( P_{\dt}^{N-n} \e^{n \dt \mathcal L} \varphi(x_0) - P_{\dt}^{N-(n+1)} \e^{(n+1) \dt \mathcal L} \varphi (x_0) \right) \\ + &= \sum_{n=0}^{N-1} P_{\dt}^{N-(n+1)} \left( P_{\dt} - \e^{\dt \mathcal L} \right) \e^{n \dt \mathcal L} \varphi (x_0) + \end{align*} + \item Since $u(t, x) := \e^{t \mathcal L} \varphi(x)$ solves the backward Kolmogorov equation \begin{align*} @@ -1297,14 +1323,6 @@ where $(x^n)$ is an approximation of $(x_{n \dt})$ \[ \e^{\dt \mathcal L} \varphi = \I + \dt \mathcal L \varphi + \frac{\dt^2}{2} \mathcal L^2\varphi + \dotsb \] - \item - Introduce a telescopic sum - \begin{align*} - \expect \bigl[\varphi(x^N)\bigr] - \expect \bigl[\varphi(x_{N \dt})\bigr] - &= P_{\dt}^N \varphi (x_0) - \e^{N \dt \mathcal L} \varphi(x_0) \\ - &= \sum_{n=0}^{N-1} \left( P_{\dt}^{N-n} \e^{n \dt \mathcal L} \varphi(x_0) - P_{\dt}^{N-(n+1)} \e^{(n+1) \dt \mathcal L} \varphi (x_0) \right) \\ - &= \sum_{n=0}^{N-1} P_{\dt}^{N-(n+1)} \left( P_{\dt} - \e^{\dt \mathcal L} \right) \e^{n \dt \mathcal L} \varphi (x_0) - \end{align*} \end{itemize} \end{frame} @@ -1348,12 +1366,10 @@ where $(x^n)$ is an approximation of $(x_{n \dt})$ \begin{frame} {Error estimates on the invariant measure (equilibrium)} -\bu {\red Assumptions} on the operators in the weak-type expansion - \begin{block}{Error estimates on $\pi_\dt$} Suppose that \begin{itemize} - \item + \item For all smooth $\varphi$, the following expansion holds \[ P_\dt \varphi = \varphi + \dt \, \mathcal A_1 \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt} @@ -1362,12 +1378,14 @@ where $(x^n)$ is an approximation of $(x_{n \dt})$ \[ \int_\cX \mathcal A_k \varphi \, d\pi = 0 \] + \item + + {\red Technical assumptions} usually satisfied \end{itemize} Then \[ - \int_\cX \varphi \, d\pi_\dt = \int_\cX \varphi \Big(1+\dt^{p}f_{p+1}\Big) d\pi + \dt^{p+1} R_{\varphi,\dt}, + \int_\cX \varphi \, \d\pi_\dt = \int_\cX \varphi \Big(1+\dt^{p}f_{p+1}\Big) \d\pi + \dt^{p+1} R_{\varphi,\dt}, \] - where $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$. + where $g_{p+1} = \mathcal A_{p+1}^* \mathbf 1$ and $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$. \end{block} Error on invariant measure can be {\blue (much) smaller} than the weak error @@ -1421,12 +1439,12 @@ C = -M^{-1} p \cdot \nabla_p + \frac1\beta \Delta_p \bu Trotter splitting $\to$ weak order 1 \[ -P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \mathcal O(\dt^2) +P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \, \mathcal O(\dt^2) \] \bu Strang splitting $\to$ {\blue weak order 2} \[ -P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \mathcal O(\dt^3) +P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \, \mathcal O(\dt^3) \] \bu Other category: {\red Geometric Langevin}\footnote{N.~Bou-Rabee and H.~Owhadi, {\em SIAM J. Numer. Anal.} (2010)} algorithms, \textit{e.g.} $P_\dt^{\gamma C,A,B_\eta,A}$ \\ @@ -1467,85 +1485,89 @@ p^{n+1} & = \alpha_{\dt/2} \widetilde{p}^{n+1} + \sqrt{\frac{1-\alpha_{\dt}}{\be \begin{frame} - {Numerical estimators and associated challenges} - \begin{itemize} - \item - Estimator of linear response (observable~$R$ with equilibrium average~0) - \[ - \widehat{A}_{\eta,t} = \frac{1}{\eta t}\int_0^t R(q_s^\eta,p_s^\eta) \, ds \xrightarrow[t\to+\infty]{\mathrm{a.s.