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

@@ -61,7 +61,7 @@ for latexmath:[c_1, c_2, c_3, \gamma_1, \gamma_2,] and latexmath:[\gamma_3] inde
 
 . 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.
+.. 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 ) = 1.53107].
 .. 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 )  -
   u_N ( \mu )|||_\mu /|||u( \mu )|||_\mu] and the maximum relative output error max latexmath:[\mu \in\Xi_{test} |{T_{root}}( \mu )  -  {T_{root}} N( \mu
   )|/{T_{root}}( \mu )] as a function of latexmath:[N] (use the Python command `+semilogy+` for plotting).