diff --git a/CMakeLists.txt b/CMakeLists.txt
index 47d5ce4497085d31c495001e4e6e2421983c534b..57b93a56b35f80de29700d6e2c8ee241a83dd947 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -13,24 +13,15 @@ macro(set_config_option STRING VARNAME)
   message(STATUS "Found " ${STRING})
 endmacro()
 
+set(CMAKE_CXX_STANDARD 17)
+
 include(CheckCXXCompilerFlag)
-check_cxx_compiler_flag("-std=c++14" HAVE_CXX14)
-if(HAVE_CXX14)
-  set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -std=c++14")
-else()
-  message(ERROR "The compiler ${CMAKE_CXX_COMPILER} has no C++14 support. Please use a different C++ compiler.")
-endif()
 
 check_cxx_compiler_flag("-Wall" HAVE_WALL)
 if(HAVE_WALL)
   set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -Wall")
 endif()
 
-check_cxx_compiler_flag("-O3" HAVE_O3)
-if(HAVE_O3)
-  set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -O3")
-endif()
-
 ####
 #   Mandatory libs
 ####
@@ -234,8 +225,8 @@ add_library( fwi ${LIB_SOURCES} ${LIB_HEADERS})
 
 add_executable( synthetics synthetics.cpp ${EXTRA_INCS})
 target_link_libraries( synthetics fwi  ${EXTRA_LIBS})
-add_executable( time_synthetics_wave time_synthetics_wave.cpp ${EXTRA_INCS})
-target_link_libraries( time_synthetics_wave fwi  ${EXTRA_LIBS})
+#add_executable( time_synthetics_wave time_synthetics_wave.cpp ${EXTRA_INCS})
+#target_link_libraries( time_synthetics_wave fwi  ${EXTRA_LIBS})
 add_executable( directional directional.cpp ${EXTRA_INCS})
 target_link_libraries( directional fwi  ${EXTRA_LIBS})
 add_executable( directional_field directional_field.cpp ${EXTRA_INCS})
diff --git a/common/model/discretization.h b/common/model/discretization.h
index 697dcfc8197f006422dd9712b56950d1cdcf46db..0492513e5f781433469947fde494bcde66318de1 100644
--- a/common/model/discretization.h
+++ b/common/model/discretization.h
@@ -5,7 +5,7 @@
 #include <string>
 //GmshFem library
 #include "GmshFem.h"
-#include "FunctionSpaceForm0.h"
+#include "FunctionSpace.h"
 #include "Message.h"
 
 namespace model
diff --git a/common/wave/correlation.h b/common/wave/correlation.h
index 554fbc2bb9fd493b7cc471e8cbbb2e15771c180d..85bce9727a2987beb393b7e73a6e78439ff9ae2d 100644
--- a/common/wave/correlation.h
+++ b/common/wave/correlation.h
@@ -52,7 +52,7 @@ namespace gmshfem
       {
       }
 
-      virtual void operator()(const WaveMultiField<T_Physic> *fieldA, const WaveMultiField<T_Physic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const = 0;
+      virtual void operator()(const WaveMultiField<T_Physic> *fieldA, const WaveMultiField<T_Physic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const = 0;
     };
 
     template< class T_Op, Physic T_Physic >
@@ -96,31 +96,23 @@ namespace gmshfem
         return false;
       }
 