}} - \alpha_\eta := \frac1\eta \int_{\mathcal E} R \, f_\eta \, d\mu = \alpha + \mathcal O(\eta) - \] - {\bf Issues with linear response methods:} - \begin{itemize} - \item Statistical error with {\red asymptotic variance $\mathcal O(\eta^{-2})$} - \item Bias $\mathcal O(\eta)$ due to $\eta \neq 0$ - \item Bias from finite integration time - \end{itemize} + {Error estimates on linear response (1/3)} + \textbf{Aim:} For observable~$R$, approximate + \[ + \alpha = \lim_{\eta \to 0} \frac{\expect_{\red \eta} [R]}{\eta} + \] + + \textbf{Estimator} of linear response (up to time discretization): + \[ + \widehat{A}_{\eta,t} = \frac{1}{\eta t}\int_0^t R(q_s^\eta,p_s^\eta) \, \d s \xrightarrow[t\to+\infty]{\mathrm{a.s.}} + \alpha_\eta := \frac1\eta \int_{\mathcal E} R \, f_\eta \, \d \mu = \alpha + \mathcal O(\eta) + \] + + {\bf Contributions to the error} + \begin{itemize} + \item Statistical error with {\red asymptotic variance $\mathcal O(\eta^{-2})$} + \item Bias $\mathcal O(\eta)$ due to $\eta \neq 0$ + \item Bias from finite integration time + \item Timestep discretization bias \end{itemize} + + \end{frame} -\begin{frame}\frametitle{Analysis of variance / finite integration time bias} +\begin{frame} + {Error estimates on linear response (2/3)} - \bu {\bf Statistical error} dictated by {\blue Central Limit Theorem}: - \[ - \sqrt{t} \left(\widehat{A}_{\eta,t} - \alpha_\eta \right) \xrightarrow[t \to +\infty]{\mathrm{law}} \mathcal{N}\left(0,\frac{\sigma_{R,\eta}^2}{\eta^2}\right), - \qquad - \sigma_{R,\eta}^2 = \sigma_{R,0}^2 + \mathcal O(\eta) - \] - so $\dps \widehat{A}_{\eta,t} = \alpha_\eta + \mathcal O_{\rm P}\left(\frac{1}{\eta \sqrt{t}}\right)$ $\to$ requires {\red long simulation times} $t \sim \eta^{-2}$ - - \bigskip - - \bu {\bf Finite time integration bias}: $\dps \left| \mathbb{E}\left(\widehat{A}_{\eta,t}\right) - \alpha_\eta \right| \leq \frac{K}{\eta t}$ \\ - Bias due to $t < +\infty$ is $\dps \mathcal O\left(\frac{1}{\eta t}\right)$ $\to$ typically {\red smaller than statistical error} + \bu {\bf Statistical error} dictated by {\blue Central Limit Theorem}: + \[ + \sqrt{t} \left(\widehat{A}_{\eta,t} - \alpha_\eta \right) \xrightarrow[t \to +\infty]{\mathrm{law}} \mathcal{N}\left(0,\frac{\sigma_{R,\eta}^2}{\eta^2}\right), + \qquad + \sigma_{R,\eta}^2 = \sigma_{R,0}^2 + \mathcal O(\eta) + \] + so $\dps \widehat{A}_{\eta,t} = \alpha_\eta + \mathcal O_{\rm P}\left(\frac{1}{\eta \sqrt{t}}\right)$ $\to$ requires {\red long simulation times} $t \sim \eta^{-2}$ -%\bigskip - %\bu Bias~$\mathcal O(\eta)$ and statistical error equilibrated for~$t \sim \eta^{-3}$ + \bigskip -\bigskip + \bu {\bf Finite time integration bias}: $\dps \left| \expect\left(\widehat{A}_{\eta,t}\right) - \alpha_\eta \right| \leq \frac{K}{\eta t}$ \\ + Bias due to $t < +\infty$ is $\dps \mathcal O\left(\frac{1}{\eta t}\right)$ $\to$ typically {\red smaller than statistical error} -\bu Key equality for the proofs: introduce $\dps -\left(\mathcal{L}+\eta\widetilde{\mathcal{L}}\right) \mathscr{R}_\eta = R - \int_\mathcal{E} R f_\eta \, d\mu$ -\[ -\widehat{A}_{\eta,t} - \frac1\eta \!