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Climbing Image method

Having construct a minimum energy path joining two states, we often want to find the saddle point joining them. Here, we have implemented the climbing image method as described in E, Ren, and Vanden-Eijnden (2007).

Identify Candidate Saddle

The first step of the climbing image method is to identify a vector that points from one of the images, roughly, towards the saddle. This can be done by:

  • Finding the highest energy image along the string. This is neareest the highest energy saddle.
  • Using the highest energy image along with two neighbors to construct an estimate of the vector pointing in the unstable direction of the saddle point. Or, more practically, construct a vector that is sufficiently collinear to the unstable direction.

For example, for string U, the following computes this vector, storing it in τ:

max_idx = argmax(V.(U));
τ = upwind_tangent(U[max_idx-1:max_idx+1], V)

Set up Climbing Image Solver

Having identified the direction $\tau$, we then sep the climbing image solver with

StringMethod.ClimbingImage — Type

ClimbingImage(∇V, τ, ∇V_climb,integrate!, dist, Δt) - Set up the Climbing Image Method

Fields

  • ∇V - In place gradient of the potential
  • τ - Approximate tangent vector of the unstable direction
  • ∇V_climb- In place gradient of the potential with reflector
  • integrate! - In place integrator
  • Δt - Time step
source

Set Options

Analogously to the string method, we can set solver options with respect to number of iterations, toleranace, etc. with

StringMethod.SaddleOptions — Type

SaddleOptions(;nmax = 10^3, tol = 1e-6, verbose = false, save_trajectory = true, print_iters = 10^3) - Set up climbing image method saddle point options

Fields

  • nmax = 10^3 - maximum number of solver iterations
  • tol = 1e-6 - stopping tolerance
  • verbose = false - print diagnostic and convergence information
  • save_trajectory = true - save the entire string at each algorithmic step
  • print_iters = 10^3 - print diagonstic information at this frequency
source

Solve

Having set the desired parameters, we execute

StringMethod.climbing_image — Function

climbing_image(u₀, C; options=SaddleOptions()): Run the climbing image method

Fields

  • u₀ - Initial guess for saddle
  • C - Climbing image data structure

Optional Fields

  • options - String method options
source

Example

using Test
using LinearAlgebra
using StringMethod
using TestLandscapes
using ForwardDiff
using Printf
using Plots

# set potential and get its gradient
V = x -> Muller(x);
∇V = x -> ForwardDiff.gradient(V, x);

U0 = linear_string([0, -0.25], [0, 1.5], 15)

Δt = 1e-4;
dist = (u, v) -> norm(u - v, 2);
pin = false;

string = SimplifiedString(∇V, stepRK4!, spline_reparametrize!, dist, pin, Δt);
opts = StringOptions(verbose=false, save_trajectory=false)

# solve
U_trajectory = simplified_string(U0, string, options=opts);
U=U_trajectory[end];

# saddle point search
max_idx = argmax(V.(U));
τ = upwind_tangent(U[max_idx-1:max_idx+1], V);
climb = ClimbingImage(∇V,τ, stepRK4!, 1e-4);
opts = StringMethod.SaddleOptions(save_trajectory=false);
u_trajectory = climbing_image(U[max_idx], climb, options=opts);
saddle_pt = u_trajectory[end];

# visualize
xx =LinRange(-1.5, 1.5,150);
yy = LinRange(-0.5, 2.0,150);
V_vals = [V([x,y]) for y in yy, x in xx];

plt = contour(xx,yy,min.(V_vals,500),
levels = LinRange(-150,500,50),color=:viridis,colorbar_title="V")
scatter!(plt, [x_[1] for x_ in U],
        [x_[2] for x_ in U],color=:red, label="String")
scatter!(plt, [saddle_pt[1]], [saddle_pt[2]],
    color=:orange, markershape=:star5, label="Saddle")
xlabel!("x")
ylabel!("y")
Example block output