-      virtual void evaluate(std::vector< typename MathObject< typename T_Op::D_Scalar, T_Op::D_Degree >::Object > &values, const std::vector< scalar::Precision< typename T_Op::D_Scalar > > &points, const std::vector< scalar::Precision< typename T_Op::D_Scalar > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override
+      virtual void evaluate(OutputVector< typename T_Op::D_Scalar, T_Op::D_Degree > &values, const std::vector< scalar::Precision< typename T_Op::D_Scalar >, numa::allocator< scalar::Precision< typename T_Op::D_Scalar > > > &points, const std::vector< scalar::Precision< typename T_Op::D_Scalar > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override
       {
-#pragma omp single
+        if(values.size() < points.size() / 3)
         {
-          if(this->_memorize) {
-            this->getFromMemory(values, gaussPoints, elementType, entity);
-          }
-        }
-        if(!this->_isInMemory) {
-#pragma omp single
-          values.resize(points.size() / 3);
-
-          _op(_fieldA, _fieldB, values, points, gaussPoints, elementType, entity);
 #pragma omp barrier
-          if(this->_memorize) {
-#pragma omp single
-            this->setInMemory(values, gaussPoints, elementType, entity);
-          }
+            FunctionAllocator< typename MathObject< typename T_Op::D_Scalar, T_Op::D_Degree >::Object >::initalizeNUMAMemory(points.size() / 3);
+#pragma omp master
+            values.resize(points.size() / 3);
+#pragma omp barrier
         }
+        _op(_fieldA, _fieldB, values, points, gaussPoints, elementType, entity);
+#pragma omp barrier
       }
 
       virtual BiWaveMultiFieldNode< T_Op,T_Physic > *copy() const override
       {
         BiWaveMultiFieldNode< T_Op,T_Physic > *cp = new BiWaveMultiFieldNode< T_Op,T_Physic >(_op, _fieldA, _fieldB);
-        if(this->_memorize) cp->activateMemorization();
         return cp;
       }
 
@@ -161,7 +153,7 @@ namespace gmshfem
       {
       }
 
-      virtual void operator()(const WaveMultiField<T_Physic>* field, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const = 0;
+      virtual void operator()(const WaveMultiField<T_Physic>* field, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const = 0;
     };
 
     template< class T_Op, Physic T_Physic >
@@ -203,31 +195,24 @@ namespace gmshfem
         return false;
       }
 
-      virtual void evaluate(std::vector< typename MathObject< typename T_Op::D_Scalar, T_Op::D_Degree >::Object > &values, const std::vector< scalar::Precision< typename T_Op::D_Scalar > > &points, const std::vector< scalar::Precision< typename T_Op::D_Scalar > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override
+      virtual void evaluate(OutputVector< typename T_Op::D_Scalar, T_Op::D_Degree > &values, const std::vector< scalar::Precision< typename T_Op::D_Scalar >, numa::allocator< scalar::Precision< typename T_Op::D_Scalar > > > &points, const std::vector< scalar::Precision< typename T_Op::D_Scalar > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override
       {
-#pragma omp single
-        {
-          if(this->_memorize) {
-            this->getFromMemory(values, gaussPoints, elementType, entity);
-          }
-        }
-        if(!this->_isInMemory) {
-#pragma omp single
-          values.resize(points.size() / 3);
-
-          _op(_field, values, points, gaussPoints, elementType, entity);
+          if(values.size() < points.size() / 3)
+          {
+#pragma omp barrier
+            FunctionAllocator< typename MathObject< typename T_Op::D_Scalar, T_Op::D_Degree >::Object >::initalizeNUMAMemory(points.size() / 3);
+#pragma omp master
+            values.resize(points.size() / 3);
 #pragma omp barrier
-          if(this->_memorize) {
-#pragma omp single
-            this->setInMemory(values, gaussPoints, elementType, entity);
-          }
-        }
       }
 
+      _op(_field, values, points, gaussPoints, elementType, entity);
+#pragma omp barrier
+    }
+
       virtual WaveMultiFieldNode< T_Op,T_Physic > *copy() const override
       {
         WaveMultiFieldNode< T_Op,T_Physic > *cp = new WaveMultiFieldNode< T_Op,T_Physic >(_op, _field);
-        if(this->_memorize) cp->activateMemorization();
         return cp;
       }
 