\int_{\mathcal{E}} \!R f_\eta \, d\mu = \frac{\mathscr{R}_\eta(q_0^\eta,p_0^\eta) - \mathscr{R}_\eta(q_t^\eta,p_t^\eta)}{\eta t} + \frac{\sqrt{2\gamma}}{\eta t\sqrt{\beta}} \int_0^t \!\!\nabla_p \mathscr{R}_\eta(q_s^\eta,p_s^\eta)^T dW_s -\] + %\bigskip + %\bu Bias~$\mathcal O(\eta)$ and statistical error equilibrated for~$t \sim \eta^{-3}$ + + \bigskip + + \bu Key equality for the proofs: introduce $\dps -\left(\mathcal{L}+\eta\widetilde{\mathcal{L}}\right) \mathscr{R}_\eta = R - \int_\mathcal{E} R f_\eta \, \d\mu$ + \[ + \widehat{A}_{\eta,t} - \frac1\eta \!\int_{\mathcal{E}} \!R f_\eta \, \d\mu = \frac{\mathscr{R}_\eta(q_0^\eta,p_0^\eta) - \mathscr{R}_\eta(q_t^\eta,p_t^\eta)}{\eta t} + \frac{\sqrt{2\gamma}}{\eta t\sqrt{\beta}} \int_0^t \!\!\nabla_p \mathscr{R}_\eta(q_s^\eta,p_s^\eta)^T \d W_s + \] \end{frame} -\begin{frame}\frametitle{Error estimates on linear response} +\begin{frame} + {Error estimates on linear response (3/3)} -\begin{block}{Error estimates for nonequilibrium dynamics} -There exists a function $f_{\alpha,1,\gamma} \in H^1(\mu)$ such that -\vspace{-0.3cm} -\[ -\int_{\mathcal E} \psi \, d{\mu}_{\gamma,\eta,\dt} = \int_{\mathcal E} \psi \Big(1+ \eta f_{0,1,\gamma} + \dt^\alpha f_{\alpha,0,\gamma} + \eta \dt^\alpha f_{\alpha,1,\gamma} \Big) d{\mu} + r_{\psi,\gamma,\eta,\dt}, -\] -where the remainder is compatible with linear response -\vspace{-0.1cm} -\[ -\left|r_{\psi,\gamma,\eta,\dt}\right| \leq K(\eta^2 + \dt^{\alpha+1}), -\qquad -\left|r_{\psi,\gamma,\eta,\dt} - r_{\psi,\gamma,0,\dt}\right| \leq K \eta (\eta + \dt^{\alpha+1}) -\] -\end{block} + \begin{block} + {Finite integration time bias and timestep bias} + There exist functions $f_{0,1}$, $f_{\alpha,0}$ and $f_{\alpha,1}$ such that + \[ + \int_{\mathcal E} R \, \d{\mu}_{\eta,\dt} = \int_{\mathcal E} R \Big(1+ \eta f_{0,1} + \dt^\alpha f_{\alpha,0} + \eta \dt^\alpha f_{\alpha,1} \Big) \d{\mu} + r_{\psi,\eta,\dt}, + \] + where the remainder is compatible with linear response + \vspace{-0.1cm} + \[ + \left|r_{\psi,\eta,\dt}\right| \leq K(\eta^2 + \dt^{\alpha+1}), + \qquad + \left|r_{\psi,\eta,\dt} - r_{\psi,0,\dt}\right| \leq K \eta (\eta + \dt^{\alpha+1}) + \] + \end{block} \medskip \bu Corollary: error estimates on the {\blue numerically computed mobility} \[ \begin{aligned} -\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^\t M^{-1} p \, \mu_{\gamma,\eta,\dt}(d{q}\,d{p}) - \int_{\mathcal E} F^\t M^{-1} p \, \mu_{\gamma,0,\dt}(d{q}\,d{p}) \right) \\ -& = \rho_{F} + \dt^\alpha \int_{\mathcal E} F^\t M^{-1} p \, f_{\alpha,1,\gamma} \, d{\mu} + \dt^{\alpha+1} r_{\gamma,\dt} +\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^\t p \, \mu_{\eta,\dt}(\d{q}\,\d{p}) - \int_{\mathcal E} F^\t p \, \mu_{0,\dt}(\d{q}\,\d{p}) \right) \\ +& = \rho_{F} + \dt^\alpha \int_{\mathcal E} F^\t p \, f_{\alpha,1} \, \d{\mu} + \dt^{\alpha+1} r_{\dt} \end{aligned} \] -\bu Results in the {\red overdamped} limit\footnote{B.