---
abstract: 'Developing effective descriptions of the microscopic dynamics of many physical phenomena can both dramatically enhance their computational exploration and lead to a more fundamental understanding of the underlying physics. Previously, an effective description of a driven interface in the presence of mobile impurities @mikko, based on an Ising variant model @mend2001 and a single empirical coarse variable, was partially successful; yet it underlined the necessity of selecting additional coarse variables in certain parameter regimes. In this paper we use a data mining approach to help identify the coarse variables required. We discuss the implementation of this *diffusion map* approach, the selection of a similarity measure between system snapshots required in the approach, and the correspondence between empirically selected and automatically detected coarse variables. We conclude by illustrating the use of the diffusion map variables in assisting the atomistic simulations, and we discuss the translation of information between fine and coarse descriptions using lifting and restriction operators.'
author:
- 'Benjamin E. Sonday'
- Mikko Haataja
- 'Ioannis G. Kevrekidis'
bibliography:
- 'Paper1008.bib'
title: 'Coarse-graining the Dynamics of a Driven Interface in the Presence of Mobile Impurities: Effective Description via Diffusion Maps'
---

Introduction\[Intro\]
=====================

Driving an extended defect or domain wall in a system that contains stationary or mobile impurities lies at the heart of many physical phenomena, ranging from the classic example of the motion of grain boundaries in polycrystalline materials @cahn62 to the more recent modeling of ferroelectric domain wall dynamics @tybell2002, dislocation dynamics in metals containing stationary and mobile impurities @hahner, and stick-slip phenomena in tribology @cule96. While the interactions between isolated impurities and the domain wall are rather well understood at the microscopic level, translating the impact of such interactions to effective dynamics at mesoscopic or macroscopic scales remains a challenging problem.

In this work, we employ a microscopic model, a variant of the Ising model in which the domain wall moves under the influence of an external magnetic field while the mobile interstitial impurities are attracted to the domain wall. Our goal is to extract an effective description of the system based on a small number of coarse variables. In previous work @mikko, a one-dimensional description was proposed based on empirical evidence and physical intuition. While this reduced description accurately represents the Ising variant model in some regimes, it fails to capture the more salient features of the coupled impurity-domain wall dynamics. Clearly, a more systematic dimensionality reduction (i.e., coarse-graining) tool, which relies less on intuition, is desired.

To this end, diffusion maps have recently emerged as an effective nonlinear dimensionality reduction tool @belkin2003 [@coifman1; @coifman2], see also @tenen2000. Diffusion maps find a "best" (possibly nonlinear) low-dimensional underlying manifold that (approximately) contains the data set of interest. Given a data set of system states, the diffusion map approach first requires a scalar *similarity measure* between each pair of system states. When the similarity measure is based upon dynamic proximity, it is expected that the resulting manifold parameterizes the meaningful dynamics of the system states provided. When the data are in the form of vectors with real entries (e.g. concentrations of chemical species in a chemical reaction network) the standard Euclidean distance may be used to construct such a similarity measure (presumably, system states with similar concentration profiles are dynamically close as well). Typically (see the Appendix) the similarity measure becomes negligible beyond a local neighborhood of each data vector: large Euclidean distances do not provide useful similarity information. This is related to the notion that on a general nonlinear manifold, large Euclidean distances can not be expected to reliably approximate geodesic distances. These pairwise similarities are used in the diffusion map algorithm to construct a matrix (the discretization of a diffusion operator) whose leading eigenvectors (discretizations of the corresponding eigenfunctions) are computed. If a spectral gap is detected between a few leading eigenvalues and the remaining ones, the corresponding few eigenvectors can help *nonlinearly* embed the data set in $\mathbf{R}^n$, thereby parameterizing the underlying manifold. The Euclidean distance between snapshots in the new, reduced space, is the *diffusion distance*, a notion similar to geodesic distance on a manifold (see, however, @lafonlee); the process thus transforms our local understanding of similarity between data to a global geometry.

When this data-mining approach is applied to the Ising variant model simulations, we find that we can extend the previously obtained one-dimensional coarse variable description to a more quantitative description involving two coarse variables. Treating the Ising model variant as a stochastic dynamical system in these two coarse variables, we postulate a two-dimensional Langevin (and the corresponding Fokker-Planck) equation description. The effect of all other degrees of freedom is incorporated in the drift and diffusion coefficients of the Fokker-Planck equation; we extract these coefficients through either long-term or multiple short-burst Monte Carlo simulations, and use them to extract system-level statistics for the problem.

Dimensionality reduction of physical systems and their models can be useful for many reasons. An effective reduced description provides intuition about the physical system, grants accelerated retrieval of dynamical quantities of interest, and allows for increased computational efficiency. The Ising model variant discussed in this paper has high underlying dimensionality and no transparent effective description. Even when there is no clear correspondence between diffusion map coordinates and physical variables, the computer-assisted detection of coarse variables can still be a useful step in gaining intuition about which physical features of the model are dynamically important.

The rest of this paper is organized as follows. Key features of the Ising variant model and the dynamics it exhibits are presented in section \[keyfeat\]. The choice of a similarity measure between nearby system snapshots and its use in the data-mining process is described in section \[distmet\]. The extraction of a two-dimensional effective stochastic description follows in section \[resultdescr\]. The selection of a physically meaningful second coarse variable is also explored here. Finally, the translation of information between the coarse and the physical system descriptions is discussed in section \[lifting\], where we also show how to accelerate the calculation of dynamical quantities of interest using our effective description. The diffusion map algorithm is discussed in the Appendix.

The kinetic Monte Carlo simulation and its dynamics\[keyfeat\]
==============================================================

Our kinetic Monte Carlo simulations are performed on a $32\times32$ square lattice (grid of domain spins) with associated interstitial sites, each of which may or may not contain an impurity; this is the same model simulated in @mikko, and we describe it here only briefly for completeness. Evolution on this lattice is governed by the Hamiltonian $$\mathcal{H} / k_BT = -J/2 \sum s_i s_j - h \sum s_i + E_0/4 \sum
\epsilon_\alpha |\sum s_j|,$$ where $h$ is the external magnetic field, acting normal to the lattice plane. Individual spins ($``+1"$ or $``-1"$) are denoted as $s_i$, $J$ accounts for the domain wall energy, $E_0 > 0$ quantifies the attraction of the impurities by the domain wall, and $\epsilon_\alpha = 1$ if an impurity lies in interstitial site $\alpha$ and $0$ otherwise; $c_{imp}$ is the impurity concentration.

