asd

kalman/graphics/figure3.py

@@ -6,11 +6,12 @@ import pickle
 import yaml
 
 
+#import matplotlib.font_manager
 from matplotlib import rc
 rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']})
 rc('text', usetex=True)
 
-import matplotlib.font_manager
+
 
 def is_ipython():
     ''' Check if script is run in IPython.
@@ -49,10 +50,14 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
     if is_ipython():
         plt.ion()
 
+
+
+    resultsname =  'results_slices_15dB/'
+
     idx_a = input_file.find('/')
     idx_b = input_file[idx_a+1::].find('/')
     name_file = input_file[idx_a+1:idx_b+idx_a+1]
-    inputfile_path = 'results/' + name_file + '/input.yaml'
+    inputfile_path = resultsname + name_file + '/input.yaml'
 
 
     with open(inputfile_path) as file:
@@ -76,11 +81,16 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
             }
 
 
-
+    dim = dat['theta'].shape[-1]
 
     meas_flag = False
-    RC_flag = False
+    if dim==7:
+        RC_flag = True
+    else:
+        RC_flag = False
     line_split = 1.5
+
+    rec_values = {}
     current_val = []
     current_val_C = []
     ids_type = []
@@ -95,10 +105,10 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
                     ids.append(bnd_c['id'])
                     ids_type.append('windkessel')
                     current_val.append(bnd_set['parameters']['R_d'])
-                    labels.append('$R_' + str(bnd_c['id']))
+                    labels.append('$R_' + str(bnd_c['id']-3))
                     if RC_flag:
                         current_val_C.append(bnd_set['parameters']['C'])
-                        labels.append('$C_' + str(bnd_c['id']))
+                        labels.append('$C_' + str(bnd_c['id']-3))
 
 
         elif 'dirichlet' in bnd_c['type']:
@@ -109,11 +119,9 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
 
 
 
-
-    dim = dat['theta'].shape[-1]
-    fig1, axes1 = plt.subplots(1,1,figsize=(12,6))
+    fig1, axes1 = plt.subplots(1,1,figsize=(12,7))
     if RC_flag:
-        fig2, axes2 = plt.subplots(1,1,figsize=(12,6))
+        fig2, axes2 = plt.subplots(1,1,figsize=(12,7))
     
 
     t = dat['times']
@@ -144,50 +152,53 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
 
 
     for i in range(len(ids)):
+
         cur_key = ids[i]
-
-        true_level =  np.log(true_values[ids[i]]/current_val[i])/np.log(2)
         rec_value = np.round(2**theta[-1, idx]*current_val[i],2)
-
-        
-        #curve = theta[:,idx] + line_split*idx - true_level
-        #dash_curve = line_split*idx + t*0
-
         curve = 2**theta[:, idx]*current_val[i]
         std_down =  2**(-np.sqrt(P[:, idx, idx]))*curve
         std_up =  2**np.sqrt(P[:, idx, idx])*curve
         dash_curve = true_values[ids[i]]  + t*0
-        
+        rec_values[cur_key] = rec_value
+
+
 
         if ids_type[i] == 'dirichlet':
-            pass
-            #axes3.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key])  + '$')
-            #axes3.fill_between(t, curve - np.sqrt(P[:, idx, idx]), curve + np.sqrt(P[:, idx, idx]), alpha=0.3, color=col_)
-            #legends_=next(legends)
-            #axes3.plot(t, dash_curve , color=col_,ls='--')
+            fig3, axes3 = plt.subplots(1,1,figsize=(12,5))
+            axes3.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key])  + '$', linewidth = 4)
+            axes3.fill_between(t, std_down, std_up, alpha=0.3, color=col_)
+            legends_=next(legends)
+            axes3.plot(t, dash_curve , color=col_,ls='--' , linewidth = 3)
+            axes3.set_ylabel(r'$U$',fontsize=36)
+            axes3.legend(fontsize=36,loc='upper right')
+            axes3.set_xlim([-0.01,0.81])
+            axes3.set_xlabel(r'$t  (s)$',fontsize=36)
+            axes3.set_box_aspect(1/4)
+            plt.xticks(fontsize=28)
+            plt.yticks(fontsize=28)
+            plt.savefig(resultsname + name_file + '/U.png')
         else:
-            axes1.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key])  + '$', linewidth = 2)
+
+            axes1.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key])  + '$', linewidth = 3)
             axes1.fill_between(t, std_down, std_up, alpha=0.3, color=col_)
-            axes1.plot(t, dash_curve , color=col_,ls='--')
+            axes1.plot(t, dash_curve , color=col_,ls='--',linewidth = 3)
             legends_=next(legends)
 
+            if RC_flag:
+                if i<len(current_val_C):
+                    rec_value_C = np.round(2**theta[-1, idx+1]*current_val_C[idc],6)
 
-        if RC_flag:
-            if i<len(current_val_C):
-                true_level_C = np.log(true_values_C[ids[i]]/current_val_C[i])/np.log(2)
-                rec_value_C = np.round(2**theta[-1, idc]*current_val_C[idc],6)
-
-                curve_C = 2**theta[:, idx+1]*current_val_C[idc]
-                dash_curve_C = true_values_C[ids[i]] + t*0
-                std_C_down =  2**(-np.sqrt(P[:, idx+1, idx+1]))*curve_C
-                std_C_up =  2**np.sqrt(P[:, idx+1, idx+1])*curve_C
-                
-                axes2.plot(t, curve_C , '-', color=col_,label= legends_ + '= ' + str(rec_value_C) + '/' + str(true_values_C[cur_key])  + '$', linewidth = 2)
-                axes2.fill_between(t, std_C_down, std_C_up, alpha=0.3, color=col_)
-                axes2.plot(t, dash_curve_C , color=col_,ls='--')
-                legends_=next(legends)
-                idx +=1
-                idc +=1
+                    curve_C = 2**theta[:, idx+1]*current_val_C[idc]
+                    dash_curve_C = true_values_C[ids[i]] + t*0
+                    std_C_down =  2**(-np.sqrt(P[:, idx+1, idx+1]))*curve_C
+                    std_C_up =  2**np.sqrt(P[:, idx+1, idx+1])*curve_C
+                    
+                    axes2.plot(t, curve_C , '-', color=col_,label= legends_ + '= ' + str(rec_value_C) + '/' + str(true_values_C[cur_key])  + '$', linewidth = 3)
+                    axes2.fill_between(t, std_C_down, std_C_up, alpha=0.3, color=col_)
+                    axes2.plot(t, dash_curve_C , color=col_,ls='--',linewidth = 3)
+                    legends_=next(legends)
+                    idx +=1
+                    idc +=1
 
 
         if meas_flag:
@@ -197,21 +208,56 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
         idx +=1
 
 
-    axes1.set_ylabel(r'$R_d$',fontsize=22)
-    axes1.legend(fontsize=18,loc='upper right')
+    axes1.set_ylabel(r'$R_d$',fontsize=36)
+    axes1.legend(fontsize=36,loc='upper right')
     axes1.set_xlim([-0.01,0.81])
-    axes1.set_xlabel(r'$t  (s)$',fontsize=22)
-    plt.savefig('C.png')
+    axes1.set_ylim([1700,55000])
+    axes1.set_box_aspect(1/2)
+    plt.xticks(fontsize=28)
+    plt.yticks(fontsize=28)
+    axes1.set_xlabel(r'$t  (s)$',fontsize=36)
+    plt.savefig('Rd.png')
+
 
     if RC_flag:
 
-        axes2.set_ylabel(r'$C$',fontsize=22)
-        axes2.legend(fontsize=18,loc='upper right')
+        axes2.set_ylabel(r'$C$',fontsize=36)
+        axes2.legend(fontsize=36,loc='upper right')
         axes2.set_xlim([-0.01,0.81])
-        axes2.set_xlabel(r'$t  (s)$',fontsize=22)
+        plt.xticks(fontsize=28)
+        plt.yticks(fontsize=28)
+        axes2.set_xlabel(r'$t  (s)$',fontsize=36)
         fig2.savefig('C.png')
 
-    fig1.savefig('Rd.png')
+    fig1.savefig(resultsname + name_file + '/Rd.png')
+
+
+    print('Final value theta: \t {}'.format(theta[-1, :]))
+    print('Deparameterized: 2^theta_end: \t {}'.format(2**theta[-1, :]))
+    print('Real values: \t {}'.format(true_values))
+    print('Recon. values: \t {}'.format(rec_values))
+
+
+    # saving reconstructed values
+    np.save(resultsname + name_file + '/recon_values.npy',rec_values)
+
+    print('----- paper ----')
+    rec_values_new = {}
+    epsilon = []
+
+    for rk in rec_values.keys():
+        eps = 100*rec_values[rk]/true_values[rk]-100
+        epsilon.append(eps)
+        if rk == 2:
+            rec_values_new[rk] = np.round(rec_values[rk],1)
+        else:
+            rec_values_new[rk] = np.round(rec_values[rk]/1000,2)
+
+    print('Recon. values: \t {}'.format(rec_values_new))
+    print('epsilon =' , np.round(np.mean(epsilon),2))
+    
+
+
     if not is_ipython():
         plt.show()
 

kalman/graphics/figureRd.py

@@ -5,6 +5,14 @@ import argparse
 import pickle
 import yaml
 
+
+#import matplotlib.font_manager
+from matplotlib import rc
+rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']})
+rc('text', usetex=True)
+
+
+
 def is_ipython():
     ''' Check if script is run in IPython.
 