~Leimkuhler, C.~Matthews and G.~Stoltz, {\em IMA J. Numer. Anal.} (2015)} - -\bigskip - \end{frame} @@ -1561,172 +1583,203 @@ Scaling of the mobility for the first order scheme $P_\dt^{A,B_\eta,\gamma C}$ a \end{frame} +\begin{frame} + {Error estimates on the Green--Kubo formula (1/3)} + \textbf{Aim:} For observable~$R$, approximate + \[ + \alpha = \int_0^{+\infty} \!\! \expect_0\Big(R(q_t,p_t)S(q_0,p_0) \Big) \, \d t + \] + -%----------------------------------------------------------- -\begin{frame}\frametitle{Error estimates on Green-Kubo formulas (1)} - -\bu For methods of {\bf weak order}~1, {\red Riemman sum} ($\phi,\varphi$ average 0 w.r.t. $\pi$) -\vspace{-0.2cm} -\[ -\begin{aligned} -& -\int_0^{+\infty} \expect \Big( \phi(x_t) \varphi(x_0) \Big) d{t} = \dt \sum_{n=0}^{+\infty} \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) + \mathrm{O}(\dt) \\[-7pt] -& \mathrm{where} \ \Pi_\dt \phi = \phi - \int_\cX \phi \, d\pi_\dt -\end{aligned} -\] - -\bu Correlation approximated in practice using $K$ independent realizations -%\bi -%\item truncating the integration (decay estimates) -%\item using empirical averages ($K$ independent realizations) -\vspace{-0.2cm} -\[ -\begin{aligned} -& \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) -\simeq -\frac{1}{K}\sum_{m=1}^{K} \left( \phi(x^{n,k}) - \overline{\phi}^{n,K} \right)\left( \varphi(x^{n,k}) - \overline{\varphi}^{n,K} \right) \\[-10pt] -& \mathrm{where} \ \overline{\phi}^{n,K} = \frac1K \sum_{m=1}^{K} \phi(x^{n,k}) -\end{aligned} -\] - -\bu For methods of {\bf weak order} 2, {\blue trapezoidal rule} -\vspace{-0.1cm} -\[ -\begin{aligned} -\int_0^{+\infty} \expect \Big( \phi(x_t) \varphi(x_0) \Big) d{t} & = \frac{\dt}{2} \expect_\dt \left(\Pi_\dt \phi\left(x^{0}\right)\varphi\left(x^0\right)\right) \\ -& \ \ + \dt \sum_{n=1}^{+\infty} \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) + \mathrm{O}(\dt^2) -\end{aligned} -\] - - -%\bu Allows to quantify the variance $\dps \frac{\sigma^2_{A,\dt}}{N_{\rm iter}\dt} \simeq \frac{\dps 2 \int_0^{+\infty} \expect\left[\delta A(x_t)\delta A(x_0)\right] \, dt}{T}$ where $T = N_{\rm iter}\dt$ + \textbf{``Natural'' estimator} (up to time discretization) + \[ + \widehat{A}_{K,T} = \frac1K \sum_{k=1}^K \int_0^T R(q_t^k,p_t^k)S(q_0^k,p_0^k)\, \d t + \] + \bu {\bf Contributions to the error:} + \begin{itemize} + \item Truncature of time (exponential convergence of $\e^{t \mathcal L}$) + \item The {\red statistical error} increases linearly with $T$. + \item {\blue Timestep bias and quadrature formula} + \end{itemize} \end{frame} -%----------------------------------------------------------- -\begin{frame}\frametitle{Error estimates on Green-Kubo formulas (2)} +\begin{frame} + {Error estimates on the Green--Kubo formula (2/3)} -\bu Error of {\red order~$\alpha$ on invariant measure}: $\dps \int_\cX \psi \, d{\pi}_\dt = \int_\cX \psi \, d{\pi} + \mathrm{O}(\dt^\alpha)$ + \bu {\bf Truncation bias}: {\blue small} due to generic exponential decay