stochastic_integrals_2013_fsu

stochastic_integrals_2013_fsu, a series of talks about the approximation of integrals associated with a problem defined by ordinary or partial differential equations with a stochastic component, presented to the class on "Numerical Methods for Stochastic Differential Equations", ISC 5936-01, for the FSU Department of Scientific Computing, 26/28 February, 5/7/19/21 March 2013.

• stochastic_integrals_2013_fsu.tex
• stochastic_integrals_2013_fsu.pdf

Files that were used include:

• bang.png, an image of the ignition of a match.
• burgers_base.png, the "base" solution of the steady Burgers equation.
• burgers_convergence.png, study of the approximation of the quantity of interest, Q=U(X=0,T=3), for the time dependent Burgers equation, using sparse grids and the Monte Carlo method, for three values of viscosity.
• burgers_grid.png, a schematic of the time and space grid for Burgers equation with an initial condition and periodic boundaries.
• burgers_ic.png, a plot suggestion how an initial condition for the time dependent Burgers equation could be defined by 9 data values and a spline.
• burgers_ic_perturb.png, a plot showing how the solution profile at time = 3 of the Burgers equation would be affected by changing any one of the 9 data values that define the initial condition.
• burgers_shock.png, a 3D plot showing how the inviscid Burgers equation can develop a shock, causing the solution process to break down.
• burgers_time3.png, the "base" solution of the time dependent Burgers equation, at T = 3.
• cc_3d.png, images of the first 6 sparse grids in 3D, based on the Clenshaw Curtis points.
• cc_grid_5x9.png, a 5 by 9 Clenshaw Curtis product grid.
• cc_sequence.png, the locations of the nodes in a nested sequence of Clenshaw Curtis rules.
• cc_sparse_2d.png, a 2d sparse grid based on Clenshaw Curtis rules.
• chebyshev.png, illustrates that the Clenshaw Curtis nodes are the cosines of equally spaced angles.
• d6_ppeak.png, integration test with Genz "Product Peak" function for dimension 6.
• d10_con.png, integration test with Genz "Continous" function for dimension 10.
• d10_cpeak.png, integration test with Genz "Corner Peak" function for dimension 10.
• d10_discon.png, integration test with Genz "Discontinous" function for dimension 10.
• d10_gauss.png, integration test with Genz "Gaussian" function for dimension 10.
• d10_oscill.png, integration test with Genz "Oscillatory" function for dimension 10.
• d10_ppeak.png, integration test with Genz "Product Peak" function for dimension 10.
• diffusion_mc.png, error plots for the diffusion equation, Monte Carlo estimates.
• diffusion_mc_and_sg.png, error plots for the diffusion equation, Monte Carlo and sparse grid estimates.
• diffusion_mc.png, error plot for the diffusion equation, Monte Carlo estimate.
• exponential_cdf.png, the Cumulative Density Function for the exponential density.
• exponential_invcdf.png, the inverse Cumulative Density Function for the exponential density.
• exponential_pdf.png, the Probability Density Function for the exponential density.
• fireball_base_run.png, a plot of the time-dependent solution of the Fireball problem, with the base value of the parameters.
• fsu_logo.pdf, a logo;
• gl_grid_10x20.png, Gauss-Legendre 10x20 2D product grid.
• level4_o1x17.png, a 1x17 product grid.
• level4_o3x9.png, a 3x9 product grid.
• level4_o5x5.png, a 5x5 product grid.
• level4_o9x3.png, a 9x3 product grid.
• level4_o17x1.png, a 17x1 product grid.
• linear_sample.png, histogram of samples from the linear PDF.
• log_plot1.png, uniform PDF for S = Log(U).
• log_plot2.png, uniform PDF for U = Exp(S).
• mc_2d_test.png, LogLog plot of Monte Carlo error for a 2D integral.
• mc_test.png, LogLog plot of Monte Carlo error for a 1D integral.
• ncc_3d.png, a 5x5x5 3D product grid of equally spaced points.
• normal_cdf.png, the Cumulative Density Function for the normal density.
• normal_invcdf.png, the inverse Cumulative Density Function for the normal density.
• normal_pdf.png, the Probability Density Function for the normal density.
• normal_sample1.png, sampling from the normal density, result 1.
• normal_sample2.png, sampling from the normal density, result 2.
• normal_sample3.png, sampling from the normal density, result 3.
• part12.png, end part 1, begin part 2.
• part23.png, end part 2, begin part 3.
• part34.png, end part 3, begin part 4.
• part45.png, end part 4, begin part 5.
• part56.png, end part 5, begin part 6.
• ppeak_plot.png, Genz's product peak integrand function.
• probability_box.png, illustrates the 2D probability distribution formed by the product of two 1D discrete probability functions.
• rho_x.png, the X component of a 2D product PDF, example 1.
• rho_y.png, the Y component of a 2D product PDF, example 1.
• rho_xy.png, the 2D product PDF, example 1.
• rho2_x.png, the X component of a 2D product PDF, example 2.
• rho2_y.png, the Y component of a 2D product PDF, example 2.
• rho2_xy.png, the 2D product PDF, example 2.
• rho3_x.png, the X component of a 2D product PDF, example 3.
• rho3_y.png, the Y component of a 2D product PDF, example 3.
• rho3_xy.png, the 2D product PDF, example 3.
• sensitivity_a.png, a rough exploration of the sensitivity of the "base" solution of the steady Burgers equation with respect to the location of the left boundary condition.
• sensitivity_alpha.png, a rough exploration of the sensitivity of the "base" solution of the steady Burgers equation with respect to the value of the left boundary condition.
• sensitivity_nu.png, a rough exploration of the sensitivity of the "base" solution of the steady Burgers equation with respect to the viscosity.
• tanh_plot.png, a plot of the hyperbolic tangent function, which is essentially the solution to our simple steady Burgers equation.
• sparse1.png, step 1 of "covering the black boxes".
• sparse2.png, step 2 of "covering the black boxes".
• sparse3.png, step 3 of "covering the black boxes".
• sparse4.png, step 4 of "covering the black boxes".
• sparse5.png, step 5 of "covering the black boxes".
• sparse6.png, step 6 of "covering the black boxes".
• sparse7.png, step 7 of "covering the black boxes".
• sparse8.png, step 8 of "covering the black boxes".
• sparse9.png, step 9 of "covering the black boxes".
• tanh_plot.png, a plot of the hyperbolic tangent function.
• th01.png, a plot of the forebody simulator.
• th02.png, a plot of the initial conditions for idealized flow past an obstacle.
• th03.png, a plot of a closeup of the flow past an obstacle.
• th04.png, a plot of the desired flow along a profile line.
• th05.png, a plot of the difference between the desired and achieved profile error as a parameter is varied.
• the_end.png, a graphic for the end of the talk.
• uniform_cdf.png, the Cumulative Density Function for the uniform density.
• uniform_invcdf.png, the inverse Cumulative Density Function for the uniform density.
• uniform_pdf.png, the Probability Density Function for the uniform density.
• uniform_qoi.png, ignition time as a function of the parameter delta for the blowup problem.
• uniform_run.png, sample realizations of the blowup problem.
• x0_alpha.png, a plot of the location where the solution to the steady Burgers equation changes sign, as a function of the value of alpha, the value specified at the left boundary.