The domain is periodic in the vertical direction and extends, in principle, infinitely far to the left and right @mikko: all domain spins to the left of the simulation grid are assumed to be $+1$ and all domain spins to the right to be $-1$. We can think of the system as lying on an infinitely long cylinder of circumference $32$. Our simulations always involve a *domain wall* (a boundary between $+1$ spins and $-1$ spins), and our chosen cylinder *length* of $32$ is sufficient to contain the extent of this domain wall. Three typical system snapshots are shown in Fig.\[typsiss\] (in our figures we will typically "crop" the domain in length, but not circumference, so that only the portion containing the domain wall is shown). As the simulation evolves, the domain wall moves to the right under the influence of the external magnetic field; to "keep up" with this motion, we continually shift the computation along the axis of the cylinder, keeping the domain wall close to the middle of our $32
\times 32$ simulation grid. The impurities diffuse interstitially and are attracted to the domain wall @mend2001; the boundary conditions for impurity motion are periodic in both the $x$ and the $y$ directions. At each time step of the kinetic Monte Carlo simulation, we select either a domain spin to flip or an impurity to move. The resulting energy change is calculated, and the move is accepted if the energy change is negative, or with probability $e^{-\frac{\Delta \mathcal{H}}{k_BT}}$ if it is positive. The simulation clock advances by $\frac{1}{32 \times 32}$ units whenever a spin is sampled. In our simulations, we used the parameters $h= 0.12$, $c_{imp} =
0.01$, $J = 2.0$, and $E_0 = 5.75$.

![Three typical system snapshots (above) arising in a sequence of "stick-slip" events. Impurities are black dots, $+1$ spins are red, and $-1$ spins are blue; the domain extends infinitely to the left(right) of the the part shown and it is periodic in the vertical direction. The domain wall in snapshot X is practically stationary; in snapshot Y it is moving (more or less) smoothly to the right; in snapshot Z it just started moving smoothly to the right after "engulfing" its domain wall impurities (see text).\[typsiss\]](typsis2_RB.eps){height="170mm" width="130mm"}

Diffusing impurities impede domain wall movement, while the external magnetic field promotes domain wall motion. Large external fields and small impurity-wall attraction result in smoothly traveling domain walls, while small external fields and large impurity-wall attraction result in walls which hardly move. The parameters in Fig.\[typsiss\] correspond to a compromise between these two competing forces. As the snapshots in Fig.\[typsiss\] summarize, the model is capable of exhibiting a type of bistability: it switches between a (more or less) constant speed motion to the right and (more or less) stationarity. One metastable state corresponds to an impurity-rich, stationary domain wall, while the second metastable state corresponds to an impurity-deficient, mobile domain wall. The pattern of behavior is as follows. A domain wall which is relatively flat and impurity-deficient moves to the right norman60@example.net. As it moves, however, it "collects" impurities and eventually becomes so overwhelmed that it practically stops moving. Even so, the domain wall may become rough when fluctuations result in an uneven impurity distribution *along the wall* in the vertical direction. Portions of the wall which are impurity-rich move very little, while impurity-deficient portions continue moving to the right. As the resulting wall roughness increases, mobile, impurity-deficient regions of the wall are often seen to "engulf" impurity-rich regions, leaving behind a pocket of impurities and a relatively flat, impurity-deficient wall which travels to the right virtually uninhibited (see Fig. \[rarEvent\]). The cycle continues.

Rationalizing hysteresis observations as a function of parameters (such as the impurity-domain wall interaction strength $E_0$) was the main goal of @mikko; indeed, a rationalization was obtained by using the number of impurities on the domain wall, $N$, as a coarse variable. The selection of this variable was based on long observations of system transient simulations and intuition. Model reduction, in the form of an effective Fokker-Planck description of the domain wall motion dynamics in terms of this single coarse variable $$\begin{aligned}
\frac{\partial P(N,t)}{\partial t} = -\frac{\partial}{\partial
N}\left[V(N)\, P(N,t) \right] + \\
\frac{\partial^2}{\partial N^2}\left[D(N) \, P(N,t) \right],
\nonumber\end{aligned}$$ yielded an effective free energy surface with two "wells": one corresponding to domain wall motion, and one to stationarity. $P(N,t)$ is here the probability that at time $t$, the number of impurities on the domain wall is $N$; the quantities $V(N)$ and $D(N)$ appearing in the effective Fokker-Planck equation are not available in closed form, and were estimated from simulation data.

Changing parameters changed the relative depth and barrier height between these wells, making hysteresis more or less apparent, depending on the typical time scale of an observer. This type of effective description was comparably successful over a wide set of atomistic model parameter values @mikko. This apparent success was, however, partially mitigated by the fact that the one-dimensional reduced dynamic description was less quantitative for small values of $N$ (see Fig.3 in @mikko). This gradual deterioration of the predictive strength of the reduced one-dimensional stochastic model strongly suggested the existence of additional "hidden variables", variables whose statistics do not become quickly slaved to $N$ during simulation, in the low-$N$ regime.

In attempting to guess what such additional important coarse observables might be, one might return to the observation of long simulations, especially in the moderate wall impurity regime, when the domain wall transitions from stationarity to motion (where, as we discussed above, a type of roughening of the domain wall is often observed).

![Three system snapshots just before (left), during (middle), and just after (right) a transition from trapped domain wall to smoothly moving domain wall. Transitions like this one suggest that the number of impurities on the domain wall may not completely describe the system state, and that domain wall shape may also become important.\[rarEvent\]](rare_event_RB.eps){height="170mm" width="92mm"}

Data mining, diffusion maps and a distance metric \[distmet\]
=============================================================

While physical understanding and intuition may suggest additional physical observables for model reduction purposes (we already mentioned observations of interface roughness, and we will return to it below), a systematic, computer-assisted procedure for detecting/suggesting such additional variables is clearly desirable. The use of manifold-learning techniques (such as Isomap or Diffusion Maps) has started to be explored in the context of reducing atomistic models (see, e.g. @cecilia [@frewenchem; @laing]).