of correlations + \[ + \left|\expect\left(\widehat{A}_{K,T}\right)-\alpha\right| \leq C \e^{-\kappa T} + \] -\medskip + \bigskip -\bu Expansion of the evolution operator ($p+1 \geq \alpha$ and $\mathcal A_1 = \mathcal L$) -\[ -P_\dt \varphi = \varphi + \dt \, \mathcal L \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt} -\] + \bu {\bf Statistical error}: {\red large}, increases with the integration time + \[ + \forall T \geq 1, \qquad \mathrm{Var}\left(\widehat{A}_{K,T}\right) \leq C \frac{T}{K} + \] -\begin{block}{Ergodicity of the numerical scheme} -\centerequation{ -\forall n \in \mathbb{N}, \qquad \left\| P_\dt^n \right \|_{\mathcal{B}(L^\infty_{\Li_s,\dt})} \leq C_s \e^{-\lambda_s n\dt} -} -where $\mathcal{K}_s$ is a Lyapunov function ($1+|p|^{2s}$ for Langevin) and -\[ -L^\infty_{\Li_s,\dt} = \left\{ \frac{\varphi}{\mathcal{K}_s} \in L^\infty(\cX), \ \int_\cX \varphi \, d\pi_\dt = 0\right\} -\] -\end{block} -\bu Proof: Lyapunov condition + uniform-in-$\dt$ minorization condition\footnote{M. Hairer and J. Mattingly, \emph{Progr. Probab.} (2011)} + \bu {\bf Time discretization and quadrature bias}: if + \begin{itemize} + \item {\red uniform-in-$\Delta t$ convergence} + \item error on the invariant measure of order~$\dt^a$ + \item $P_\dt = \mathrm{Id} + \dt \mathcal L + \dt^2 L_2 + \dots + \dt^{a} L_a + \dots$ + \end{itemize} + Then for $R,S$ with average~0 w.r.t.~$\mu$, + \[ + \hspace{-0.1cm}\int_0^{+\infty} \expect \Big( R(X_t) S(X_0) \Big) d{t} = \dt \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{R}_{\dt}\left(X^{n}\right)S\left(X^0\right)\right) + \mathrm{O}(\dt^a) \vspace{-0.5cm} + \] + with + \[ + \widetilde{R}_{\dt} = \Big(\mathrm{Id} + \dt \,L_2 \mathcal L^{-1} + \dots + \dt^{a-1} L_a \mathcal L^{-1} \Big)R - \mu_\dt(\dots) + \] \end{frame} -%----------------------------------------------------------- -\begin{frame}\frametitle{Error estimates on Green-Kubo formulas (3)} - -\begin{block}{Error estimates on integrated correlation functions} -Observables $\varphi,\psi$ with average~0 w.r.t. invariant measure~$\pi$ -\[ -\int_0^{+\infty} \expect \Big( \psi(x_t) \varphi(x_0) \Big) d{t} = \dt \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{\psi}_{\dt,\alpha}\left(x^{n}\right)\varphi\left(x^0\right)\right) + \dt^\alpha r^{\psi,\varphi}_\dt, -\] -where $\expect_\dt$ denotes expectations w.r.t. initial conditions $x_0 \sim \pi_\dt$ and over all realizations of the Markov chain $(x^n)$, and -\[ -\widetilde{\psi}_{\dt,\alpha} = \psi_{\dt,\alpha} - \int_\cX \psi_{\dt,\alpha} \, d\pi_\dt\] -with $\dps \psi_{\dt,\alpha} = \Big(\I + \dt \,\mathcal A_2 \mathcal L^{-1} + \dots + \dt^{\alpha-1} \mathcal A_{\alpha}\mathcal L^{-1} \Big)\psi$ -\end{block} - -\bu Useful when $\mathcal A_k \mathcal L^{-1}$ can be computed, \emph{e.g.