New test and example scripts.
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77
Estimations/CustomCylindricalCutOff.m
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77
Estimations/CustomCylindricalCutOff.m
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@ -0,0 +1,77 @@
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% Define the size of the cube using NumberOfGridPoints
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NumberOfGridPoints = [32, 32, 16]; % [X, Y, Z] dimensions of the cube
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Dimensions = [30, 30, 15]; % Real-world dimensions for the grid
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% Generate a random 3D cube based on the NumberOfGridPoints
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cube = rand(NumberOfGridPoints(1), NumberOfGridPoints(2), NumberOfGridPoints(3));
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% Define the real-world distance range for each axis (X, Y, Z)
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x_range = linspace(-Dimensions(1)/2, Dimensions(1)/2, NumberOfGridPoints(1)); % 32 points along X mapped to [-15, 15]
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y_range = linspace(-Dimensions(2)/2, Dimensions(2)/2, NumberOfGridPoints(2)); % 32 points along Y mapped to [-15, 15]
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z_range = linspace(-Dimensions(3)/2, Dimensions(3)/2, NumberOfGridPoints(3)); % 16 points along Z mapped to [-7.5, 7.5]
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% Create a 3D grid of coordinates for the cube using meshgrid
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[x, y, z] = meshgrid(x_range, y_range, z_range);
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% Define the cylinder's radius and height (in real-world units)
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radius = 10; % Radius in real-world units (e.g., meters)
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height = 4; % Height in real-world units (e.g., meters)
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% Get the center of the cube (in real-world coordinates)
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center_x = mean(x_range); % Center of x_range
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center_y = mean(y_range); % Center of y_range
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center_z = mean(z_range); % Center of z_range
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% Define the cylindrical mask
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cylinder_mask = ((x - center_x).^2 + (y - center_y).^2 <= radius^2) & ...
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(abs(z - center_z) <= height / 2);
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% Set everything inside the cylinder to 1 and outside to 0
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mask = double(cylinder_mask);
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% Multiply the mask by the random cube to apply the cut-off
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masked_cube = cube .* mask;
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% Visualize the cube and the cylindrical mask using isosurfaces
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figure(1);
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clf
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figure_width = 1200;
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set(gcf, 'Position', [100, 100, figure_width, round(figure_width / 1.618)]);
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set(gca, 'FontSize', 16, 'Box', 'On', 'Linewidth', 2);
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% Set the axis limits to be the same for both subplots
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x_limits = [min(x_range), max(x_range)];
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y_limits = [min(y_range), max(y_range)];
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z_limits = [min(z_range), max(z_range)];
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% Visualize the random cube
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subplot(1, 2, 1);
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isosurface(x, y, z, cube, 0.5); % Isosurface for the random cube
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axis([x_limits, y_limits, z_limits]);
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title('Original 3D Numerical Grid', 'FontSize', 16);
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xlabel('X (\mum)', 'Interpreter', 'tex', 'FontSize', 16);
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ylabel('Y (\mum)', 'Interpreter', 'tex', 'FontSize', 16);
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zlabel('Z (\mum)', 'Interpreter', 'tex', 'FontSize', 16);
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grid on;
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axis equal;
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view(3);
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camlight;
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lighting gouraud;
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% Visualize the masked cube
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subplot(1, 2, 2);
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isosurface(x, y, z, masked_cube, 0.5); % Isosurface for the masked cube
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axis([x_limits, y_limits, z_limits]);
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title('Grid with Custom Cylindrical Cut-off', 'FontSize', 16);
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% Add subtitle with radius and height with adjusted position
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text('Units', 'normalized', 'Position', [0.5, -0.2], 'String', ...
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['Radius = ', num2str(radius), ' \mum, Height = ', num2str(height), ' \mum'], ...