We will employ the diffusion map approach here. As described in the Appendix, we start with a reasonably extensive ensemble of simulation snapshots, and with a scalar similarity measure $W_{ij}$ of the "closeness" between each pair ${i,j}$ of these snapshots. To achieve a low-dimensional embedding for a data set of $M$ individual $d$-dimensional vectors $\vec X_1$,...,$\vec X_M$ with real entries, the Euclidean distance (possibly weighted) can help construct such a similarity measure: $$W_{ij} = \text{exp} \left[ - \left(\frac{||\vec X_i - \vec
X_j||}{\epsilon} \right)^2 \right].$$ Here $\epsilon$ provides a separation scale beyond which the data are considered effectively dissimilar. An interpretation of this scale is that the Euclidean distance between data points is "meaningful" (meaningless) in determining data similarity when it is smaller (larger) than $\epsilon$. An "inner" and an "outer" distance play a role in the diffusion map approach. The inner distance (e.g. the Euclidean distance above) can be the basis of a meaningful similarity measure between data points when it is small; the diffusion map approach builds on such locally meaningful similarity measures to construct a *global* distance, the so-called *diffusion distance*, that meaningfully quantifies data similarity even when the data are far away from each other. As a byproduct, the procedure also provides a new set of coordinates (based on eigenfunctions of a diffusion operator, or, in discretized form, on eigenvectors of a Markov matrix) which may prove useful in embedding the data. If we are lucky, a spectral gap will appear in the spectrum of the diffusion operator, and a relatively small number of these new coordinates can be used in reducing the dimensionality of the data (and, thus, hopefully, of the model itself). If the dimensionality of the data can indeed be reduced, the diffusion distance is the Euclidean distance in the new space in which the data is embedded. The essence of the process is therefore to transform a "trustworthy" local distance into a trustworthy global one in a new set of coordinates; and while the Euclidean distance is an obvious inner distance candidate, it is by no means the only one.

If physical understanding or intuition can suggest a locally meaningful inner distance (beyond the obvious Euclidean norm), it can serve as the basis of a successful reduction process. Our data (our system snapshots) come in the form of 2048 "pixels": the 1024 spin lattice sites have either $+1$ or $-1$ entries, while the 1024 interstitial sites have either $1$ (impurity present) or $0$ entries. At first thought, one might consider vectorizing each of the system snapshots and using the Euclidean distance as an inner distance; yet the entry values are clearly arbitrary, and one needs to devise some relative weight for the spin and impurity entries. Fig. \[dynclose\] shows an example of two snapshots that have large pixel-by-pixel differences, yet are close dynamically, in the sense that a brief simulation can easily transform the one to the other.

![Two snapshots exhibiting large pixel-by-pixel differences which, however, are close dynamically: they are likely to transform to each other via a small number of impurity diffusion steps and a few quick, energetically favorable spin flips.\[dynclose\]](dynsim_RB.eps){height="102mm" width="130mm"}

Clearly, coming up with a good inner distance for our data is nontrivial. We could possibly consider smoothing the lattice into spatial fields (with the help of localized bump functions at each lattice and interstitial site), or using Wasserstein-type (earthmover's distance) metrics. However, since we are interested in constructing a reduced *dynamic* model of the process, we decided to search for a local distance that would incorporate some form of dynamic similarity of snapshot pairs: how easy/likely it is for the direct simulation to transform one to the other.

We started by exploring the set of likely system snapshots through "long" kinetic Monte Carlo simulations; we regularly sampled the simulation, obtaining $M$ system snapshots, denoted $\{\mathbf{P}_i
\}^M_{i=1}$. Inspection of the Hamiltonian shows that each impurity is free to diffuse along horizontal or vertical straight segments of the domain wall without energetic penalty and that these diffusion steps occur over times very short relative to the domain wall evolution; hence, two snapshots that differ only by such diffusion steps can be considered dynamically close. In a process we call *simulated diffusion* (and only for the purpose of "off-line" computing our proposed inner distance) we replace each impurity on a straight segment of the domain wall by a "smeared out" version of itself. Specifically, we divide the domain wall into distinct straight segments of varying lengths and determine the number of impurities on each segment. Then, on each segment, we distribute these impurities uniformly on the interstitial sites (see Fig. \[simdiff\]). After this simulated diffusion, we "summarize" our system by a vector of length $32$, $\vec z$, adding up the fractions of *diffused impurity* in each row (again, see Fig. \[simdiff\]). Entries of the vector with value greater than one correspond to horizontal segments of the domain wall, and entries with values between zero and one correspond to vertical segments; zero entries may correspond to either. We have found that even though this summary appears, at first sight, to be an enormous loss of information about the system, it is nevertheless a useful one: the behavior of the system over medium time scales is not determined by the specific positions of domain spins and impurities, but rather, by the overall shape of the domain wall, i.e. its horizontal and vertical segments and how much diffused impurity is present on each of these. The vector contains exactly this type of information.

![The *simulated diffusion* used in our similarity measure construction. In a time interval that is short relative to the domain wall motion timescale, impurities diffuse along the boundary resulting in an apparent short-term equilibrium. The only energetic barrier to this short-term motion is the presence of domain wall corners (see text) . After simulated diffusion, we add up the total amount of *diffused impurity* in each row and place the answer in the corresponding entry of a vector (rightmost in the figure).\[simdiff\]](simdiff_new_RB.eps){height="170mm" width="130mm"}