} $\mathcal A_k = a_k \mathcal L^{k}$ +\begin{frame} + {Error estimates on Green-Kubo formulas (1/3)} -\medskip + \bu For methods of {\bf weak order}~1, {\red Riemman sum} ($\phi,\varphi$ average 0 w.r.t. $\pi$) + \vspace{-0.2cm} + \[ + \begin{aligned} +& +\int_0^{+\infty} \expect \Big( \phi(x_t) \varphi(x_0) \Big) d{t} = \dt \sum_{n=0}^{+\infty} \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) + \mathrm{O}(\dt) \\[-7pt] +& \mathrm{where} \ \Pi_\dt \phi = \phi - \int_\cX \phi \, d\pi_\dt + \end{aligned} + \] -\bu Reduces to trapezoidal rule for second order schemes + \bu For methods of {\bf weak order} 2, {\blue trapezoidal rule} + \vspace{-0.1cm} + \[ + \begin{aligned} + \int_0^{+\infty} \expect \Big( \phi(x_t) \varphi(x_0) \Big) d{t} & = \frac{\dt}{2} \expect_\dt \left(\Pi_\dt \phi\left(x^{0}\right)\varphi\left(x^0\right)\right) \\ + & \ \ + \dt \sum_{n=1}^{+\infty} \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) + \mathrm{O}(\dt^2) + \end{aligned} + \] \end{frame} -%----------------------------------------------------------- -\begin{frame}\frametitle{Sketch of proof (1)} +% \begin{frame}\frametitle{Error estimates on Green-Kubo formulas (2)} +% +% \bu Error of {\red order~$\alpha$ on invariant measure}: $\dps \int_\cX \psi \, d{\pi}_\dt = \int_\cX \psi \, d{\pi} + \mathrm{O}(\dt^\alpha)$ +% +% \medskip +% +% \bu Expansion of the evolution operator ($p+1 \geq \alpha$ and $\mathcal A_1 = \mathcal L$) +% \[ +% P_\dt \varphi = \varphi + \dt \, \mathcal L \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt} +% \] +% +% \begin{block}{Ergodicity of the numerical scheme} +% \centerequation{ +% \forall n \in \mathbb{N}, \qquad \left\| P_\dt^n \right \|_{\mathcal{B}(L^\infty_{\Li_s,\dt})} \leq C_s \e^{-\lambda_s n\dt} +% } +% where $\mathcal{K}_s$ is a Lyapunov function ($1+|p|^{2s}$ for Langevin) and +% \[ +% L^\infty_{\Li_s,\dt} = \left\{ \frac{\varphi}{\mathcal{K}_s} \in L^\infty(\cX), \ \int_\cX \varphi \, d\pi_\dt = 0\right\} +% \] +% \end{block} +% +% \bu Proof: Lyapunov condition + uniform-in-$\dt$ minorization condition\footnote{M. Hairer and J. Mattingly, \emph{Progr. Probab.} (2011)} +% +% \end{frame} +% +% %----------------------------------------------------------- +% \begin{frame}\frametitle{Error estimates on Green-Kubo formulas (3)} +% +% \begin{block}{Error estimates on integrated correlation functions} +% Observables $\varphi,\psi$ with average~0 w.r.t. invariant measure~$\pi$ +% \[ +% \int_0^{+\infty} \expect \Big( \psi(x_t) \varphi(x_0) \Big) d{t} = \dt \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{\psi}_{\dt,\alpha}\left(x^{n}\right)\varphi\left(x^0\right)\right) + \dt^\alpha r^{\psi,\varphi}_\dt, +% \] +% where $\expect_\dt$ denotes expectations w.r.t. initial conditions $x_0 \sim \pi_\dt$ and over all realizations of the Markov chain $(x^n)$, and +% \[ +% \widetilde{\psi}_{\dt,\alpha} = \psi_{\dt,\alpha} - \int_\cX \psi_{\dt,\alpha} \, d\pi_\dt\] +% with $\dps \psi_{\dt,\alpha} = \Big(\I + \dt \,\mathcal A_2 \mathcal L^{-1} + \dots + \dt^{\alpha-1} \mathcal A_{\alpha}\mathcal L^{-1} \Big)\psi$ +% \end{block} +% +% \bu Useful when $\mathcal A_k \mathcal L^{-1}$ can be computed, \emph{e.g.