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'Interpreter', 'tex', 'FontSize', 14, 'FontWeight', 'normal', 'HorizontalAlignment', 'center');
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xlabel('X (\mum)', 'Interpreter', 'tex', 'FontSize', 16);
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ylabel('Y (\mum)', 'Interpreter', 'tex', 'FontSize', 16);
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zlabel('Z (\mum)', 'Interpreter', 'tex', 'FontSize', 16);
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grid on;
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axis equal;
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view(3);
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camlight;
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lighting gouraud;
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@ -1,6 +1,6 @@
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%%
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% Parameters
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alpha = 0.5; % Example value for alpha
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alpha = 8; % Example value for alpha
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N = linspace(1e4, 1e6, 100); % Atom number range
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OD = linspace(0, 3, 100); % Optical depth range
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N1 = 100; % Example parameter for N1 controlling transition steepness
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@ -37,13 +37,10 @@ title(['|C(X)| = OD^3 * N^{\alpha - 9/5} * Smoothed Heaviside Function with Alph
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% title('Cost Function: OD^3 * N^{\alpha - 9/5} with Smoothed Heaviside Function');
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%%
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% Parameters
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N = linspace(1e3, 1e6, 100); % Atom number range
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OD = linspace(0, 3, 100); % Optical depth range
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N = linspace(1e3, 1e6, 100); % Atom number range
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OD = linspace(0, 3, 100); % Optical depth range
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N1 = 1e5; % Example parameter for N1 controlling transition steepness
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% Smoothed Heaviside function as given in the appendix
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% Create figure and initial plot
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fig = figure(2);
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clf
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@ -51,17 +48,26 @@ set(gcf,'Position',[50 50 950 750])
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set(gca,'FontSize',16,'Box','On','Linewidth',2);
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alpha = 0.5; % Initial value for alpha
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function cf = getSmoothedHeaviside(x, N1)
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if x > 0
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cf = (2 ./ (1 + exp(N1 ./ x)));
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else
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cf = 0;
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end
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end
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% Nested function to calculate and update the cost function plot
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function update_plot(alpha_value, N, N1, OD)
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SmoothedHeaviside = @(x) (x > 0) .* (2 ./ (1 + exp(N1 ./ x)));
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% Calculate the cost function for the given alpha
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CostFunction = zeros(length(N), length(OD));
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for i = 1:length(N)
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for j = 1:length(OD)
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H_N = SmoothedHeaviside(N(i)); % Smoothed Heaviside function
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H_N = getSmoothedHeaviside(N(i), N1); % Smoothed Heaviside function
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CostFunction(i, j) = H_N * OD(j)^3 * N(i)^(alpha_value - 9/5);
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end
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end
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% Create a meshgrid for plotting
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[OD_grid, N_grid] = meshgrid(OD, N);
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% Plot the cost function as a surface plot
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@ -1,3 +1,165 @@
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%% Direction solution to the unconstrained optimization problem with the Conjugate Gradient technique
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clc;
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clear;
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format long;
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% Define the function (now accepts a 3D array)
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f = @(X) sum(X(:).^4) - 4*sum(X(:).^3) + 6*sum(X(:).^2) + 5;
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% Initial Guess: Now we use a 3D array
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x_o = rand(3, 3, 3); % Random 3D array as the initial guess
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fprintf('Initial Objective Function Value: %.6f\n\n', f(x_o));
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% Convergence Criteria:
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Epsilon = 1E-5;
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% Gradient Calculation (numerical)
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J = compute_gradient(f, x_o); % Gradient at initial point
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S = -J; % Initial search direction
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% Iteration Counter:
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i = 1;
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% Minimization
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while norm(S(:)) > Epsilon % Halting Criterion
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% Step Size Optimization (Line Search)
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lambda = optimize_step_size(f, x_o, S);
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% Update the solution
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x_o_new = x_o + lambda * S;
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% Update the gradient and search direction
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J_old = J;
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J = compute_gradient(f, x_o_new);
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S = update_search_direction(S, J, J_old);
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% Update for next iteration
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x_o = x_o_new;
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i = i + 1;
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end
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% Output
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fprintf('Number of Iterations for Convergence: %d\n\n', i);
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fprintf('Objective Function Minimum Value: %.6f\n\n', f(x_o));
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%% Visualize the optimization