diff --git a/error.cpp b/error.cpp
index 903e8b118a738a08211d557d55a0bac6b41223ca..c4896285ffbb29f60eebcf8478413957d63c1534 100644
--- a/error.cpp
+++ b/error.cpp
@@ -4,6 +4,8 @@
 #include "GmshFem.h"
 #include "Post.h"
 #include "Function.h"
+#include "CSVio.h"
+
 using namespace gmshfem;
 using namespace gmshfem::common;
 using namespace gmshfem::function;
diff --git a/input/paper2/extended/common.txt b/input/paper2/extended/common.txt
index 42ea1a3118820f9a6a3d8ee581955db8f6569e43..8d8545fcb16f5742e29556360a50551dc92e4707 100644
--- a/input/paper2/extended/common.txt
+++ b/input/paper2/extended/common.txt
@@ -4,7 +4,6 @@ configuration = surface_acquisition
 #
 #Parametrization
 parametrization = slowness2_real
-model_reference_frequency = 1.
 #
 xoffset = 0
 Ler = 8.712
diff --git a/input/paper2/extended/innerproduct_sst.txt b/input/paper2/extended/innerproduct_sst.txt
index 19a42e35d1c309430817927ae69c3b713f509afd..5e92c82283fee5a8923dd1a2b0a3208bbbac5701 100644
--- a/input/paper2/extended/innerproduct_sst.txt
+++ b/input/paper2/extended/innerproduct_sst.txt
@@ -2,6 +2,6 @@
 scale0 = 0.0
 innerproduct = sobolev_l2
 innerproduct_weight_bnd = 0.
-innerproduct_diag_preconditioner = 1
+innerproduct_diag_preconditioner_scale = 1
 innerproduct_preconditioner_ref0 = 5400
 innerproduct_stabilization = 5400
diff --git a/input/paper2/extended/inversion.txt b/input/paper2/extended/inversion.txt
index 78c11cf5bff218022c5b82994d21a12452ac1aee..e3e68cee2d4a732b02d9bba2d37bb55b3b4fed4a 100644
--- a/input/paper2/extended/inversion.txt
+++ b/input/paper2/extended/inversion.txt
@@ -48,5 +48,5 @@ objective = l2distance
 #
 #Global minimum search
 globalminimum_n_scale=1
-globalminimumsearch = sequentialscale
+globalminimumsearch = mono
 globalminimum_relDecreaseThreshold = 0.0
diff --git a/input/thesis_part1/concrete3/common.txt b/input/thesis_part1/concrete3/common.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5efd909ddf1b45e3cb57dc568aea23bf96207ed1
--- /dev/null
+++ b/input/thesis_part1/concrete3/common.txt
@@ -0,0 +1,94 @@
+prename = concrete3_
+#Configuration
+configuration = rectangular_acquisition
+#
+#Parametrization
+parametrization = slowness2_real
+#
+Re(mbc0) = 11.111
+Im(mbc0) = 0.
+#
+L = 0.0312
+H = 0.0042
+Ler = 0.03
+Her = 0.003
+hx = 0.00015
+hy = 0.00015
+em_offset = 0
+em_step = 1
+re_offset = 0
+re_step = 1
+exclude_bottom_points = 1
+n_excluded_points = 12
+p_ex0 = 329
+p_ex1 = 330
+p_ex2 = 331
+p_ex3 = 332
+p_ex4 = 333
+p_ex5 = 334
+p_ex6 = 349
+p_ex7 = 350
+p_ex8 = 351
+p_ex9 = 352
+p_ex10 = 353
+p_ex11 = 354
+ReceiverOnEmitter=0
+#
+#Equation
+physic = acoustic
+equation = simplewave
+#
+#Discretization
+integration_type = Gauss
+#Wave
+wave_FunctionSpaceType = HierarchicalH1
+#
+equation_integration_degree_blk0 = 5
+equation_integration_degree_bnd0 = 6
+equation_integration_degree_blk1 = 5
+equation_integration_degree_bnd1 = 6
+equation_integration_degree_blk2 = 5
+equation_integration_degree_bnd2 = 6
+equation_integration_degree_blk3 = 7