@@ -45,8 +53,7 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
     idx_a = input_file.find('/')
     idx_b = input_file[idx_a+1::].find('/')
     name_file = input_file[idx_a+1:idx_b+idx_a+1]
-    inputfile_path = 'results/' + name_file + '/input.yaml'
-
+    inputfile_path = 'results_slices_15dB/' + name_file + '/input.yaml'
 
     with open(inputfile_path) as file:
         inputfile = yaml.full_load(file)
@@ -61,21 +68,10 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
             2: 75
             }
 
-    true_values_C = {
-            3: 0.0004,
-            4: 0.0004,
-            5: 0.0003,
-            6: 0.0003,
-            }
 
+    dim = dat['theta'].shape[-1]
 
-
-
-    meas_flag = False
-    RC_mod = True
-    line_split = 1.5
     current_val = []
-    current_val_C = []
     ids_type = []
     labels = []
     ids = []
@@ -88,10 +84,7 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
                     ids.append(bnd_c['id'])
                     ids_type.append('windkessel')
                     current_val.append(bnd_set['parameters']['R_d'])
-                    labels.append('$R_' + str(bnd_c['id']))
-                    if RC_mod:
-                        current_val_C.append(bnd_set['parameters']['C'])
-                        labels.append('$C_' + str(bnd_c['id']))
+                    labels.append('$R_' + str(bnd_c['id']-3))
 
 
         elif 'dirichlet' in bnd_c['type']:
@@ -101,93 +94,89 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
             labels.append('$U')
 
 
+    if 'windkessel' in ids_type:
+        fig1, axes1 = plt.subplots(1,1,figsize=(12,7))
+    if 'dirichlet' in ids_type: 
+        fig3, axes3 = plt.subplots(1,1,figsize=(12,7))
+
 
 
-    dim = dat['theta'].shape[-1]
-    fig1, axes = plt.subplots(1,1,figsize=(8,6))
 
-    axes.set_ylabel(r'$\theta$',fontsize=18)
 
     t = dat['times']
     theta = dat['theta']
     P = dat['P_theta']
 
-    col = cycle(['C0', 'C1', 'C2', 'C3','C4'])
-    ls = cycle(['-', '-', '--', '--', ':', ':', '-.', '-.'])
+
+    color_list = ['tomato', 'springgreen' , '#2CBDFE']
+    col = cycle(color_list)
     legends = cycle(labels)
 
-    if meas_flag:
-        t_und = t[0::30]
-        t_und = np.append( t_und , [t[-1]])
-        meas_mark = t_und*0
-
     col_ = next(col)
-    ls_ = next(ls)
     legends_=next(legends)
 
     if dim == 1:
         theta = theta.reshape((-1, 1))
         P = P.reshape((-1, 1, 1))
     
-
-
     idx = 0
-    idc = 0
 
     for i in range(len(ids)):
+
         cur_key = ids[i]
-
-        true_level =  np.log(true_values[ids[i]]/current_val[i])/np.log(2)
         rec_value = np.round(2**theta[-1, idx]*current_val[i],2)
-
-        #curve = theta[:,i] + line_split*i
-        #dash_curve = line_split*i + t*0 + true_level
-        
-        curve = theta[:,idx] + line_split*idx - true_level
-        dash_curve = line_split*idx + t*0
-        
-        axes.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key])  + '$')
-        axes.fill_between(t, curve - np.sqrt(P[:, idx, idx]), curve + np.sqrt(P[:, idx, idx]), alpha=0.3, color=col_)
-        legends_=next(legends)
-        axes.plot(t, dash_curve , color=col_,ls='--')
+        curve = 2**theta[:, idx]*current_val[i]
+        std_down =  2**(-np.sqrt(P[:, idx, idx]))*curve
+        std_up =  2**np.sqrt(P[:, idx, idx])*curve
+        dash_curve = true_values[ids[i]]  + t*0
 
 
-        if RC_mod:
-        
-            if i<len(current_val_C):
-                true_level_C = np.log(true_values_C[ids[i]]/current_val_C[i])/np.log(2)
-                rec_value_C = np.round(2**theta[-1, idc]*current_val_C[idc],6)
-                curve_C = theta[:,idx+1] + line_split*(idx+1) - true_level_C
-                dash_curve_C = line_split*(idx+1) + t*0
-                #print(true_values_C[cur_key_C])
-                axes.plot(t, curve_C , '-', color=col_,label= legends_ + '= ' + str(rec_value_C) + '/' + str(true_values_C[cur_key])  + '$')
-                axes.fill_between(t, curve_C - np.sqrt(P[:, idx+1, idx+1]), curve_C + np.sqrt(P[:, idx+1, idx+1]), alpha=0.3, color=col_)
-                axes.plot(t, dash_curve_C , color=col_,ls='--')
-                legends_=next(legends)
-                idx +=1
-                idc +=1
+        if ids_type[i] == 'dirichlet':
+            #axes3.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key])  + '$', linewidth = 5)
+            axes3.plot(t, curve , '-', color=col_,label= legends_ + '$', linewidth = 5)
+            axes3.fill_between(t, std_down, std_up, alpha=0.3, color=col_)
+            legends_=next(legends)
+            axes3.plot(t, dash_curve , color=col_,ls='--' , linewidth = 3)
+            axes3.set_ylabel(r'$U$',fontsize=36)
+            axes3.legend(fontsize=36,loc='upper right')
+            axes3.set_xlim([0,0.45])
+            axes3.set_ylim([8,180])
+            axes3.set_xlabel(r'$t  (s)$',fontsize=36)
+            axes3.set_box_aspect(1/2)
+            plt.xticks(fontsize=28)
+            plt.yticks(fontsize=28)
+            #plt.savefig('U_' + name_file + '.png')
+            plt.close(fig3)
+        else:
+            #axes1.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key])  + '$', linewidth = 4)
+            axes1.plot(t, curve , '-', color=col_,label= legends_ +  '$', linewidth = 4)
+            axes1.fill_between(t, std_down, std_up, alpha=0.3, color=col_)
+            axes1.plot(t, dash_curve , color=col_,ls='--',linewidth = 3)
+            legends_=next(legends)
 
 
-        if meas_flag:
-            axes.plot(t_und, meas_mark + line_split*idx, marker = 'x', color='red')
-
         col_ = next(col)
         idx +=1
 
-    axes.legend(fontsize=14,loc='lower right')
-    axes.set_xlim([-0.01,0.81])
-    axes.set_xlabel(r'time (s)',fontsize=18)
-    # print('theta_peak: \t {}'.format(theta[round(len(theta)/2), :]))
-    print('Final value theta: \t {}'.format(theta[-1, :]))
-    print('Deparameterized: 2^theta_end: \t {}'.format(2**theta[-1, :]))
-    print('Real values: \t {}'.format(true_values))
-    #print('Recon values: \t {a}:{b} '.format(a=ids[:],b=np.round(2**theta[-1, :]*current_val,2)))
 
 
 