Two system snapshots $\mathbf{P}_1$ and $\mathbf{P}_2$ are (through simulated diffusion) mapped to two $32$-long vectors $\vec z_1$ and $\vec z_2$. Because of the geometry (periodicity in the vertical ($y$) direction, reflection symmetry around the $x$-"axis") each vector $\vec z_i$ is equivalent to (has zero distance from) 63 other vectors obtained by index reflection (e.g. $[\vec z_i^{(1)}, \vec
z_i^{(2)} ,..., \vec z_i^{(31)}, \vec z_i^{(32)} ] \sim [\vec
z_i^{(32)}, \vec z_i^{(31)}, ..., \vec z_i^{(2)}, \vec z_i^{(1)}]$) and cyclic index shifts (e.g. $[\vec z_i^{(1)}, \vec z_i^{(2)}, ...,
\vec z_i^{(31)}, \vec z_i^{(32)}] \sim [\vec z_i^{(k+1)},\vec
z_i^{(k+2)},...,\vec z_i^{(32)}, \vec z_i^{(1)},\vec
z_i^{(2)},...,\vec z_i^{(k)}]$); the set of all these equivalent vectors is denoted as $\mathbf{Z}_i$. The problem is also translationally invariant in the horizontal direction; yet, our vector snapshot summary has automatically accounted for this symmetry. Our proposed inner distance $\mathbf{d}$ is then $$\mathbf{d}(\mathbf{P}_1,\mathbf{P}_2) = \text{min}_{\vec z_2 \in
\mathbf{Z}_2} ||\vec z_1-\vec z_2|| = \text{min}_{\vec z_1 \in
\mathbf{Z}_1} ||\vec z_1-\vec z_2||,$$ where for the rest of this paper, $||\,\,\, ||$ denotes the $L_2$ norm unless otherwise specified. We will show below how we used this inner distance; we found it quite robust in that small variations in its definition (e.g. smearing impurities as a Gaussian, rather than uniformly along straight segments) led to comparable diffusion map embeddings.

The resulting two-dimensional description \[resultdescr\]
=========================================================

Using our inner distance, we computed (along the lines of the Appendix) the eigenvalues of the Markov matrix resulting from the ensemble of system snapshots. In addition to the trivial eigenvalue at 1, two eigenvalues appear significant, suggesting that we might be able to parameterize the snapshots using their components in the corresponding two eigenvectors, yielding a two-dimensional description. The data plotted in terms of these two diffusion map coordinates are shown in Figs. \[jac2d\] A and \[jac2d\] B.

![Coarse variables: data mining and/or intuition. Figures A and B show the data ensemble plotted in terms of their components in the two leading (nontrivial) diffusion map eigenvectors. Figures C and D show the same data plotted now in terms of their respective number of impurities on the domain wall $N$ (x-axis) and their "roughness" $\mathbf{J}$ (y-axis, see text). To assist in correlating the two representations, figures A and C are colored by "roughness"; figures B and D are colored by number of impurities on the domain wall. These figures visually suggest that the determinant of the Jacobian of the transformation from physical variables to diffusion map coordinates is everywhere nonzero.\[jac2d\]](jac2d_2.eps){height="170mm" width="130mm"}

Relating diffusion map coordinates to physical observables
----------------------------------------------------------

Although the diffusion map approach is capable of discovering a low-dimensional parameterization of our system snapshot data ensemble which is mathematically optimal in a certain sense (e.g. see @coifman1 or @belkin2003), it does not provide physical intuition about the embedding. Thus, it makes sense to explore how the selected coordinates correlate with physical observables. Based on the intuition in @mikko, we test whether the diffusion map variables correlate with our original single coarse variable: the number of impurities on the domain wall. Figure \[jac2d\] D colors the data in terms of their domain wall impurity number; a visual inspection of the "color layering" in the plot suggest a strong, but not perfect, correlation of this observable with the first detected diffusion map variable. Notice that the layering becomes increasingly slanted towards the right, that is, towards low impurity counts. This is consistent with @mikko: the number of impurities provided there a good coarse variable for high impurity counts (towards the left in the figure). Yet the one-dimensional model deteriorated significantly at low impurity counts (towards the right in the figure); this is precisely what prompted our search for additional coarse variables.

Based on physical intuition, we expect a good second observable to contain information associated not only with the number of domain wall impurities, but also with the domain wall shape. One way to quantify this might use the nonuniformities in the impurity distribution along the domain wall length; we have repeatedly observed that impurity-rich domain wall regions exhibit high local curvature, forming apparent "bubbles" that may be engulfed, thus releasing the newly impurity-deficient wall. We instead constructed a more direct measure of the wall roughness associated with its high local curvature; we denote this measure as $\mathbf{J}$ and compute it as follows.

Consider the sequence $\{\mathbf{F}_i\}_{i=1}^{31}$ of $32 \times
32$ spin lattices with perfectly vertical domain walls: $\mathbf{F}_1$ has only one (the first, leftmost) column of $+1$ spins, and all its other columns are $-1$ spins; $\mathbf{F}_2$ has two leftmost $+1$ spin columns, and so on. For a system snapshot $\mathbf{P}$ with domain spin lattice entries $\mathbf{S}$, $\mathbf{J}$ is found by minimizing (over the set of lattices $\{\mathbf{F}_i\}_{i=1}^{31}$) the square of the Frobenius norm $$\mathbf{J}(\mathbf{S}) = \text{min}_{\{\mathbf{F}_i\}_{i=1}^{31}}
||\mathbf{F}_i-\mathbf{S}||_F^2.$$ It might be, at first thought, surprising that the computation of our roughness measure involves no impurity information whatsoever; yet the placement of domain wall impurities and the overall domain wall shape are highly correlated: a rough domain wall which contains no impurities will very quickly (in the simulation) become flat. The combination of the four panels (considering also their shading) strongly suggests that there exists a bijection between the two coordinates discovered through the diffusion map process and the two coordinates arising from physical considerations. Indeed, the Jacobian of the transformation between the two pairs of coordinates does not become singular over the data ensemble, as we have confirmed numerically.