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clc;
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clear;
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format long;
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% Define the function (now accepts a 3D array)
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f = @(X) sum(X(:).^4) - 4*sum(X(:).^3) + 6*sum(X(:).^2) + 5;
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% Initial Guess: Now we use a 3D array
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x_o = rand(3, 3, 3); % Random 3D array as the initial guess
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fprintf('Initial Objective Function Value: %.6f\n\n', f(x_o));
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% Convergence Criteria:
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Epsilon = 1E-5;
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% Gradient Calculation (numerical)
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J = compute_gradient(f, x_o); % Gradient at initial point
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S = -J; % Initial search direction
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% Iteration Counter:
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i = 1;
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% Prepare for visualization
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fig = figure;
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clf
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set(gcf,'Position',[50 50 950 750])
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set(gca,'FontSize',16,'Box','On','Linewidth',2);
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hold on;
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% Create a mesh grid for visualization
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[X1, X2] = meshgrid(-2:0.1:2, -2:0.1:2); % Adjust domain based on your function
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Z = arrayfun(@(x1, x2) f([x1, x2, 0]), X1, X2); % Example 2D slice: f([X1, X2, 0])
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% Plot the surface
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surf(X1, X2, Z);
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xlabel('X1');
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ylabel('X2');
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zlabel('f(X)');
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title('Conjugate Gradient Optimization Process');
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colorbar;
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scatter3(x_o(1), x_o(2), f(x_o), 'r', 'filled'); % Initial point
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% Set the 3D view
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view(75, 30); % Example 3D view: azimuth 45°, elevation 30°
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% Minimization with visualization
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while norm(S(:)) > Epsilon % Halting Criterion
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% Step Size Optimization (Line Search)
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lambda = optimize_step_size(f, x_o, S);
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% Update the solution
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x_o_new = x_o + lambda * S;
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% Update the gradient and search direction
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J_old = J;
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J = compute_gradient(f, x_o_new);
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S = update_search_direction(S, J, J_old);
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% Update for next iteration
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x_o = x_o_new;
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i = i + 1;
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% Visualization Update
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scatter3(x_o(1), x_o(2), f(x_o), 'g', 'filled'); % Current point
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drawnow; % Update the plot
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end
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% Output
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fprintf('Number of Iterations for Convergence: %d\n\n', i);
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fprintf('Objective Function Minimum Value: %.6f\n\n', f(x_o));
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%%
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% Numerical Gradient Calculation using the finite differences method for 3D
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function J = compute_gradient(f, X)
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epsilon = 1E-5; % Step size for numerical differentiation
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J = zeros(size(X)); % Gradient with the same size as the input
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% Loop over all elements of the 3D array
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for i = 1:numel(X)
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X1 = X;
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X2 = X;
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% Perturb the element X(i) in both directions
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X1(i) = X1(i) + epsilon;
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X2(i) = X2(i) - epsilon;
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% Central difference formula for gradient
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J(i) = (f(X1) - f(X2)) / (2 * epsilon);
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end
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end
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% Line Search (Step Size Optimization)
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function lambda = optimize_step_size(f, X, S)
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alpha = 1; % Initial step size
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rho = 0.5; % Step size reduction factor
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c = 1E-4; % Armijo condition constant
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max_iter = 100; % Max iterations for backtracking
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for k = 1:max_iter
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% Check if the Armijo condition is satisfied
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grad = compute_gradient(f, X); % Compute gradient
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if f(X + alpha * S) <= f(X) + c * alpha * (S(:)' * grad(:))
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break;
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else
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alpha = rho * alpha; % Reduce the step size
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end
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end
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lambda = alpha; % Return the optimized step size
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end
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% Update Search Direction (Polak-Ribiere method)
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function S_new = update_search_direction(S, J_new, J_old)
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beta = (norm(J_new(:))^2) / (norm(J_old(:))^2);
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S_new = -J_new + beta * S;
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end
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%{