-    plt.savefig('windk_res')
-    if not is_ipython():
-        plt.show()
+    axes1.set_ylabel(r'$R_d$',fontsize=30)
+    axes1.legend(fontsize=36,loc='upper right')
+    axes1.set_xlim([0,0.8])
+    axes1.set_ylim([-1000,66000])
+    axes1.set_box_aspect(1/2)
+    plt.xticks(fontsize=28)
+    plt.yticks(fontsize=28)
+    axes1.set_xlabel(r'$t  (s)$',fontsize=36)
+
+    path_paper = '/home/yeye/A_aliasing_kalman/latex/0_preprint/Figures/'
+    path_paper = '/home/yeye/Desktop/'
+    #fig1.savefig('Rd_'+ name_file +'.png')
+    fig1.savefig(path_paper + 'Rd_'+ name_file +'.png')
+    
+
+
 
 
 def get_parser():

kalman/graphics/figureU.py

@@ -0,0 +1,153 @@
+import matplotlib.pyplot as plt
+import numpy as np
+from itertools import cycle
+import argparse
+import pickle
+import yaml
+
+
+#import matplotlib.font_manager
+from matplotlib import rc
+rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']})
+rc('text', usetex=True)
+
+
+
+
+
+def plot_parameters():
+    ''' Plot the parameters in separate subplots with uncertainties.
+
+    Args:
+        dat (dict):  data dictionary
+        deparameterize (bool):  flag indicating if parameters should be
+          deparameterized via 2**theta
+        ref:    reference value to be plotted with parameters
+    '''
+ 
+
+    
+    name_file = ['SNR12V120_Pf','SNR12V70_Pf','SNR12V30_Pf']
+    #name_file = ['SNR12V120_Pf_MAG','SNR12V70_Pf_MAG','SNR12V30_Pf_MAG']
+    #name_file = ['SNR12V120_Pb_MAG','SNR12V70_Pb_MAG','SNR12V30_Pb_MAG']
+
+    name_file = ['slice2.3_Pa']
+    vencs = ['180','105','45']
+    path0 = '/home/yeye/Desktop/kalman/results/'
+
+
+    fig, axes = plt.subplots(1,1,figsize=(12,7))
+    col = cycle(['orangered', 'dodgerblue', 'limegreen', 'C3','C4'])
+
+    true_values = {
+            3: 4800,
+            4: 7200,
+            5: 11520,
+            6: 11520,
+            2: 75
+            }
+
+
+
+    for nn,name in enumerate(name_file):
+
+
+
+        path1 = path0 + name + '/'
+        inputfile_path = path1 + 'input.yaml'
+        dat = np.load(path1 + 'theta_stats.npz')
+
+
+        with open(inputfile_path) as file:
+            inputfile = yaml.full_load(file)
+
+        col_ = next(col)
+    
+
+        dim = dat['theta'].shape[-1]
+
+        current_val = []
+        ids_type = []
+        labels = []
+        ids = []
+
+        for bnd_c in inputfile['estimation']['boundary_conditions']:
+            
+            if 'windkessel' in bnd_c['type']:
+                for bnd_set in inputfile['boundary_conditions']:
+                    if bnd_c['id'] == bnd_set['id']:
+                        ids.append(bnd_c['id'])
+                        ids_type.append('windkessel')
+                        current_val.append(bnd_set['parameters']['R_d'])
+                        
+
+
+            elif 'dirichlet' in bnd_c['type']:
+                current_val.append(inputfile['boundary_conditions'][0]['parameters']['U'])
+                ids.append(bnd_c['id'])
+                ids_type.append('dirichlet')
+                labels.append('$U')
+
+
+
+
+        t = dat['times']
+        theta = dat['theta']
+        P = dat['P_theta']
+
+        legends = cycle(labels)
+
+        legends_=next(legends)
+
+        if dim == 1:
+            theta = theta.reshape((-1, 1))
+            P = P.reshape((-1, 1, 1))
+        
+        idx = 0
+
+        for i in range(len(ids)):
+
+            cur_key = ids[i]
+            rec_value = np.round(2**theta[-1, idx]*current_val[i],2)
+            curve = 2**theta[:, idx]*current_val[i]
+            std_down =  2**(-np.sqrt(P[:, idx, idx]))*curve
+            std_up =  2**np.sqrt(P[:, idx, idx])*curve
+            dash_curve = true_values[ids[i]]  + t*0
+
+
+            if ids_type[i] == 'dirichlet':
+                axes.plot(t, curve , '-', color=col_,label= '$(venc \ '+ vencs[nn] + ' \ cm/s) \ U = ' + str(rec_value) + '/' + str(true_values[cur_key])  + '$', linewidth = 5)
+                #axes.plot(t, curve , '-', color=col_,label= legends_ + vencs[nn] + ' \ cm/s$', linewidth = 5)
+                axes.fill_between(t, std_down, std_up, alpha=0.3, color=col_)
+                legends_=next(legends)
+                axes.plot(t, dash_curve , color='black',ls='--' , linewidth = 3)
+
+
+            idx +=1
+
+
+
+    axes.set_ylabel(r'$U$',fontsize=36)
+    axes.legend(fontsize=30,loc='upper right')
+    axes.set_xlim([0,0.35])
+    axes.set_ylim([10,160])
+    axes.set_xlabel(r'$t  (s)$',fontsize=36)
+    axes.set_box_aspect(1/2)
+    plt.xticks(fontsize=28)
+    plt.yticks(fontsize=28)
+    plt.savefig('U.png')   
+    plt.show()
+
+
+
+    #path_paper = '/home/yeye/A_aliasing_kalman/latex/0_preprint/Figures/'
+    #fig1.savefig('Rd_'+ name_file +'.png')
+    
+
+
+
+
+
+if __name__ == '__main__':
+
+    plot_parameters()

kalman/graphics/figure_func.py

@@ -0,0 +1,39 @@
+import matplotlib.pyplot as plt
+import numpy as np
+from itertools import cycle
+import argparse
+import pickle
+import yaml
+
+
+from matplotlib import rc
+rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']})
+rc('text', usetex=True)
+
+
+u = np.linspace(-2,2,100)
+utrue = 1
+venc1 = 0.9*utrue
+venc2 = 0.6*utrue
+
+fig1, ax1 = plt.subplots(1,1,figsize=(8, 5))
+
+J1 = 1 - np.cos((utrue-u)/venc1*np.pi)
+J2 = 1 - np.cos((utrue-u)/venc2*np.pi)
+
+
+lwidth = 2
+font_size = 28
+
+ax1.plot(u, J1, color = 'orangered', label = '$venc = 0.9 u_{true}$', linestyle='-',linewidth=lwidth)
+ax1.plot(u, J2, color = 'dodgerblue', label = '$venc = 0.6 u_{true}$', linestyle='-',linewidth=lwidth)
+ax1.axvline(x=1,color = 'black',linewidth = lwidth , label = '$u_{true}$')
+
+ax1.legend(fontsize=20, loc= 'upper right')
+ax1.tick_params(axis='both', which='major', labelsize=22)
+ax1.set_yticks([])
+ax1.set_xlabel('$u$',fontsize=font_size)
+ax1.set_ylabel('$J(u)$',fontsize=font_size)
+
+plt.show()
+fig1.savefig('functionals.png', dpi=500, bbox_inches='tight')