An effective dynamic model \[mathEffDescr\]
-------------------------------------------

Along the lines of @mikko, we attempt to describe the reduced system dynamics in terms of an effective two-dimensional Fokker-Planck equation,

$$\begin{aligned}
\frac{\partial P(\psi_1,\psi_2,t)}{\partial t} = - \sum_{i=1}^2
\frac{\partial}{\partial \psi_i}
\left[D_i^1(\psi_1,\psi_2)P(\psi_1,\psi_2,t) \right] + \\
\sum_{i,j=1}^2 \frac{\partial^2}{\partial \psi_i \partial \psi_j}
\left[D^2_{ij}(\psi_1,\psi_2)P(\psi_1,\psi_2,t) \right]. \nonumber\end{aligned}$$

Here the drifts and diffusion coefficients in the equation can be estimated by $D_i^1(\psi_1,\psi_2) = \text{lim}_{\Delta t
\rightarrow 0} \mbox{$<\psi_i(t+\Delta t)-\psi_i(t)>/\Delta t$}$ and $2 D_{ij}^2(\psi_1,\psi_2) = \text{lim}_{\Delta t \rightarrow 0}
\mbox{$<\left(\psi_i(t+\Delta t)-\psi_i(t)
\right)\left(\psi_j(t+\Delta t) - \psi_j \right)>/\Delta t$}$. As in the one-dimensional case, setting $\frac{\partial
P(\psi_1,\psi_2,t)}{\partial t} = 0$ gives the steady state probability distribution $P_{FP}(\psi_1,\psi_2)$. If the effective Fokker-Planck model is accurate, $P_{FP}$ will be well approximated by histograms constructed using long kinetic Monte-Carlo simulation data, $P_{MC}(\psi_1,\psi_2)$; see Fig. \[eqdistt\].

![Two views of the steady-state probability distribution obtained by long Monte Carlo simulation viewed from two different angles; the distribution is truncated at $P = 4 \times 10^{-3}$ but extends approximately to the height $P = 8 \times 10^{-3}$. \[eqdistt\]](eqdisttttt.eps){height="170mm" width="80mm"}

The $D_{ij}^2$ and $D_i^1$ may be estimated in several ways, two of which are discussed here. The first is to choose a particular point $(\psi_1,\psi_2)$ in diffusion map space (actually, a small "box" surrounding it), locate several instances it arises in long kinetic Monte-Carlo simulations, and then for each of these instances, record the observed diffusion map coordinates $\Delta t$ later. Averaging these results as shown above will provide a numerical estimate of the $D_{ij}^2$ and $D_i^1$.

Fig. \[flowmap\] shows the "drift vector field", the plot of $D_i^1(\psi_1,\psi_2)$ estimated this way.

![The estimated effective drift vector field in diffusion map coordinate space, along with representative system snapshots at five selected locations to assist rationalizing the dynamics. Snapshots 1 and 2 lie in a high-impurity, low-roughness region, while 5 is located in a low-impurity, low-roughness region. The domain wall of snapshot 3 will evolve towards the apparent saddle point, while the wall in 4 appears to be increasing its roughness, preparing to engulf its impurities. \[flowmap\]](flowmapp_RB.eps){height="170mm" width="130mm"}

At each point $(\psi_1,\psi_2)$ on the diffusion map plane, the vector $\left[D_1^1(\psi_1,\psi_2),D_2^1(\psi_1,\psi_2)\right]$ is plotted. The estimation has been performed in diffusion map coordinate space; we are, however, able to directly transform to the $\mathbf{J}$/$N$ space as illustrated in Fig. \[jac2d\]; we have included in Fig. \[flowmap\] representative snapshots at selected points to assist physical discussion of the drift dynamics. Visual inspection of the drift vector field (ignoring, for the moment, the stochastic component of the dynamics) shows exactly what we might expect for this system: starting on the right (somewhere in the circle, shown in the blowup) the domain wall remains relatively flat as it gathers impurities; it eventually engulfs these impurities by first becoming more rough. There are two apparent steady states in this drift vector field: one to the right, where an impurity-deficient domain wall "slips" with some apparent speed; and, one to the left, where an impurity-rich domain wall "sticks". A saddle-type steady state can be seen for the drift vector field; it lies in a region characterized by moderate roughness and moderate number of wall impurities. Here domain walls (again ignoring, for the moment, the stochastic component of the dynamics) will, under small perturbations, either go on to gather more impurities and "stick", or they may become more rough, engulf their impurities, and cycle back to low-impurity fast-moving states. Fig. \[speedFig\] summarizes much of this information by plotting estimated domain wall velocity vs. diffusion map coordinate. $(\psi_1,\psi_2)$.

![Estimated domain wall speed as a function of the diffusion map coordinates. Velocity is in units of length per unit of time, where one unit of length is defined as the width of a domain spin and the unit of time was defined above. High speeds occur when the domain wall is free of impurities (at the center of the pink circle) and/or it has high roughness (taking Fig. \[jac2d\] into account). \[speedFig\]](speedFig.eps){height="170mm" width="92mm"}

The Ising model variant we study here does not appear to be a gradient system in two effective dimensions; observe, for instance, the curl-intense region of the drift vector field in Fig. \[flowmap\]. This is precisely the region which suggested the need for a second coarse variable in @mikko. This, unfortunately, does not allow us to simply and directly obtain the steady-state probability distribution in terms of an effective free energy that can be explicitly constructed from the estimated Fokker-Planck coefficients. We can, however, proceed to numerically solve, using standard discretization techniques (e.g. finite elements), for the steady state of the (numerically estimated) Fokker-Planck equation in two-dimensional diffusion map coordinate space.

A second approach to estimating the Fokker-Planck equation coefficients is motivated by the corresponding Langevin stochastic differential equation and uses short bursts of kinetic Monte-Carlo simulations appropriately initialized at particular values of the diffusion map coordinates. Indeed, if the diffusion map coordinates we constructed are dynamically meaningful (which we certainly expect because our similarity measure was dynamically motivated), initializing an ensemble of short simulation bursts on demand at specified diffusion map coordinates allows us to estimate the drift and diffusion coefficients by processing the results of short burst simulations through the above formulas. A new computational challenge arises now: we may wish to initialize the kinetic Monte Carlo simulations at points in diffusion map space not included in our original data ensemble. Furthermore, the simulations will produce new snapshots in physical space which are also not included in our original data ensemble. Clearly, for this approach to be viable, we need operators that successfully translate information between physical and diffusion map space. These are the "lifting" (coarse to fine, diffusion map to spin/impurity lattice) and "restriction" (fine to coarse, spin/impurity lattice to diffusion map) operators of equation-free computation @man1 [@man2].

Lifting and Restriction \[lifting\]
===================================

The *restriction* operator, which finds the diffusion map coordinates of a new system snapshot $\mathbf{P}_{\text{new}}$ that was not part of the original data ensemble, is implemented via Nyström extension (see the Appendix). It involves the computation of the similarity of the new data point to the points of the original ensemble (possibly only the nearby points).