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function [psi] = minimizeEnergyFunctional(this,psi,Params,Transf,VDk,V,Observ)
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format long;
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@ -168,4 +330,5 @@ function S_new = update_search_direction(S, J_new, J_old)
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% (Polak-Ribiere method)
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beta = max(0, (J_new(:)' * (J_new(:) - J_old(:))) / (norm(J_old(:))^2));
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S_new = -J_new + beta * S;
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end
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end
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%}
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91
Estimations/SplitStepFourierMethod.m
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91
Estimations/SplitStepFourierMethod.m
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% Constants
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hbar = 1.0545718e-34; % Reduced Planck constant (Joule second)
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m_e = 9.10938356e-31; % Mass of electron (kg)
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wavelength = 5e-9; % de Broglie wavelength (meters)
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k0 = 2 * pi / wavelength; % Wave number
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% Grid parameters
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Lx = 1e-6; % Simulation domain size (meters)
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Nx = 1024; % Number of grid points
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dx = Lx / Nx; % Grid spacing
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x = linspace(-Lx / 2, Lx / 2, Nx); % Spatial grid
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% Time parameters
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dt = 3e-15; % Time step (seconds)
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Nt = 1001; % Number of time steps
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% Wave packet parameters
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sigma_x = 60e-9; % Width of the Gaussian packet (meters)
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x0 = -2e-7; % Initial position of the packet (meters)
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kx0 = k0; % Initial wave vector
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% Barrier parameters
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barrier_width = 4e-8; % Width of the square barrier (meters)
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barrier_height = 0.8 * hbar^2 * k0^2 / (2 * m_e); % Height of the barrier (Joules)
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% Initial Gaussian wave packet
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psi0 = exp(-(x - x0).^2 / (2 * sigma_x^2)) .* exp(1i * kx0 * x);
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psi0 = psi0 / sqrt(sum(abs(psi0).^2) * dx); % Normalization
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% Potential energy (Square barrier in the center)
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V = zeros(size(x));
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V(abs(x) < barrier_width / 2) = barrier_height;
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% Fourier space components
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kx = (2 * pi / Lx) * (-Nx/2:Nx/2-1); % Frequency domain representation
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kx = fftshift(kx); % Shift zero frequency component to center
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K2 = kx.^2; % Squared wave numbers
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% Split-step Fourier method
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psi = psi0; % Initialize wave function
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transmission_prob = 0; % Initialize transmission probability
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figure(1); % Create a figure for visualization
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clf
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set(gcf,'Position',[100 100 950 750])
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set(gca,'FontSize',16,'Box','On','Linewidth',2);
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for t = 1:Nt
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% (a) 1/2 Evolution for the potential energy in real space
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psi = psi .* exp(-1i * V * dt / (2 * hbar));
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% (b) Forward transform to Fourier space
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psi_k = fft(psi);
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% (c) Full evolution for the kinetic energy in Fourier space
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psi_k = psi_k .* exp(-1i * hbar * K2 * dt / (2 * m_e));
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% (d) Inverse Fourier transform back to real space
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psi = ifft(psi_k);
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% (e) 1/2 Evolution for the potential energy in real space
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psi = psi .* exp(-1i * V * dt / (2 * hbar));
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% Calculate transmission probability
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Rho = abs(psi).^2; % Probability density
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transmission_prob = sum(Rho(x > barrier_width / 2)) * dx;
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% Visualization of wave packet evolution
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if mod(t, 50) == 0
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clf; % Clear figure before drawing
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plot(x * 1e9, Rho / max(Rho), 'r', 'LineWidth', 2); % Plot normalized density
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hold on;
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plot(x * 1e9, V / max(V), 'k', 'LineWidth', 2); % Plot barrier potential
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title(sprintf('Time step: %d\nTransmission Probability: %.3f', t, transmission_prob),'FontSize',16);
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xlabel('x (nm)', 'FontSize', 16);
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ylabel('Probability Density', 'FontSize', 16);
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grid on
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drawnow; % Update the figure dynamically
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end
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end
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% Final wave packet distribution
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figure(2);
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clf
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set(gcf,'Position',[100 100 950 750])
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set(gca,'FontSize',16,'Box','On','Linewidth',2);
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plot(x * 1e9, abs(psi).^2, 'b', 'LineWidth', 2);
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xlabel('x (nm)','FontSize',16);
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ylabel('Probability Density','FontSize',16);
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title('Final Wave Packet Distribution','FontSize',16);
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grid on
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%%
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