kalman/graphics/figure_func2.py

@@ -0,0 +1,169 @@
+import matplotlib.pyplot as plt
+import numpy as np
+from itertools import cycle
+import argparse
+import pickle
+import yaml
+
+
+from matplotlib import rc
+rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']})
+rc('text', usetex=True)
+
+font_size = 28
+
+################ Flow Parameters
+Rd      =   2.5
+Rt      =   0.5
+GradP   =   4
+mu      =   0.5
+fac     =   1
+nr      =   50
+VENC    =   0.6
+VENC2    =   0.4
+
+gamma   =   267.513e6     #  rad/Tesla/sec   Gyromagnetic ratio for H nuclei
+Bo      =   1.5              #  Tesla           Magnetic Field Strenght
+TE      =   5e-3             #  Echo-time
+phi0    =   gamma*Bo*TE       # Reference phase
+phi02   =   phi0%3.14
+r       =   np.linspace(-Rd, Rd, nr)
+dr      =   r[2]-r[1]
+vmax    =   1
+v       =   vmax/Rt**2*( Rt**2 - r**2 )*(np.abs(r)<Rt);  # Poiseuille Formula
+ai      =   v/vmax
+
+
+theta       =   np.linspace(-4,5,2000)
+vtest       =   np.linspace(-5,5,2000)
+jv          =   0*theta
+JV          =   0*theta
+JV2          =   0*theta
+
+Mjv         =   np.zeros([len(theta),len(ai)])
+Mjv2        =   np.zeros([len(theta),len(ai)])
+jv0         =   0*theta
+JV0         =   0*theta
+Mjv0        =   np.zeros([len(theta),len(ai)])
+#################################### MAGNETIZACION FROM V
+phiv    =   phi02 + v*np.pi/VENC
+phiv2    =   phi02 + v*np.pi/VENC2
+
+modv    =   np.ones(phiv.shape) 
+M1  =   modv*np.cos(phi02)  + 1j*modv*np.sin(phi02)
+M2  =   modv*np.cos(phiv)   + 1j*modv*np.sin(phiv)
+M2_2  =   modv*np.cos(phiv2)   + 1j*modv*np.sin(phiv2)
+
+
+################################### FFT to COMPLEX M
+S1      =   np.fft.fft(M1)
+S2      =   np.fft.fft(M2)
+S2_2      =   np.fft.fft(M2_2)
+MR1         =   np.fft.ifft(S1)
+MR2         =   np.fft.ifft(S2)
+MR2_2         =   np.fft.ifft(S2_2)
+
+vrec1       =   (np.angle(MR2)-phi02)*VENC/(np.pi)
+vrec2       =   (np.angle(MR2_2)-phi02)*VENC2/(np.pi)
+
+
+
+for k in range(len(ai)):
+    # v func
+    jv0 = 1-np.cos(np.pi*(vrec1[k]-vtest)/VENC)
+    Mjv0[:,k] = jv0[:]
+    JV0 += jv0  
+    
+    # theta func
+    jv = 1-np.cos(np.pi*(vrec1[k]-theta*ai[k])/VENC)
+    Mjv[:,k] = jv[:]
+    JV += jv
+    
+    jv2 = 1-np.cos(np.pi*(vrec2[k]-theta*ai[k])/VENC2)
+    Mjv2[:,k] = jv2[:]
+    JV2 += jv2 
+
+NJV1    =   JV#*100/np.max(JV)
+NJV2    =   JV2#*110/np.max(JV)
+MV = Mjv0
+V =NJV1
+V2 =NJV2
+
+
+
+
+fig     =   plt.figure(figsize=(12, 6), dpi=100)
+ax1     =   plt.subplot(1,2,1)
+
+ch1     =   20
+ch2     =   23
+color1 = 'xkcd:coral'
+color2 = 'xkcd:azure'
+color3 = 'darkviolet'
+lwidth = 2
+
+
+# Miniplot
+left, bottom, width, height = [0.18, 0.17, 0.1, 0.1]
+ax0 = fig.add_axes([left, bottom, width, height])
+ax0.plot(r,v,'b-')
+ax0.plot([r[ch1]],[v[ch1]],color=color1,marker='o')
+ax0.plot([r[ch2]],[v[ch2]],color=color2,marker='o')
+ax0.set_xlim((-1.5,1.5))
+ax0.set_xticks([])
+ax0.set_ylabel(r'$u$',fontsize=20)
+
+
+
+# Figure 1
+#ax1.plot(vtest, MV[:,ch1],color='xkcd:coral',label='$v_1$')
+#ax1.plot(vtest, MV[:,ch2],color='xkcd:azure',label='$v_2$')
+ax1.plot(vtest, MV[:,ch1],color=color1,linewidth=lwidth)
+ax1.plot(vtest, MV[:,ch2],color=color2,linewidth=lwidth)
+
+m1x     =   vtest[np.where( np.abs(MV[:,ch1] - np.min(MV[:,ch1]))<0.001  )]
+m1y     =   np.min(MV[:,ch1])
+m2x     =   vtest[np.where( np.abs(MV[:,ch2] - np.min(MV[:,ch2]))<0.001  )]
+m2y     =   np.min(MV[:,ch2])
+
+#ax1.plot([m1x],[m1y],color='xkcd:coral',marker='o')
+#ax1.plot([m2x],[m2y],color='xkcd:azure',marker='o')
+ax1.axvline(x=v[ch1], color=color1, linestyle='--',label='$v_{1,true}$')
+ax1.axvline(x=v[ch2], color=color2, linestyle='--',label='$v_{2,true}$')
+ax1.set_ylabel('$individual \ functional$',fontsize=20)
+#ax1.legend(loc='upper right', bbox_to_anchor=(0.5, 1.05),ncol=2, fancybox=True, shadow=True,fontsize=15)
+ax1.set_yticks([])
+ax1.tick_params(axis='both', which='major', labelsize=22)
+#ax1.set_xticks([])
+#ax1.legend(fontsize=20, loc= 'upper right')
+ax1.set_xlim((-3.5,3.5))
+ax1.set_ylim((-1.1,2.9))
+ax1.set_xlabel('$u$',fontsize=font_size)
+ax1.tick_params(axis='both', which='major', labelsize=22)
+ax1.set_yticks([])
+ax1.text(-1,2.4,'$u_{true}$',fontsize=22, color = color1)
+ax1.text(1.1,2.4,'$u_{true}$',fontsize=22, color = color2)
+
+
+# Figure 2
+ax2     =   plt.subplot(1,2,2)
+ax2.plot(theta,V,color=color3,linestyle = '-',linewidth=lwidth, label = '$venc=0.6u_{true}$')
+ax2.plot(theta,V2,color='darkorange',linestyle = '-',linewidth=lwidth, label = '$venc=0.4u_{true}$')
+
+ax2.axvline(x=1, color='black', linestyle='--')
+ax2.set_xlabel(r'$\theta$',fontsize=font_size)
+ax2.set_ylabel(r'$total \  functional$',fontsize=20)
+plt.yticks([])
+ax2.legend(fontsize=17, loc= 'upper left',frameon=False)
+ax2.set_ylim((-3,20))
+ax2.text(1.2,17,r'$ \theta _{true}$',fontsize=22, color = 'black')
+ax2.tick_params(axis='both', which='major', labelsize=22)
+#ax2.set_xticks([])
+#plt.title(r'$\theta_{true}=1$' + '\n' +'$venc < v_{max}$',fontsize=15)
+plt.xlim((-3.5,3.5))
+
+
+
+
+plt.show()
+fig.savefig('functionals2.png', dpi=500, bbox_inches='tight')

kalman/input_files/aorta.yaml

@@ -5,15 +5,15 @@ fluid:
     density: 1.2
     dynamic_viscosity: 0.035
     stokes: False
-    implicit_windkessel: True
+    state_velocity: 'update'
 
 io:
-    write_path: 'results/Rz_Pa_vnoise'
+    write_path: 'results/aorta'
     restart:
         path: ''  # './projects/nse_coa3d/results/test_restart2/' 
         time: 0
     write_xdmf: True
-    write_checkpoints: True
+    write_checkpoints: True 
     write_hdf5_timeseries: False
     write_velocity: 'update'     # update or tentative
 
@@ -26,10 +26,7 @@ boundary_conditions:
         #                 -U*sin(DOLFIN_PI*(t-1.6)/Th)*(t<= 1.6+Th )*(t>1.6) + (t<2.4)*(1.6+Th<t)*(U*DOLFIN_PI/Th*(t-1.6-Th)*exp(-(t-1.6-Th)*beta))' ]
         value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (Th<t)*(U*DOLFIN_PI/Th*(t-Th)*exp(-(t-Th)*beta))']
         parameters:
-            #U: 75  #P0
-            U: 150 #Pa
-            #U: 100 #Pg
-            #U: 40 #Pc
+            U: 75  #REFERENCE
             Th: 0.36
             beta: 70
             t: 0
@@ -51,15 +48,10 @@ boundary_conditions:
         type: 'windkessel'
         parameters:
             R_p: 480
-            #C: 0.0004  # P0
-            C: 0.0005   # Pa
-            #C: 0.0010   # Pb
-            #C: 0.0001   # Pc
-            #C: 0.0008   # Pg
-            #R_d: 7200  #P0
+            C: 0.0004  # REFERENCE
+            #R_d: 7200  # REFERENCE
             R_d:  8760  #Pa
             #R_d: 17520 #Pb x2
-            #R_d: 10000  #Pg 
             #R_d: 4000  #Pc
             p0: 85
             conv: 1333.223874   
@@ -68,16 +60,10 @@ boundary_conditions:
         type: 'windkessel'
         parameters:
             R_p: 520
-            #C: 0.0003 # REFERENCE
-            C: 0.0005 # Pa
-            #C: 0.0010 # Pb
-            #C: 0.0001 # Pc
-            #C: 0.0008 # Pg
+            C: 0.0003 # REFERENCE
             #R_d: 11520 # REFERENCE
             R_d:  8760  #Pa
             #R_d: 17520 #Pb x2
-            #R_d: 26280 #Pc x3
-            #R_d: 10000 #Pg
             #R_d: 4000 #Pc
             p0: 85
             conv: 1333.223874   
@@ -86,16 +72,10 @@ boundary_conditions:
         type: 'windkessel'
         parameters:
             R_p: 520
-            #C: 0.0003 # REFERENCE
-            C: 0.0005 #Pa
-            #C: 0.0010 #Pb
-            #C: 0.0001 #Pc
-            #C: 0.0008 #Pg
+            C: 0.0003 # REFERENCE
             #R_d: 11520 # REFERENCE
             R_d:  8760 #Pa
             #R_d: 17520 #Pb x2
-            #R_d: 26280 #Pc x3
-            #R_d: 10000 #Pg
             #R_d: 4000 #Pc
             p0: 85
             conv: 1333.223874   
@@ -129,6 +109,13 @@ timemarching:
     checkpoint_dt:  0.03 # <= 0: only last; else value + last
     report: 1   # 0: print nothing, 1: print time step and writeout, 2: 1 + flux
 