*Lifting*, the inverse problem of finding a new physical initial condition with prescribed diffusion map coordinates, is significantly more computationally involved than restriction; because of the nonlinear nature of the diffusion-map based model reduction, it is significantly more complicated than, say, reduction using Principal Component Analysis. Lifting is a "one-to-many" operation; there are many physical configurations that reduce to (practically) the same two diffusion coordinate values. Here the lifting problem is solved through a simulated annealing algorithm @sciencesimann, which is modified by linking to the kinetic Monte Carlo simulation as we now describe. We note that each lift requires *multiple* applications of the restriction operator.

The lifting procedure
---------------------

The only inputs to the lifting procedure are the two target diffusion map coordinate values, $\psi_1^{\text{targ}}$ and $\psi_2^{\text{targ}}$. We start with an arbitrary trial system snapshot (possibly a snapshot of the original ensemble whose diffusion map coordinates are close to the target values) and evaluate its diffusion map coordinates using Nyström extension. We attempt to modify this trial system snapshot by either changing the number of wall impurities (preserving the total lattice impurity count) or by flipping a small neighborhood, or "block", of spins close to the domain wall; we compute the modified diffusion map coordinates again through Nyström extension. The modification is accepted if its diffusion map coordinates are closer to the target values; if not, the change is accepted with some probability that depends exponentially on the diffusion map coordinate change. The concrete steps are as follows (see Fig. \[figsimann\]):

1.  Start with a random initial system snapshot.

2.  Evolve (through kinetic Monte Carlo) the system briefly, allowing it to *heal* for a relatively short amount of simulation time. Here, we choose 20 units of time. Denote the healed system snapshot as $\mathbf{P}_{\text{curr}}$. Healing is intended to get rid of improbable snapshot features artificially generated by the simulated annealing steps. Determine the diffusion map coordinates of this healed snapshot using Nyström extension. In order to decide about accepting the modification or not, we need to prescribe an *annealing temperature*, which we choose here to be $T = ||\Psi_2(\mathbf{P}_{\text{curr}})-[\psi_1^{\text{targ}},
    \psi_2^{\text{targ}}]||$ (see Appendix for meaning of $\Psi_2$). Depending on how far away from the desired coordinate values we are (i.e., depending on the annealing temperature) we allow for bigger perturbations; we set the height of the block of spins to be flipped to $H = \text{min}\left(18,\text{max}(3,\text{ceil}(18\,
    U_1([0,1])\, \sqrt T))\right)$, and its width to $W =
    \text{min}\left(6,\text{max}(1,\text{ceil}(6\, U_2([0,1])\, \sqrt
    T))\right)$, where $U_1$ and $U_2$ are uniform random variables. This way, for moderate $T$, $HW \approx 27T$. $H$ is chosen longer than $W$ on average, in order to keep the perturbed domain wall flatter in the steps that follow. These choices are motivated by observing statistics of fluctuations during the system dynamic evolution; the procedure is quite robust to the details of such choices.

3.  \[trialsnap\] Generate a new trial snapshot $\mathbf{P}_{\text{trial}}$ by either

    1.  Adding an impurity to an empty spot on the domain wall boundary, or by removing an impurity on the domain wall boundary (keeping the total impurity count constant).

    2.  Replacing a block of $+1$ spins intersecting the domain wall with $-1$ spins. The block is $H$ rows high and $W$ rows wide, and its placement is selected randomly and uniformly from the set of all possible placements that intersect the domain wall. Without intersection with the domain wall, flipping the spin block would have no effect; blocks taken in regions entirely to the right are already comprised of $-1$ spins, while blocks taken in regions entirely to the left of the domain wall would flip back almost immediately. If the selected block contains impurities, these impurities are projected to the left, so that they remain on the new domain wall. The system is allowed to heal for $HW$ units of time.

4.  The new trial snapshot is accepted (we set $\mathbf{P}_{\text{curr}} = \mathbf{P}_{\text{trial}}$) if its diffusion map coordinates (computed through Nyström extension) are closer to the target diffusion map coordinates, or with probability $$\begin{aligned}
    P = \text{exp} [-\frac{1}{T}(||\Psi_2(\mathbf{P}_{\text{trial}})-[\psi_1^{\text{targ}},\psi_2^{\text{targ}}]||-\\
    ||\Psi_2(\mathbf{P}_{\text{curr}})-[\psi_1^{\text{targ}},\psi_2^{\text{targ}}]||)]
    \nonumber\end{aligned}$$ otherwise.

5.  The current annealing temperature is set to $T =
    ||\Psi_2(\mathbf{P}_{\text{curr}})-[\psi_1^{\text{targ}},\psi_2^{\text{targ}}]||$, and $H$ and $W$ are reset accordingly. If $T<\rho$, exit; $\mathbf{P}_{\text{curr}}$ is the desired snapshot; otherwise, go back to step \[trialsnap\]. In our experience $\rho = 0.05$ gives satisfactory snapshots, close enough to the target diffusion map variables.

The healing steps, interspersed with the simulated annealing steps, appear to be important in giving plausible physical snapshots with the desired diffusion map coordinates (i.e. snapshots whose diffusion map coordinates evolve typically slowly upon further simulation). There are many physical snapshots which (through Nyström extension) possess the same two diffusion map coordinates (we have already said that lifting is a one-to-many operation). What our simulations suggest is that, among these snapshots, we can find both "mature" ones, for which further diffusion map coordinate evolution is slow, and "unhealed" ones, for which further diffusion map coordinate evolution quickly brings us to a different "mature" point on our two-dimensional manifold with different diffusion map coordinates. Our lifting process typically takes about 30 cycles, each requiring a Nyström extension.

There are a number of more or less *ad hoc* choices in the above algorithm, such as the type and scale of perturbations chosen, the selection of the annealing temperature, the size of $H$ and $W$, the healing time etc. Having to make such choices is part of any simulated annealing algorithm; we have found that our results are quite robust to modifications in the specific choices presented above.