Files and Programs:

• cc_precision.m, investigate precision of a CC rule.
• cc_test.m, test a CC rule.
• clenshaw_curtis_compute.m, compute the points and weights of a CC rule.
• exponential_plots.m, plot PDF, CDF, ICDF for the exponential density.
• fish_sample.m, estimates the mean and variance of a fish population.
• fish_sample2.m, estimates the mean and variance of a fish population.
• gl_test.m, test a GL rule.
• histogram.m, tries 3 ways to compute N normal samples, and histograms results.
• imtqlx.m, an eigenvalue solver needed to compute the GL rule.
• legendre_ek_compute.m, compute points and weights of a GL rule.
• linear_sample.m, sample from the linear PDF.
• log_plots.m, displays the PDF for Log(U) and U for the blowup problem.
• mc_2d_test.m, test the Monte Carlo method as a 2D quadrature rule.
• mc_test.m, test the Monte Carlo method as a 1D quadrature rule.
• midpoint_error.m, compute the error of the composite midpoint rule as the number of intervals increases.
• ncc_set.m, set the points and weights of a closed Newton-Cotes rule.
• ncc_test.m, test a closed Newton-Cotes rule.
• normal_plots.m, plot PDF, CDF, ICDF for the normal density.
• normal_variance.m, sample from the normal density, estimate the variance.
• quad_2d_compute.m, compute the points and weights of a 2D quadrature rule that is the product of two 1D rules.
• quad_2d.m, use a 2D product rule that is the product of two 1D rules.
• rule_adjust.m, adjust a quadrature rule, defined on [a,b], to a new interval [c,d].
• seeds.m, look at how the seed affects a random number sequence.
• simpson_error.m, compute the error of the composite Simpson rule as the number of intervals increases.
• sphere_surface.m, estimate the area of the surface of a sphere using the Monte Carlo method.
• sum_estimate.m, determines the average sum of a pair of dice.
• uniform_plots.m, plot PDF, CDF, ICDF for the uniform density.
• uniform_variance.m, sample from the uniform density, estimate the variance.