+windkessel:
+    implicit: True
+    # low rank update formulation can run in parallel.
+    # note that fgmres or similar has to be selected for the pressure solve, as
+    # matrix is not explicitly formed.
+    low_rank_update: False
+
 # solver setup
 fem:
     velocity_space: p1  # p1 p1b/p1+ p2
@@ -145,7 +132,7 @@ fem:
         streamline_diffusion:
             enabled: False
             parameter: 'standard'   # standard, shakib, codina, klr
-            length_scale: 'metric' # average, max, metric
+            length_scale: 'average' # average, max, metric
             parameter_element_constant: True
             Cinv: ~
         monolithic:
@@ -159,52 +146,52 @@ fem:
 
 linear_solver:
     method: 'lu'
+    #inputfile: 'input_files/fracstep_p-gmres_u-LU_WK-LRC.yaml'
 
 estimation:
     boundary_conditions:
         #- 
         #    id: 3
         #    type: 'windkessel'
+        #    mode: 'Rd'
         #    initial_stddev: 1
         - 
             id: 4
             type: 'windkessel'
             mode: 'Rd'
-            initial_stddev: 1
-        - 
-            id: 5
-            type: 'windkessel'
-            mode: 'Rd'
-            initial_stddev: 1
-        - 
-            id: 6
-            type: 'windkessel'
-            mode: 'Rd'
-            initial_stddev: 1
-        - 
-            id: 2
-            type: 'dirichlet'
-            parameters: 'U'
-            initial_stddev: 1
+            initial_stddev: 0.5
+        #- 
+        #    id: 5
+        #    type: 'windkessel'
+        #    mode: 'Rd'
+        #    initial_stddev: 0.5
+        #- 
+        #    id: 6
+        #    type: 'windkessel'
+        #    mode: 'Rd'
+        #    initial_stddev: 0.5
+        #- 
+        #    id: 2
+        #    type: 'dirichlet'
+        #    parameters: 'U'
+        #    initial_stddev: 0.5
 
 
     measurements:
         -
-            mesh: '/home/yeye/NuMRI/kalman/meshes/coaortaH3_leo2.0.h5'
-            #mesh: './meshes/coaortaH1.h5'
-            fe_degree: 1
-            #xdmf_file: 'measurements/aorta_zdir/Perturbation/Mg15V120/u_all.xdmf'
-            #file_root: 'measurements/aorta_zdir/Perturbation/Mg15V120/u{i}.h5'
-            xdmf_file: 'measurements/aorta_zdir_vnoise/u_all.xdmf'
-            file_root: 'measurements/aorta_zdir_vnoise/u{i}.h5'
+            mesh: '/home/yeye/NuMRI/kalman/meshes/slice_Hz2.3.h5'
+            fe_degree: 0
+            xdmf_file: 'measurements/slice/u_all.xdmf'
+            file_root: 'measurements/slice/u{i}.h5'
             indices: 0 # indices of checkpoints to be processed. 0 == all
-            velocity_direction: [0,0,1]
-            noise_stddev: 45     # standard deviation of Gaussian noise
+            velocity_direction: ~
+            noise_stddev: 10
+            VENC: 0
+            module_meas_file_root: ''
+
     
     roukf:
         particles: 'simplex' # unique or simplex
         observation_operator: 'postprocessing'         #state or postprocessing
         reparameterize: True
-        ODV_functional:
-            enable: False
-            VENC: 244 
+        MAG_functional: False

kalman/input_files/fracstep_p-gmres.yaml

@@ -0,0 +1,62 @@
+config_name: test
+petsc_options:
+    # - u_ten_ksp_view
+    - u_ten_ksp_converged_reason
+    - u_ten_ksp_monitor_true_residual
+    - u_ten_ksp_type bcgs
+    - u_ten_ksp_rtol 1.0e-6
+    - u_ten_ksp_initial_guess_nonzero
+
+    - u_ten_pc_type jacobi
+
+    - d_ksp_type cg
+    - d_ksp_rtol 1.0e-6
+
+    # - p_ksp_view
+    - p_ksp_converged_reason
+    - p_ksp_monitor_true_residual
+    - p_ksp_type fgmres
+    - p_ksp_rtol 1.0e-6
+    - p_ksp_initial_guess_nonzero
+
+    #- p_pc_type gamg
+    #- p_pc_gamg_type agg
+    #- p_pc_gamg_threshold 0.03
+    #- p_pc_gamg_square_graph 10     # no effect ??
+    #- p_pc_gamg_sym_graph
+    - p_mg_levels_ksp_type richardson
+    - p_mg_levels_pc_type sor
+
+    - p_mass_ksp_type cg
+    - p_mass_ksp_converged_reason
+    - p_mass_ksp_monitor_true_residual
+    - p_mass_ksp_rtol 1.0e-8
+    - p_mass_ksp_initial_guess_nonzero
+    - p_mass_pc_type jacobi
+
+    - u_upd_ksp_type cg
+    - u_upd_ksp_rtol 1.0e-8
+    - u_upd_ksp_initial_guess_nonzero
+    - u_upd_pc_type jacobi
+
+    - p_pc_type hypre
+    - p_pc_hypre_type boomeramg
+    - p_pc_hypre_boomeramg_strong_threshold 0.75
+    - p_pc_hypre_boomeramg_max_levels 25
+    - p_pc_hypre_boomeramg_smooth_type Euclid
+    - p_pc_hypre_boomeramg_eu_bj
+
+    # - p_pc_type hypre
+    # - p_pc_hypre_type boomeramg
+    # - p_pc_hypre_boomeramg_agg_nl 4
+    # - p_pc_hypre_boomeramg_agg_num_paths 2
+    # # Truncation factor for interpolation (note: increasing towrds 1
+    # # appears to reduce memory useage
+    # - p_pc_hypre_boomeramg_truncfactor 0.9
+    # # Max elements per row for interpolation operator
+    # - p_pc_hypre_boomeramg_P_max 5
+    # # - p_pc_hypre_boomeramg_max_levels 10
+    # # Strong threshold (BoomerAMG docs recommend 0.5-0.6 for 3D
+    # #   Poisson
+    # - p_pc_hypre_boomeramg_strong_threshold 0.5
+    