![Lifting using simulated annealing alternating with simulation-induced healing. Only accepted simulated annealing steps are shown. The perturbation scale (flipped spin block size), annealing temperature, and healing time become increasingly smaller as the target is approached. Brief healing paths tend to flow along drift vector field trajectories. \[figsimann\]](simannfig_RB.eps){height="170mm" width="130mm"}

Testing the lifting operator
----------------------------

The variables detected by the diffusion map process are useful in providing a good parameterization of the snapshot data; plotting the data in terms of these coordinates can provide very helpful visual summaries of the overall dynamics (e.g. Fig. \[flowmap\]), and can even help develop intuition and useful hypotheses about the system. If a good lifting operator can be constructed, however, these coordinates can be even more useful: they can assist in the design of computational experiments that extract system-level information from the problem. One might wait, for example, for a long kinetic Monte Carlo simulation to sample enough domain wall pinning/depinning events to estimate the corresponding characteristic time; alternatively, short bursts of kinetic Monte Carlo simulations can be designed, by appropriately initializing them through lifting at selected diffusion map coordinates, to extract this information exploiting the effective Langevin (or Fokker-Planck) model discussed above. Our variables are dynamically meaningful if a Langevin (or associated Fokker-Planck) dynamic model in terms of only these variables is successful in describing the problem (if, for example, the Fokker-Planck coefficients estimated through long kinetic Monte Carlo simulations and the ones estimated through short bursts following a lifting procedure are practically the same). We expect this to be the case since our similarity measure was dynamically motivated. Tables \[liftingTableDrift\] and \[liftingTableDiff\] show precisely such a test of our two diffusion map variables along with the correct performance of our lifting operator: Fokker-Planck coefficient estimates using each of the two approaches at seven selected locations are favorably compared.

   $\psi_1^{\text{targ}}$   $\psi_2^{\text{targ}}$       $D_{1\,MC}^1$           $D_{1\,SB}^1$
  ------------------------ ------------------------ ----------------------- ------------------------
           -1.65                    -1.00            -1.76e-2 $\pm$ 4.3e-4   -1.80e-2 $\pm$ 2.1e-4
            1.25                     1.00            -7.81e-3 $\pm$ 3.2e-4   -7.63e-2 $\pm$ 1.6e-4
           -0.20                    -0.60            -4.43e-3 $\pm$ 5.1e-4   -4.20e-3 $\pm$ 2.5e-4
            0.75                     1.60            5.20e-2 $\pm$ 6.9e-4     5.29e-2 $\pm$ 3.4e-4
            1.00                     0.20            -8.75e-3 $\pm$ 2.7e-4   -8.81e-3 $\pm$ 1.3e-4
            0.90                    -0.40            -8.52e-3 $\pm$ 5.4e-5   -8.49e-3 $\pm$ 3.2e-5
            1.50                    -0.60            -1.49e-1 $\pm$ 1.6e-3   -1.51e-1 $\pm$ 7.76e-4

  : Drift coefficients computed for certain target diffusion map coordinates via the two methods discussed in the text. Only one of the two drift coefficients is shown. "MC" stands for "Monte Carlo", while "SB" stands for "short burst".\[liftingTableDrift\]

   $\psi_1^{\text{targ}}$   $\psi_2^{\text{targ}}$      $D^2_{11\,MC}$         $D^2_{11\,SB}$
  ------------------------ ------------------------ ---------------------- ----------------------
           -1.65                    -1.00            4.52e-4 $\pm$ 6.2e-6   4.39e-4 $\pm$ 2.8e-6
            1.25                     1.00            2.53e-4 $\pm$ 3.3e-6   2.46e-4 $\pm$ 1.4e-6
           -0.20                    -0.60            6.48e-4 $\pm$ 8.5e-6   6.29e-4 $\pm$ 4.3e-6
            0.75                     1.60            1.20e-3 $\pm$ 1.6e-5   1.14e-3 $\pm$ 5.7e-6
            1.00                     0.20            1.84e-4 $\pm$ 2.2e-6   1.78e-4 $\pm$ 1.2e-6
            0.90                    -0.40            7.25e-6 $\pm$ 9.5e-7   1.03e-5 $\pm$ 7.2e-7
            1.50                    -0.60            6.20e-3 $\pm$ 8.1e-5   6.02e-3 $\pm$ 3.8e-5

  : Diffusion coefficients computed for certain target diffusion map coordinates via the two methods explained above. To save space, only one of the three diffusion coefficients is shown. Again, "MC" stands for "Monte Carlo", while "SB" stands for "short burst".\[liftingTableDiff\]

The agreement is good, but not perfect: possible reasons for the discrepancies range from the need to collect longer data and more short replica simulations, to the details of the estimation algorithm and even the possible inadequacy of a white noise model in the effective Langevin description. Tightening the $\rho = 0.05$ requirement in the simulated annealing, or keeping more than two variables might improve the agreement. Systematically testing the adequacy of the effective Fokker-Planck description and improving the features of the above approach is a vast mathematical and computational task that we do not attempt here; our main point has been to suggest the use of diffusion map coordinates as the variables of choice in this reduced modeling context.

Summary and Conclusions
=======================

In this work we demonstrated the use of nonlinear manifold learning techniques, and, in particular, diffusion maps, to obtain an effective description for a complex, high-dimensional, stochastic dynamical system. Use of physical intuition and symmetries helped us formulate a notion of a good local similarity measure between detailed system snapshots. Using a diffusion map, we obtained and validated a two-dimensional effective description, and then we implemented a lifting procedure that allowed us to obtain detailed physical system states consistent with prescribed diffusion map coordinate values. We also searched for and proposed two physical variables whose relation to the corresponding diffusion map coordinates is a bijection. This helped the development of intuition about the important features of the system. Finally, we discussed the qualitative behavior of the system exploiting the new low-dimensional description.

The computation of the diffusion map coordinates was performed "off-line", before this entire process began. The long-term goal is to use diffusion maps adaptively in an on-line setting. The idea is that a more local diffusion map parameterization, based on local system observations, can be used to guide (i.e. bias) further system simulation and collection of information by suggesting "interesting" new coarse initial conditions. Such new simulations would then be used to continually modify and extend local diffusion map coordinates into unexplored regions of phase space. For such a program to be implemented, a robust, automated and computationally efficient lifting algorithm is clearly necessary.