} $\mathcal A_k = a_k \mathcal L^{k}$ +% +% \medskip +% +% \bu Reduces to trapezoidal rule for second order schemes +% +% \end{frame} -\bu Define $\dps \Pi_\dt \varphi = \varphi - \int_\cX \varphi \, d\pi_\dt$ +%\begin{frame} +% {Sketch of proof (1)} -\smallskip +%\bu Define $\dps \Pi_\dt \varphi = \varphi - \int_\cX \varphi \, d\pi_\dt$ -\bu Since $\mathcal L^{-1}\psi$ has average~0 w.r.t.~$\pi$, introduce $\pi_\dt$ as -\begin{align*} -\int_\cX \left(-\mathcal L^{-1} \psi\right) \varphi \, d{\pi} & = \int_\cX \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi} \nonumber \\ -%& = \int_\cX \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi}_\dt + \dt^\alpha r^{\psi,\varphi}_\dt, \nonumber \\ -& = \int_\cX \Pi_\dt \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi}_\dt + \dt^\alpha r^{\psi,\varphi}_\dt, -\end{align*} +%\smallskip -\bu Rewrite $-\Pi_\dt \mathcal L^{-1}$ in terms of $P_\dt$ as -\[ -\begin{aligned} -& -\Pi_\dt \mathcal L^{-1} \psi = -\Pi_\dt \left(\dt\sum_{n=0}^{+\infty} P_\dt^n \right) \Pi_\dt \left(\frac{\I - P_\dt}{\dt}\right) \mathcal L^{-1} \psi \\ -& \ \ = \dt \left(\sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \right) \left(\mathcal L + \dots + \dt^{\alpha-1} S_{\alpha-1} + \dt^\alpha \widetilde{R}_{\alpha,\dt}\right) \mathcal L^{-1} \psi, \\ -& \ \ = \dt \sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \widetilde{\psi}_{\dt,\alpha} + \dt^\alpha \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \Pi_\dt \widetilde{R}_{\alpha,\dt} \mathcal L^{-1} \psi. -\end{aligned} -\] +%\bu Since $\mathcal L^{-1}\psi$ has average~0 w.r.t.~$\pi$, introduce $\pi_\dt$ as +%\begin{align*} +%\int_\cX \left(-\mathcal L^{-1} \psi\right) \varphi \, d{\pi} & = \int_\cX \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi} \nonumber \\ +%%& = \int_\cX \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi}_\dt + \dt^\alpha r^{\psi,\varphi}_\dt, \nonumber \\ +%& = \int_\cX \Pi_\dt \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi}_\dt + \dt^\alpha r^{\psi,\varphi}_\dt, +%\end{align*} -\end{frame} +%\bu Rewrite $-\Pi_\dt \mathcal L^{-1}$ in terms of $P_\dt$ as +%\[ +%\begin{aligned} +%& -\Pi_\dt \mathcal L^{-1} \psi = -\Pi_\dt \left(\dt\sum_{n=0}^{+\infty} P_\dt^n \right) \Pi_\dt \left(\frac{\I - P_\dt}{\dt}\right) \mathcal L^{-1} \psi \\ +%& \ \ = \dt \left(\sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \right) \left(\mathcal L + \dots + \dt^{\alpha-1} S_{\alpha-1} + \dt^\alpha \widetilde{R}_{\alpha,\dt}\right) \mathcal L^{-1} \psi, \\ +%& \ \ = \dt \sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \widetilde{\psi}_{\dt,\alpha} + \dt^\alpha \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \Pi_\dt \widetilde{R}_{\alpha,\dt} \mathcal L^{-1} \psi. +%\end{aligned} +%\] -%----------------------------------------------------------- -\begin{frame}\frametitle{Sketch of proof (2)} +%\end{frame} -\bu Uniform resolvent bounds $\dps \left\| \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \right \|_{\mathcal{B}(L^\infty_{\Li_s,\dt})} \leq \frac{C_s}{\lambda_s}$ +%%----------------------------------------------------------- +%\begin{frame}\frametitle{Sketch of proof (2)} -\medskip +%\bu Uniform resolvent bounds $\dps \left\| \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \right \|_{\mathcal{B}(L^\infty_{\Li_s,\dt})} \leq \frac{C_s}{\lambda_s}$ -\bu Coming back to the initial equality, -\[ -\int_\cX \left(-\mathcal L^{-1} \psi\right) \varphi \, d{\pi} = \dt \int_\cX \sum_{n=0}^{+\infty} \left( \Pi_\dt P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right) \left( \Pi_\dt \varphi \right) d{\pi}_\dt + \mathrm{O}\left(\dt^\alpha\right) -\] - -\bu Rewrite finally -\[ -\begin{aligned} -\int_\cX \sum_{n=0}^{+\infty} \left( \Pi_\dt P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right)\left( \Pi_\dt \varphi \right) d{\pi}_\dt & = \int_\cX \sum_{n=0}^{+\infty} \left(P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right) \varphi \, d{\pi}_\dt \\ -& = \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{\psi}_{\dt,\alpha}\left(q^{n},p^{n}\right)\varphi\left(q^0,p^0\right)\right) -\end{aligned} -\] +%\medskip -\end{frame} +%\bu Coming back to the initial equality, +%\[ +%\int_\cX \left(-\mathcal L^{-1} \psi\right) \varphi \, d{\pi} = \dt \int_\cX \sum_{n=0}^{+\infty} \left( \Pi_\dt P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right) \left( \Pi_\dt \varphi \right) d{\pi}_\dt + \mathrm{O}\left(\dt^\alpha\right) +%\] -\begin{frame} - \begin{center} -\Huge{Conclusion and perspectives} -\end{center} -\end{frame} +%\bu Rewrite finally +%\[ +%\begin{aligned} +%\int_\cX \sum_{n=0}^{+\infty} \left( \Pi_\dt P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right)\left( \Pi_\dt \varphi \right) d{\pi}_\dt & = \int_\cX \sum_{n=0}^{+\infty} \left(P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right) \varphi \, d{\pi}_\dt \\ +%& = \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{\psi}_{\dt,\alpha}\left(q^{n},p^{n}\right)\varphi\left(q^0,p^0\right)\right) +%\end{aligned} +%\] +%\end{frame} \begin{frame} - {Main points: recall the outline!} + {Summary} -\bu {\bf Definition and examples of nonequilibrium systems} + \bu {\bf Definition and examples of nonequilibrium systems} + \begin{itemize} + \item Convergence to invariant measure + \item Perturbation expansion of invariant measure + \end{itemize} -\bigskip + \bigskip -\bu {\bf Computation of transport coefficients} -\begin{itemize} -\item a survey of computational techniques -\item linear response theory -\item relationship with Green-Kubo formulas -\end{itemize} + \bu {\bf Definition and computation of transport coefficients} + \begin{itemize} + \item Mobility, heat conductivity, shear viscosity + \item Linear response theory + \item Relationship with Green-Kubo formulas + \end{itemize} -\bigskip + \bigskip -\bu {\bf Elements of numerical analysis} -\begin{itemize} -\item estimation of biases due to timestep discretization -\item {\blue (largely) open issue: variance reduction} -\item {\red (not discussed) use of non-reversible dynamics to enhance sampling} -\end{itemize} + \bu {\bf Elements of numerical analysis} + \begin{itemize} + \item estimation of biases due to timestep discretization + \item {\blue (largely) open issue: variance reduction} + \end{itemize} \end{frame} -- cgit v1.2.3