kalman/input_files/fracstep_p-gmres_u-LU_WK-LRC.yaml

@@ -0,0 +1,82 @@
+config_name: test
+petsc_options:
+    # - u_ten_ksp_view
+    # - u_ten_ksp_converged_reason
+    # - u_ten_ksp_monitor_true_residual
+    - u_ten_ksp_type preonly
+    # - u_ten_ksp_rtol 1.0e-6
+    # - u_ten_ksp_initial_guess_nonzero
+
+    # - u_ten_pc_type jacobi
+
+     # - u_ten_pc_side right
+    - u_ten_pc_type lu
+    # - u_ten_pc_gamg_type agg
+    # - u_ten_pc_gamg_threshold 0.03
+    # - u_ten_pc_gamg_square_graph 10     # no effect ??
+    # - u_ten_pc_gamg_sym_graph
+    # - u_ten_mg_levels_ksp_type richardson
+    # - u_ten_mg_levels_pc_type sor
+    # # - u_ten_pc_gamg_coarse_eq_limit 50    # ignored !?!?!
+
+
+    # - p_ksp_view
+    - p_ksp_converged_reason
+    - p_ksp_monitor_true_residual
+    # - p_ksp_type gmres
+    # - p_ksp_norm_type unpreconditioned
+    - p_ksp_type fgmres
+    - p_ksp_rtol 1.0e-8
+    - p_ksp_initial_guess_nonzero
+    - p_pc_type gamg
+    - p_pc_gamg_type agg
+    - p_pc_gamg_threshold 0.03
+    - p_pc_gamg_square_graph 10     # no effect ??
+    - p_pc_gamg_sym_graph
+    - p_mg_levels_ksp_type richardson
+    - p_mg_levels_pc_type sor
+
+
+    - p_mass_ksp_type preonly
+    # - p_mass_ksp_rtol 1.0e-8
+    # - p_mass_ksp_initial_guess_nonzero
+    - p_mass_pc_type lu
+
+    # - p_pc_type hypre
+    # - p_pc_hypre_type boomeramg
+    # - p_pc_hypre_boomeramg_strong_threshold 0.75
+    # - p_pc_hypre_boomeramg_max_levels 25
+    # - p_pc_hypre_boomeramg_smooth_type Euclid
+    # - p_pc_hypre_boomeramg_eu_bj
+
+    # - p_ksp_type preonly
+    # - p_pc_type lu
+    # - p_pc_factor_mat_solver_package mumps
+    # - p_mat_mumps_icntl_14 80
+
+    # - p_ksp_type cg
+    # - p_ksp_converged_reason
+    # - p_ksp_monitor_true_residual
+    # - p_ksp_rtol 1.0e-8
+
+    # - p_pc_type hypre
+    # - p_pc_hypre_type boomeramg
+    # - p_pc_hypre_boomeramg_agg_nl 4
+    # - p_pc_hypre_boomeramg_agg_num_paths 2
+    # # Truncation factor for interpolation (note: increasing towrds 1
+    # # appears to reduce memory useage
+    # - p_pc_hypre_boomeramg_truncfactor 0.9
+    # # Max elements per row for interpolation operator
+    # - p_pc_hypre_boomeramg_P_max 5
+    # # - p_pc_hypre_boomeramg_max_levels 10
+    # # Strong threshold (BoomerAMG docs recommend 0.5-0.6 for 3D
+    # #   Poisson
+    # - p_pc_hypre_boomeramg_strong_threshold 0.5
+    
+    # - u_upd_ksp_converged_reason
+    # - u_upd_ksp_monitor_true_residual
+    # - u_upd_ksp_view
+    - u_upd_ksp_type preonly
+    # - u_upd_ksp_rtol 1.0e-8
+    # - u_upd_ksp_initial_guess_nonzero
+    - u_upd_pc_type lu

presentations/press_8ecm/press.aux

@@ -0,0 +1,162 @@
+\relax 
+\providecommand\hyper@newdestlabel[2]{}
+\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
+\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
+\global\let\oldcontentsline\contentsline
+\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
+\global\let\oldnewlabel\newlabel
+\gdef\newlabel#1#2{\newlabelxx{#1}#2}
+\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
+\AtEndDocument{\ifx\hyper@anchor\@undefined
+\let\contentsline\oldcontentsline
+\let\newlabel\oldnewlabel
+\fi}
+\fi}
+\global\let\hyper@last\relax 
+\gdef\HyperFirstAtBeginDocument#1{#1}
+\providecommand\HyField@AuxAddToFields[1]{}
+\providecommand\HyField@AuxAddToCoFields[2]{}
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+\BOOKMARK [2][]{Outline0.1}{4D flow MRI}{}% 1
+\BOOKMARK [2][]{Outline0.2}{The mathematical model}{}% 2
+\BOOKMARK [2][]{Outline0.3}{The inverse problem}{}% 3
+\BOOKMARK [2][]{Outline0.4}{Numerical Experiments}{}% 4
+\BOOKMARK [2][]{Outline0.5}{Conclusions}{}% 5

