up

docs/modules/ROOT/pages/homework-2024/problem-set-2.adoc

@@ -27,7 +27,7 @@ and that the outputis given by
   {T_{root}}_N ( \mu ) = L^T_N u_N ( \mu ).
 ++++
 
-We derived expressions for latexmath:[A_N( \mu ) \in \mathbb{R}^{N\times N}] in terms of latexmath:[A_N( \mu )] and latexmath:[Z], latexmath:[F_N \in \mathbb{R}^N] in terms of latexmath:[F_N] and latexmath:[Z], and latexmath:[L_N \in \mathbb{R}^N] in terms of latexmath:[L_N] and latexmath:[Z]; here latexmath:[Z] is an latexmath:[\mathcal{N} \times N] matrix, the jth column of which is latexmath:[u_N ( \mu j )] (the nodal values of latexmath:[u_N ( \mu j ))]. Finally, it follows from affine parameter dependence that latexmath:[A_N ( \mu )] can be expressed as
+We derived expressions for latexmath:[A_N( \mu ) \in \mathbb{R}^{N\times N}] in terms of latexmath:[A_{\mathcal{N}}( \mu )] and latexmath:[Z], latexmath:[F_N \in \mathbb{R}^N] in terms of latexmath:[F_{\mathcal{N}}] and latexmath:[Z], and latexmath:[L_N \in \mathbb{R}^N] in terms of latexmath:[L_{\mathcal{N}}] and latexmath:[Z]; here latexmath:[Z] is an latexmath:[\mathcal{N} \times N] matrix, the jth column of which is latexmath:[u_{\mathcal{N}} ( \mu_j )] (the nodal values of latexmath:[u_{\mathcal{N} ( \mu_j ))]. Finally, it follows from affine parameter dependence that latexmath:[A_N ( \mu )] can be expressed as
 
 [latexmath#eq:1.3]
 ++++
@@ -59,8 +59,7 @@ for latexmath:[c_1, c_2, c_3, \gamma_1, \gamma_2,] and latexmath:[\gamma_3] inde
 . We first consider a one parameter (latexmath:[P = 1]) problem. To this end, we keep the Biot number fixed at latexmath:[Bi = 0.1] and assume that the conductivities of all fins are equivalent, i.e., latexmath:[k_1 = k_2 = k_3 = k_4], but are allowed to vary between latexmath:[0.1] and latexmath:[10] – we thus have latexmath:[\mu  \in D =
 [0.1, 10\].] The sample set latexmath:[S_N] for latexmath:[N_{max} = 8] is given the log equidistributed sampling.
 
-. Generate the reduced basis "`matrix`" latexmath:[Z] and all necessary reduced basis quantities. You have two options: you can use the solution "snapshots" directly in latexmath:[Z] or perform a Gram-Schmidt orthonormalization to construct latexmath:[Z] (Note that you require the latexmath:[X] – inner product to perform Gram-Schmidt; here, we use latexmath:[(\cdot, \cdot)_X = a(\cdot, \cdot; \mu )], where latexmath:[\mu = 1] – all conductivities are latexmath:[1] and the Biot number is latexmath:[0.1]). Calculate the condition number of latexmath:[A_N ( \mu )] for latexmath:[N = 8] and for latexmath:[\mu = 1] and latexmath:[\mu = 10] with and without Gram – Schmidt orthonormalization. What do you observe? Solve the reduced basis approximation (where you use the snapshots directly in latexmath:[Z]) for latexmath:[\mu_1 = 0.1] and latexmath:[N = 8]. What is latexmath:[u_N( \mu_1)]? How do you expect latexmath:[u_N( \mu_2)] to look like for latexmath:[\mu_2
-  = 10.0]? What about latexmath:[\mu_3 = 1.0975]? Solve the Gram – Schmidt orthonormalized reduced basis approximation for latexmath:[\mu_1 = 0.1] and latexmath:[\mu
+. Generate the reduced basis "`matrix`" latexmath:[Z] and all necessary reduced basis quantities. You have two options: you can use the solution "snapshots" directly in latexmath:[Z] or perform a Gram-Schmidt orthonormalization to construct latexmath:[Z] (Note that you require the latexmath:[X] – inner product to perform Gram-Schmidt; here, we use latexmath:[(\cdot, \cdot)_X = a(\cdot, \cdot; \mu )], where latexmath:[\mu = 1] – all conductivities are latexmath:[1] and the Biot number is latexmath:[0.1]). Calculate the condition number of latexmath:[A_N ( \mu )] for latexmath:[N = 8] and for latexmath:[\mu = 1] and latexmath:[\mu = 10] with and without Gram – Schmidt orthonormalization. What do you observe? Solve the reduced basis approximation (where you use the snapshots directly in latexmath:[Z]) for latexmath:[\mu_1 = 0.1] and latexmath:[N = 8]. What is latexmath:[u_N( \mu_1)]? How do you expect latexmath:[u_N( \mu_2)] to look like for latexmath:[\mu_2= 10.0]? What about latexmath:[\mu_3 = 1.0975]? Solve the Gram – Schmidt orthonormalized reduced basis approximation for latexmath:[\mu_1 = 0.1] and latexmath:[\mu
   2 = 10] for latexmath:[N = 8]. What do you observe? Can you justify the result? For the remaining questions you should use the Gram – Schmidt orthonormalized reduced basis approximation.
 .. Verify that, for latexmath:[\mu  = 1.5] (recall that Biot is still fixed at latexmath:[0.1]) and latexmath:[N = 8], the value of the output is latexmath:[{T_{root}}_N ( \mu ) = 1.61] up to 2 digits.
 .. We next introduce a regular test sample, latexmath:[\Xi_{test} \subset D], of size latexmath:[ntest = 100] (in Python you can simply use `+linspace(0.1, 10, 100)+` to generate latexmath:[\Xi_{test}]). Plot the convergence of the maximum relative error in the energy norm latexmath:[\max_{\mu \in\Xi_{test}} |||u( \mu )  -