Add links to youtube tutorials

Author: jdjakem

docs/source/figures/cvmc-many-models.png

@@ -134,6 +134,20 @@
 #%%
 #Change ``eta`` to ``eta_mc`` to see how the variance reduction changes when the covariance between models is approximated
 
+
+#%%
+#Videos
+#------
+#Click on the image below to view a video tutorial on control variate Monte Carlo quadrature with one low-fidelity model
+#
+#.. image:: ../../figures/cvmc.png
+#   :target: https://youtu.be/2GpwGG8EmNU
+#
+#Click on the image below to view a video tutorial on control variate Monte Carlo quadrature with one multiple lower fidelity models
+#
+#.. image:: ../../figures/cvmc-many-models.png
+#   :target: https://youtu.be/lBvtWo2ZwaU
+
 #%%
 #References
 #^^^^^^^^^^

docs/source/figures/cvmc.png

@@ -138,3 +138,12 @@ def plot_estimator_histrogram(nsamples, model_id, ax):
 #   \mean{\left(Q_{\alpha}(\rvset_N)-\mean{Q}\right)^2}=\underbrace{N^{-1}\var{f_\alpha}}_{I}+\underbrace{\left(Q_{\alpha}-Q\right)^2}_{II}
 #
 #From this expression we can see that the MSE can be decomposed into two terms; a so called stochastic error (I) and a deterministic bias (II). The first term is the variance of the Monte Carlo estimator which comes from using a finite number of samples. The second term is due to using an approximation of :math:`f_0`. These two errors should be balanced, however in the vast majority of all MC analyses a single model :math:`f_\alpha` is used and the choice of :math:`\alpha`, e.g. mesh resolution, is made a priori without much concern for the balancing bias and variance.
+#
+
+#%%
+#Video
+#-----
+#Click on the image below to view a video tutorial on Monte Carlo quadrature
+#
+#.. image:: ../../figures/mc.png
+#   :target: https://youtu.be/OrLsYo11kvY?si=chAXN6UqssXY8pxh

docs/source/figures/mc.png

@@ -145,6 +145,7 @@
 #
 #where :math:`W_j` denotes the cost of evaluating the jth model and :math:`W_{\max}` is the total budget.
 #
+# This optimization problem can be solved effectively using semi-definite programming [CWARXIV2023]_.
 
 target_costs = np.array([1e1, 1e2, 1e3], dtype=int)
 estimators = [
@@ -182,4 +183,4 @@
 #
 #.. [SUSIAMUQ2021] `D. Schaden, E. Ullmann. Asymptotic Analysis of Multilevel Best Linear Unbiased Estimators. SIAM/ASA Journal on Uncertainty Quantification 9 (3):953-978, 2021. <https://doi.org/10.1137/20M1321607>`_
 #
-#.. [CWARXIV2023]_ `M. Croci, K. Willcox, S. Wright. Multi-output multilevel best linear unbiased estimators via semidefinite programming. (2023)  <https://doi.org/10.1016/j.cma.2023.116130>`_
+#.. [CWARXIV2023] `M. Croci, K. Willcox, S. Wright. Multi-output multilevel best linear unbiased estimators via semidefinite programming. (2023)  <https://doi.org/10.1016/j.cma.2023.116130>`_