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+\documentclass[xcolor=dvipsnames,notheorem,mathserifs]{beamer}
+\usepackage{amsmath}
+%\documentclass{beamer}
+\usepackage[english]{babel}
+%\usepackage[latin1]{inputenc}
+\usepackage{multicol} % indice en 2 columnas
+\usepackage[utf8]{inputenc}
+
+\usepackage{helvet}
+\usefonttheme{serif} 
+%\usepackage{ccfonts}     % Font family: Concrete Math
+\usepackage[T1]{fontenc}
+
+%\usepackage{graphicx}
+%\usepackage{movie15}
+%\usepackage{media9}[2013/11/04]
+\usepackage{xcolor}
+
+
+\usepackage{graphicx}
+\usepackage{multimedia}
+\usepackage{media9}
+\usepackage{listings,xcolor,caption, mathtools, wrapfig}
+\usepackage{amsfonts}
+\usepackage{amssymb,graphicx,enumerate}
+\usepackage{subcaption}
+\usepackage{hyperref}
+
+\usepackage[normalem]{ulem} % for strike out command \sout
+
+
+
+
+
+%\usetheme{default}
+%\usetheme{AnnArbor}
+%\usetheme{Antibes}
+%\usetheme{Bergen}
+%\usetheme{Berkeley}
+%\usetheme{Berlin}
+%\usetheme{Boadilla}
+%\usetheme{CambridgeUS}
+%\usetheme{Copenhagen}
+%\usetheme{Darmstadt}
+%\usetheme{Dresden}
+%\usetheme{Frankfurt}
+%\usetheme{Goettingen}
+%\usetheme{Hannover}
+%\usetheme{Ilmenau}
+%\usetheme{JuanLesPins}
+%\usetheme{Luebeck}
+%\usetheme{Madrid}
+%\usetheme{Malmoe}
+%\usetheme{Marburg}
+%\usetheme{Montpellier}
+%\usetheme{PaloAlto}
+%\usetheme{Pittsburgh}
+%\usetheme{Rochester}
+%\usetheme{Singapore}
+%\usetheme{Szeged}
+\usetheme{Warsaw}
+
+%\usecolortheme{albatross}
+%\usecolortheme{beaver}
+%\usecolortheme{beetle}
+\usecolortheme{crane}
+%\usecolortheme{dolphin}
+%\usecolortheme{dove}
+%\usecolortheme{fly}
+%\usecolortheme{lily}
+%\usecolortheme{orchid}
+%\usecolortheme{rose}
+%\usecolortheme{seagull}
+%\usecolortheme{seahorse}
+%\usecolortheme{whale}
+%\usecolortheme{wolverine}
+
+%\useoutertheme{infolines}
+%\useoutertheme{miniframes}
+%\useoutertheme{sidebar}
+\useoutertheme{smoothbars}
+%\useoutertheme{shadow}
+%\useoutertheme{smoothtree}
+%\useoutertheme{split}
+%\useoutertheme{tree}
+
+
+\usepackage{amssymb,mathrsfs,amsmath,latexsym,amsthm,amsfonts}
+\useinnertheme{rectangles}
+
+
+\setbeamertemplate{navigation symbols}{} % quitar simbolitos
+
+
+\setbeamerfont{page number in head/foot}{size=\large}
+%\setbeamertemplate{footline}[frame number] number in footer
+\setbeamertemplate{footline}{}
+
+
+
+\title{Robust parameter estimation in fluid flow models from aliased velocity measurements}
+%\author[Jeremías Garay Labra]
+%{Jeremías Garay Labra}
+\institute[University of Groningen]
+{
+Bernoulli Institute\\
+Faculty of Sciences and Engineering\\
+University of Groningen\\[0.5cm]
+ %\includegraphics[height=1.5cm]{Imagenes/escudoU2014.pdf}   
+  %  \includegraphics[height=1cm]{Imagenes/fcfm.png} \\[0.5cm]
+  Jeremías Garay Labra \emph{join with} Cristobal Bertoglio.}
+\date{\today}
+
+
+\begin{document}
+\frame{\titlepage}
+
+
+% \onslide<1->
+
+
+
+
+\begin{frame}
+  \frametitle{Index}
+  \tableofcontents
+\end{frame}
+
+
+\section[4D flow MRI]{4D flow MRI}
+\begin{frame}
+  \frametitle{4D flow MRI}  
+ \begin{columns}[c] 
+\column{.5\textwidth} % Left column and width
+\footnotesize
+
+\begin{itemize}
+\item<2-> Velocities encoded into the magnetization phase
+\item<3-> Rich post-proccesing: derived parameters 
+\end{itemize}
+
+
+
+
+\column{.54\textwidth} % Right column and width
+\onslide<1-> 
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.9\textwidth]{images/4dflow.png}
+  \caption{\footnotesize 4D flow MRI of a human thorax}
+ \end{center}
+ \end{figure}
+\end{columns}
+\end{frame}
+
+
+
+
+
+
+\section{The mathematical model}
+
+\begin{frame}
+ \frametitle{The mathematical model}
+\begin{center}
+The mathematical model
+\end{center}
+\end{frame}
+
+
+
+\begin{frame}
+ \frametitle{The mathematical model}
+ 
+ \begin{columns}[c] 
+\column{.5\textwidth} % Left column and width
+\footnotesize
+
+\column{.54\textwidth} % Right column and width
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=1.1\textwidth]{images/full_aorta.png} 
+ \end{center}
+ \end{figure}
+\end{columns} 
+ 
+\end{frame}
+
+
+
+\begin{frame}
+ \frametitle{The mathematical model}
+ 
+ \begin{columns}[c] 
+\column{.5\textwidth} % Left column and width
+\footnotesize
+\begin{itemize}
+\item<2-> Incompressible Navier-Stokes equations:
+\begin{equation}
+\begin{cases}
+\displaystyle \rho \frac{\partial \vec{u}}{\partial t} + \rho \big ( \vec{u}  \cdot \nabla \big) \vec{u} - \mu \Delta \vec{u} + \nabla p  =  0  \\[0.2cm]
+\nabla \cdot \vec{u}  =  0 \quad \text{in} \quad \Omega \\[0.2cm]
+\vec{u}  =  \vec{u}_{inlet}  \quad \text{on} \quad \Gamma_{in} \\[0.2cm]
+\vec{u} = 0 \quad \text{on} \quad \Gamma_{walls}
+\end{cases}
+\end{equation}
+\item<3-> \emph{Three-element} Windkessel coupling at every outlet:
+\begin{equation}
+\begin{cases}
+\displaystyle C_{d,l} \frac{d \pi_l}{dt} + \frac{\pi_l}{R_{d,l}} = Q_l \\[0.2cm]
+P_l = R_{p,l} \ Q_l + \pi_l
+\end{cases}
+\end{equation}
+\end{itemize}
+
+
+\column{.54\textwidth} % Right column and width
+\onslide<1-> 
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.9\textwidth]{images/windk_model.png}
+  \caption{\footnotesize Schematic of the model}
+ \end{center}
+ \end{figure}
+\end{columns} 
+ 
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{The mathematical model}
+ \begin{columns}[c] 
+\column{.5\textwidth} % Left column and width
+\footnotesize
+\begin{itemize}
+\item Incompressible Navier-Stokes equations:
+\begin{equation}
+\begin{cases}
+\displaystyle \rho \frac{\partial \vec{u}}{\partial t} + \rho \big ( \vec{u}  \cdot \nabla \big) \vec{u} - \mu \Delta \vec{u} + \nabla p  =  0  \\[0.2cm]
+\nabla \cdot \vec{u}  =  0 \quad \text{in} \quad \Omega \\[0.2cm]
+\vec{u}  =  \vec{u}_{inlet}  \quad \text{on} \quad \Gamma_{in} \\[0.2cm]
+\vec{u} = 0 \quad \text{on} \quad \Gamma_{walls}
+\end{cases}
+\end{equation}
+ \emph{Three-element} Windkessel coupling at every outlet:
+\begin{equation}
+\begin{cases}
+\displaystyle C_{d,l} \frac{d \pi_l}{dt} + \frac{\pi_l}{R_{d,l}} = Q_l \\[0.2cm]
+P_l = R_{p,l} \ Q_l + \pi_l
+\end{cases}
+\end{equation}
+\end{itemize}
+
+
+\column{.54\textwidth} % Right column and width
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.9\textwidth]{images/ref.png}
+  \caption{\footnotesize Reference solution at peak systole}
+ \end{center}
+ \end{figure}
+\end{columns} 
+ 
+\end{frame}
+
+
+
+%\begin{frame}
+% \frametitle{The mathematical model}
+% 
+% \begin{columns}[c] 
+%\column{.5\textwidth} % Left column and width
+%\footnotesize
+%\begin{itemize}
+%\item<1-> $u_{inlet} = -U f(t) \hat{n}$, with $f(t)$ the weaveform.
+%\item<2-> Fractional step scheme.
+%\item<3-> Semi-implicit Windkessel model.
+%\item<4-> Stabilized $\mathbb{P}1/\mathbb{P}1$ finite elements.
+%\item<4-> Implemented in FEniCS.
+%\end{itemize}
+%
+%
+%\column{.54\textwidth} % Right column and width
+%\begin{figure}[!hbtp]
+% \begin{center}
+%  \includegraphics[height=0.9\textwidth]{images/ref.png}
+%  \caption{\footnotesize Reference solution at peak systole}
+% \end{center}
+% \end{figure}
+%\end{columns} 
+% 
+%\end{frame}
+
+
+
+\begin{frame}
+ \frametitle{The inverse problem}
+\begin{itemize}
+\item<1-> Upon this solution $\Longrightarrow$ build a set of measurements
+\item<2->  $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
+\item<3-> reconstructed velocity: $u \in \big ( -venc, + venc  \big )$
+\item<4-> $VNR \sim 1/venc$
+\end{itemize}
+\centering
+\onslide<5-> \textbf{The measurements:}
+\begin{itemize}
+\item<6-> Gaussian noise into the magnetization
+\item<7-> Spatial and temporal interpolation
+\item<8-> Only using the dominant component of the velocity: $u_z$
+\item<9-> Different levels of aliasing varying the $venc$ parameter
+\end{itemize}
+\end{frame}
+
+
+
+
+\begin{frame}
+ \frametitle{The inverse problem}
+\begin{itemize}
+\item Upon this solution $\Longrightarrow$ build a set of measurements
+\item $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
+\item reconstructed velocity: $u \in \big ( -venc, + venc  \big )$
+\item $VNR \sim 1/venc$
+\end{itemize}
+ \begin{figure}
+        \includegraphics[width=0.7\textwidth]{images/supra_venc.png}
+  \caption*{Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
+  \hfill
+\end{figure}
+\end{frame}
+
+
+
+\begin{frame}
+ \frametitle{The inverse problem}
+\begin{itemize}
+\item Upon this solution $\Longrightarrow$ build a set of measurements
+\item $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