Acknowledgements
================

This work was partially supported by the Department of Energy Computational Science Graduate Fellowship (BES) and the US DOE (IGK).

<span>**APPENDIX**</span>

To construct a low-dimensional embedding for a data set of $M$ individual $d$-dimensional real vectors, $\vec X_1$,...,$\vec X_M$, we start with a similarity measure between each pair of vectors $\vec X_i$, $\vec X_j$. The similarity measure is a nonnegative quantity $W_{ij} = W_{ji}$ satisfying certain additional "admissibility conditions" @coifman1. As a concrete example, consider using a Gaussian similarity measure (based on the standard $L_2$ norm): $$W_{ij} = \text{exp} \left[ - \left(\frac{||\vec X_i - \vec
X_j||}{\epsilon} \right)^2 \right].$$ A weighted Euclidean norm may be chosen over the standard $L_2$ norm in situations where the values of different components of $\vec X$ may vary over disparate orders of magnitude. $\epsilon$ defines a characteristic scale which quantifies the "locality" of the neighborhood within which Euclidean distance can be used as the basis of a meaningful similarity measure @coifman1. A systematic approach to determining appropriate $\epsilon$ values is discussed below. Next, we define the diagonal matrix $\mathbf{D}$ by $$D_{ii} = \sum^M_{k=1} W_{ik},$$ and then we compute the first few right eigenvectors and eigenvalues of the stochastic matrix $$\mathbf{K} = \mathbf{D}^{-1} \mathbf{W}.$$ In MATLAB, for instance, this can be done with the command $[\mathbf{V},\mathbf{D}] = \mbox{eigs}(\mathbf{K},N_{evl})$, where $N_{evl}$ is the number of top eigenvalues we wish to keep (we typically are only interested in the first few). It is important to note that it is not required that our data be in vector format; in the end, all we need to apply the diffusion map approach is a pairwise similarity measure $W_{ij}$.

This gives a set of real eigenvalues $\lambda_0 \geq \lambda_1 \geq
... \geq \lambda_{M-1} \geq 0$ with corresponding eigenvectors $\{\vec \psi_j\}_{j=0}^{M-1}$. Since $\mathbf{K}$ is stochastic, $\lambda_0 = 1$ and $\vec \psi_0 =
[1\, 1 \, ... \, 1]^T$. The $n$-dimensional representation of a particular $d$-dimensional data vector, $\vec X_i$, is given by the *diffusion map* $\Psi_n:\, \vec X \longrightarrow \mathbf{R}^n$, where $$\Psi_n(\vec X_i) = [\vec \psi_1^{(i)}, \vec \psi_2^{(i)}, ..., \vec
\psi_n^{(i)}],$$ a mapping which is only defined on the $M$ recorded data vectors (in order for Euclidean distance in diffusion map space to actually equal diffusion distance, mentioned above, the diffusion map must be instead scaled as $\Psi_n(\vec X_i) = [\lambda_1 \vec \psi_1^{(i)},
\lambda_2 \vec \psi_2^{(i)}, ..., \lambda_n \vec \psi_n^{(i)}]$). In other words, the vector $\vec X_i$ is mapped to the vector whose first component is the $i$th component of the first nontrivial eigenvector, whose second component is the $i$th component of the second nontrivial eigenvector, etc. If, for instance, a gap in the eigenvalue spectrum is observed between eigenvalues $\lambda_t$ and $\lambda_{t+1}$, then $\Psi_t$ gives a good low-dimensional representation of the data set@belkin2003 [@coiffp]. It is interesting that if the data come from a Markovian stochastic process, the eigenvectors and eigenvalues are approximations to the eigenfunctions and eigenvalues of the corresponding backward Fokker-Planck operator @coiffp.

We choose the value of $\epsilon$ used in the diffusion map computation by invoking the *correlation dimension* utilized in @loglog. The assumption here is that the volume of an $n$-dimensional set scales with any characteristic length $s$ as $s^n$; for relatively uniform sampling one might expect the number of neighbors less than $s$ apart to scale similarly. We first compute all pairwise distances. Fig. \[fig:loglog\] shows the total number of pairwise distances less than $\epsilon$; it is clear that an asymptote will arise at large $\epsilon$ ($M^2$, where $M$ is the number of points) and at small $\epsilon$ ($M$). In the figure, the two asymptotes are smoothly connected by an approximately straight line; the slope of this line suggests the correct dimensionality for our data set (here, two). The range of $\epsilon$ values corresponding to this straight segment are all acceptable in our diffusion map computations: here any value of $\epsilon$ between approximately $10^{-\frac{3}{2}}$ and $10^{-\frac{1}{2}}$ may be used.

![For $M$ data vectors, $\lim_{\epsilon \to \infty} L(\epsilon)$ is simply $M^2$ and $\lim_{\epsilon \to 0} L(\epsilon)$ is simply $M$. These two asymptotes are connected, however, by a sloped, approximately straight line. $\epsilon$ values chosen from this regime are appropriate.<span data-label="fig:loglog"></span>](loglog.eps){height="170mm" width="130mm"}

The problem of finding the diffusion map coordinates of a *new* $d$-dimensional vector is solved with the Nyström extension. The first step is to compute all distances $\{\mathbf{d}_{i\,\text{new}}\}_{i=1}^M$ between our new vector and the $M$ vectors in our data set, and set $W_{i\, \text{new}} =
\text{exp} \left[ - \left(
\frac{\mathbf{d}_{i\,{\text{new}}}}{\epsilon} \right)^2 \right]$; here we used the standard Euclidean norm for $\mathbf{d}$. Setting $$K_{i \, \text{new}} = \left(\sum_{k=1}^M W_{k\,
\text{new}}\right)^{-1} W_{i\, \text{new}},$$ the $j\text{th}$ diffusion map coordinate of the new vector $\mathbf{P}_{\text{new}}$ is given by $$\vec \psi_j^{\text{new}} = \frac{1}{\lambda_j} \sum_{i=1}^M K_{i \,
\text{new}} \vec \psi_j^{(i)}.$$

{'source': 'arXiv_out/0904'}