adds a function to parareal to return the maximum singular value of the error propagatin matrix

Author: Daniel Ruprecht

scripts/plot_dispersion.py

@@ -52,6 +52,7 @@ def normalise(R, T, target):
 
     phase      = np.zeros((6,Nsamples))
     amp_factor = np.zeros((6,Nsamples))
+    svds       = np.zeros((3,Nsamples))
     u0_val     = np.array([[1.0]], dtype='complex')
     targets    = np.zeros((3,Nsamples))
 
@@ -89,13 +90,13 @@ def normalise(R, T, target):
       amp_factor[2,i] = np.exp(sol_coarse.imag)
 
       # Compute Parareal phase velocity and amplification factor
-      
+      svds[0,i]         = para.get_max_svd(ucoarse=ucoarse)        
       for jj in range(0,3):
         stab_para = para.get_parareal_stab_function(k=niter_v[jj], ucoarse=ucoarse)
 
         if i==0:
           targets[jj,0] = np.angle(stab_ex)
-        
+
         stab_para_norm = normalise(stab_para[0,0], Tend, targets[jj,i])
         # Make sure that stab_norm*dt = stab
         err = abs(stab_para_norm**Tend - stab_para)
@@ -112,7 +113,6 @@ def normalise(R, T, target):
         phase[3+jj,i]      = sol_para.real/k_vec[i]
         amp_factor[3+jj,i] = np.exp(sol_para.imag)
 
-
     ###
     rcParams['figure.figsize'] = 2.5, 2.5
     fs = 8
@@ -155,3 +155,17 @@ def normalise(R, T, target):
     plt.gcf().savefig(filename, bbox_inches='tight')
     call(["pdfcrop", filename, filename])
 
+    fig  = plt.figure()
+    plt.plot(k_vec, svds[0,:], '-s', color='r', linewidth=1.5, markevery=(1,6), mew=1.0, markersize=fs/2)
+    plt.xlabel('Wave number', fontsize=fs, labelpad=0.25)
+    plt.ylabel('Maximal singular value', fontsize=fs, labelpad=0.5)
+    fig.gca().tick_params(axis='both', labelsize=fs)
+    plt.xlim([k_vec[0], k_vec[-1:]])
+    plt.ylim([0, 2.0])
+ #   plt.legend(loc='lower left', fontsize=fs, prop={'size':fs-2})
+    plt.gca().set_ylim([0.0, 2.0])
+    plt.xticks([0, 1, 2, 3], fontsize=fs)
+    #plt.show()
+    filename = 'parareal-dispersion-svd.pdf'
+    plt.gcf().savefig(filename, bbox_inches='tight')
+    call(["pdfcrop", filename, filename])

src/parareal.py

@@ -96,6 +96,12 @@ def get_parareal_stab_function(self, k, ucoarse=None):
         Mat[:,i] = R.dot(M).flatten()
       return Mat
 
+    # Returns the largest singular value of the error propagation matrix
+    def get_max_svd(self, ucoarse=None):
+      Pmat, Bmat = self.get_parareal_matrix(ucoarse)
+      svds = linalg.svds(Pmat, k=1, return_singular_vectors=False)
+      return svds[0]
+
     # Returns array containing all intermediate solutions
     def get_parareal_vector(self):
       b = np.zeros((self.u0.ndof*(self.timemesh.nslices+1),1))