+\item reconstructed velocity: $u \in \big ( -venc, + venc  \big )$
+\item $VNR \sim 1/venc$
+\end{itemize}
+ \begin{figure}
+        \includegraphics[width=0.7\textwidth]{images/coartation.png}
+  \caption*{ Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
+  \hfill
+\end{figure}
+\end{frame}
+
+
+\section{The inverse problem}
+
+
+\begin{frame}
+ \frametitle{The inverse problem}
+\begin{center}
+Parameter optimization
+\end{center}
+\end{frame}
+
+
+
+
+\begin{frame}
+ \frametitle{The Kalman Filter}
+\begin{itemize}
+\item<1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $\theta$:
+
+\onslide<2->
+\begin{equation*}
+\hat{\theta} = arg \min_{\theta} J(\theta)
+\end{equation*}
+\begin{equation}
+J(\theta) = \displaystyle \frac{1}{2} || \theta - \theta_0  ||^2_{P_0^{-1}}  + \sum_{k=1}^N \frac{1}{2}  || Z_k - \mathbb{H} X_k(\theta)   ||^2_{W^{-1}}
+\end{equation}
+\onslide<4-> Where:
+\begin{itemize}
+\item<4-> $Z$ the measurements and $X = (\vec{u} , \pi)$ the state variable
+\item<5-> $\mathbb{H}$ observation operator 
+\item<6-> $\theta_0$ is the initial guess for the parameters
+\item<7-> $P_0$ is the associated covariance matrix
+\item<8-> $W$ is the associated covariance matrix to the meas. noise
+\end{itemize}
+
+\end{itemize}
+
+\end{frame}
+
+
+
+
+\begin{frame}
+ \frametitle{The Kalman Filter}
+The parameter vector: 
+ 
+\begin{itemize}
+\item<1->  Amplitude of the inlet velocity: $U$ 
+\item<2-> Only the higher resistence: $R_d$
+\end{itemize}
+
+\onslide<4-> $$\theta =  (U,\vec{R_d})$$ \\  with $\vec{R_d} = R_{d,l}$ for $l=1,..., \color{red} n_{l-1}$ \\[0.3cm]
+\onslide<5-> \color{red} Not all the resistences can be recovered at once $\Longrightarrow$ desc. aorta fixed.
+
+\end{frame}
+
+
+
+
+\section{Numerical Experiments}
+
+
+\begin{frame}
+ \frametitle{Numerical Experiments}
+\begin{center}
+Numerical Experiments
+\end{center}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Numerical Experiments}
+ \footnotesize
+\onslide<1-> $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
+ \begin{columns}
+  \footnotesize
+ \column{.45\textwidth}
+ \begin{figure}
+ \onslide<2-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (17520,17520,17520)$
+  \onslide<3-> \includegraphics[width=1.2\textwidth]{images/U_Pb.png}
+         \includegraphics[width=1.2\textwidth]{images/Rd_Pb.png}
+\end{figure}
+\column{.45\textwidth}
+\begin{figure}
+ \onslide<2-> \textbf{Test II:} $U_0 = 40$  $\vec{R_{d,0}}= (4000,4000,4000)$
+ \onslide<4->
+        \includegraphics[width=1.2\textwidth]{images/U_Pc.png}
+         \includegraphics[width=1.2\textwidth]{images/Rd_Pc.png}
+\end{figure}
+\end{columns} 
+\end{frame}
+
+
+
+
+
+
+\begin{frame}
+ \frametitle{Aliased data}
+\begin{center}
+\onslide<1-> What happend when $venc < u_{max}$ ?
+\begin{figure}
+ \onslide<2-> \includegraphics[width=0.45\textwidth]{images/v120.png}
+ \caption{Measurement set with $venc = 120 \% u_{max}$}
+\end{figure}
+\end{center}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Aliased data}
+\begin{center}
+What happend when $venc < u_{max}$ ?
+\begin{figure}
+ \includegraphics[width=0.45\textwidth]{images/v70.png}
+ \caption{Measurement set with $venc = 70 \% u_{max}$}
+\end{figure}
+\end{center}
+\end{frame}
+
+
+
+
+\begin{frame}
+ \frametitle{Easy example: with $venc = 70 \% u_{max}$}
+ \footnotesize
+$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
+ \begin{columns}
+  \footnotesize
+ \column{.45\textwidth}
+ \begin{figure}
+  \onslide<1-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (17520,17520,17520)$
+  \onslide<2->
+        \includegraphics[width=1.2\textwidth]{images/U_Pb_V70.png}
+         \includegraphics[width=1.2\textwidth]{images/Rd_Pb_V70.png}
+\end{figure}
+\column{.45\textwidth}
+\begin{figure}
+  \onslide<1->  \textbf{Test II:} $U_0 = 40$  $\vec{R_{d,0}}= (4000,4000,4000)$
+ \onslide<2->
+        \includegraphics[width=1.2\textwidth]{images/U_Pc_V70.png}
+         \includegraphics[width=1.2\textwidth]{images/Rd_Pc_V70.png}
+\end{figure}
+\end{columns} 
+\end{frame}
+
+
+
+
+\begin{frame}
+ \frametitle{The Kalman Filter: A frequency defined functional}
+\begin{itemize}
+\item<1-> We proposed a change in the cost functional:
+\onslide<2->
+\begin{equation*}
+\hat{\theta} = arg \min_{\theta} J(\theta)
+\end{equation*}
+\begin{equation}
+J(\theta) = \displaystyle \frac{1}{2} || \theta - \theta_0  ||^2_{P_0^{-1}}  + \color{red} \sum_{k=1}^N  1- cos \big ( \frac{\pi}{venc} \cdot ( Z_k - \mathbb{H} X_k  ) \big )
+\end{equation}
+\end{itemize}
+
+\end{frame}
+
+
+
+\begin{frame}
+ \frametitle{Easy example: with $venc = 70 \% u_{max}$}
+ \footnotesize
+$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
+ \begin{columns}
+  \footnotesize
+ \column{.45\textwidth}
+ \begin{figure}
+  \onslide<1-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (17520,17520,17520)$
+  \onslide<2->
+        \includegraphics[width=1.2\textwidth]{images/HU_Pb_V70.png}
+         \includegraphics[width=1.2\textwidth]{images/HRd_Pb_V70.png}
+\end{figure}
+\column{.45\textwidth}
+\begin{figure}
+  \onslide<1->  \textbf{Test II:} $U_0 = 40$  $\vec{R_{d,0}}= (4000,4000,4000)$
+ \onslide<3->
+        \includegraphics[width=1.2\textwidth]{images/HU_Pc_V70.png}
+         \includegraphics[width=1.2\textwidth]{images/HRd_Pc_V70.png}
+\end{figure}
+\end{columns} 
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Aliased data}
+\begin{center}
+Or even higher aliasing...
+\begin{figure}
+ \includegraphics[width=0.45\textwidth]{images/v30.png}
+ \caption{Measurement set with $venc = 30 \% u_{max}$}
+\end{figure}
+\end{center}
+\end{frame}
+
+
+
+
+\begin{frame}
+ \frametitle{Easy example: with $venc = 30 \% u_{max}$}
+ \footnotesize
+$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
+ \begin{columns}
+  \footnotesize
+ \column{.45\textwidth}
+ \begin{figure}
+  \onslide<1-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (17520,17520,17520)$
+  \onslide<2->
+        \includegraphics[width=1.2\textwidth]{images/HU_Pb_V30.png}
+         \includegraphics[width=1.2\textwidth]{images/HRd_Pb_V30.png}
+\end{figure}
+\column{.45\textwidth}
+\begin{figure}
+  \onslide<1->  \textbf{Test II:} $U_0 = 40$  $\vec{R_{d,0}}= (4000,4000,4000)$
+ \onslide<3->
+        \includegraphics[width=1.2\textwidth]{images/HU_Pc_V30.png}
+         \includegraphics[width=1.2\textwidth]{images/HRd_Pc_V30.png}
+\end{figure}
+\end{columns} 
+\end{frame}
+
+
+
+
+\section{Conclusions}
+
+\begin{frame}
+\begin{center}
+Conclusions
+\end{center}
+\end{frame}
+
+
+
+\begin{frame}
+\frametitle{Conclusions}
+ \footnotesize
+\begin{itemize}
+\item<1-> 4D Flow measurements are promising for extracting data via inverse problems
+\item<2-> Noise and aliasing are the typical artifacts involved.
+\item<3-> Using a suitable Kalman filter, have shown to bypass aliasing defining the funcional in terms of the frequencies.
+\end{itemize}
+
+\onslide<4-> Future Work
+
+\begin{itemize}
+\item<5-> To include the capacitancies in the inverse problem (adding some pressure meas.)
+\item<6-> Real data!
+\end{itemize}
+
+\end{frame}
+
+
+\begin{frame}
+\begin{center}
+Thank you for your time!
+\end{center}
+
+
+\end{frame}
+
+
+
+
+ 
+\end{document}
+
+
+
+
+
+
+
+%\begin{frame}
+% \frametitle{Results}
+%\footnotesize
+%
+%\begin{figure}
+%\begin{subfigure}{.31\textwidth}
+%  \centering
+%       % \includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/u_15.png}
+%  \caption*{(a) $\vec{u}_{meas}$}
+%\end{subfigure}
+%\begin{subfigure}{.01\textwidth}
+% \hfill
+%\end{subfigure}
+%\begin{subfigure}{.31\textwidth}
+%  \centering
+%        %\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/w_15.png}
+%  \caption*{(b) $\vec{w}$}
+%\end{subfigure}
+%\begin{subfigure}{.01\textwidth}
+% \hfill
+%\end{subfigure}
+%\begin{subfigure}{.31\textwidth}
+%  \centering
+%        %\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/uc_15.png}
+%  \caption*{(c) $\vec{u}_{meas}+\vec{w}$}
+%\end{subfigure}
+%\caption{Measurements, corrector fields and corrected velocities for all the cases.}
+%\label{fig:phantom_resolution}
+%\end{figure}
+%
+%\end{frame}

presentations/press_8ecm/press.toc

@@ -0,0 +1,6 @@
+\babel@toc {english}{}
+\beamer@sectionintoc {1}{4D flow MRI}{3}{0}{1}
+\beamer@sectionintoc {2}{The mathematical model}{6}{0}{2}
+\beamer@sectionintoc {3}{The inverse problem}{23}{0}{3}
+\beamer@sectionintoc {4}{Numerical Experiments}{37}{0}{4}
+\beamer@sectionintoc {5}{Conclusions}{56}{0}{5}