+equation_integration_degree_bnd3 = 8
+equation_integration_degree_blk4 = 7
+equation_integration_degree_bnd4 = 8
+equation_integration_degree_blk5 = 7
+equation_integration_degree_bnd5 = 8
+equation_integration_degree_blk6 = 9
+equation_integration_degree_bnd6 = 10
+equation_integration_degree_blk7 = 9
+equation_integration_degree_bnd7 = 10
+equation_integration_degree_blk8 = 9
+equation_integration_degree_bnd8 = 10
+#Wave
+wave_FunctionSpaceDegree0 = 2
+wave_FunctionSpaceDegree1 = 2
+wave_FunctionSpaceDegree2 = 2
+wave_FunctionSpaceDegree3 = 3
+wave_FunctionSpaceDegree4 = 3
+wave_FunctionSpaceDegree5 = 3
+wave_FunctionSpaceDegree6 = 4
+wave_FunctionSpaceDegree7 = 4
+wave_FunctionSpaceDegree8 = 4
+#
+#Frequencies
+n_freq = 9
+#Frequency0
+frequency0 = 100.
+#Frequency1
+frequency1 = 125.
+#Frequency2
+frequency2 = 150.
+#Frequency3
+frequency3= 175.
+#Frequency4
+frequency4 = 200.
+#Frequency5
+frequency5 = 225.
+#Frequency6
+frequency6 = 250.
+#Frequency7
+frequency7 = 275.
+#Frequency8
+frequency8 = 300.
diff --git a/input/thesis_part1/concrete3/gradient.txt b/input/thesis_part1/concrete3/gradient.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6d15a6fd81d5c167fae57abd48fa9168eda34b75
--- /dev/null
+++ b/input/thesis_part1/concrete3/gradient.txt
@@ -0,0 +1,32 @@
+name = gradient
+#
+#Configuration
+unknown = all
+m0_typec0 = file
+m0_pathc0 = ../input/thesis_part1/concrete3/large_concrete_s2_initial
+#
+#Objective
+objective = l2distance
+#
+#InnerProduct
+innerproduct_diag_preconditioner_scale = 0
+innerproduct_diag_preconditioner_smooth = 0
+innerproduct_preconditioner_ref = 1.
+innerproduct_integration_degree_blk = 7
+innerproduct_integration_degree_bnd = 8
+#
+#Scale
+scale0 = 5e-4
+scaleN = 5e-2
+interval = log
+N = 2
+#
+#Frequency
+n_group=1
+#
+data = concrete3_synthetics_data
+#
+#Discretization
+#Model
+model_FunctionSpaceType = HierarchicalH1
+model_FunctionSpaceDegree = 1
diff --git a/input/thesis_part1/concrete3/large_concrete_s2.csv b/input/thesis_part1/concrete3/large_concrete_s2.csv
new file mode 100644
index 0000000000000000000000000000000000000000..07080d98a68849d220bc090afdcb5059918dc21d
--- /dev/null
+++ b/input/thesis_part1/concrete3/large_concrete_s2.csv
@@ -0,0 +1,35 @@
+211
+31
+-0.00015 0 0.00015 0.0003 0.00045 0.0006 0.00075 0.0009 0.00105 0.0012 0.00135 0.0015 0.00165 0.0018 0.00195 0.0021 0.00225 0.0024 0.00255 0.0027 0.00285 0.003 0.00315 0.0033 0.00345 0.0036 0.00375 0.0039 0.00405 0.0042 0.00435 0.0045 0.00465 0.0048 0.00495 0.0051 0.00525 0.0054 0.00555 0.0057 0.00585 0.006 0.00615 0.0063 0.00645 0.0066 0.00675 0.0069 0.00705 0.0072 0.00735 0.0075 0.00765 0.0078 0.00795 0.0081 0.00825 0.0084 0.00855 0.0087 0.00885 0.009 0.00915 0.0093 0.00945 0.0096 0.00975 0.0099 0.01005 0.0102 0.01035 0.0105 0.01065 0.0108 0.01095 0.0111 0.01125 0.0114 0.01155 0.0117 0.01185 0.012 0.01215 0.0123 0.01245 0.0126 0.01275 0.0129 0.01305 0.0132 0.01335 0.0135 0.01365 0.0138 0.01395 0.0141 0.01425 0.0144 0.01455 0.0147 0.01485 0.015 0.01515 0.0153 0.01545 0.0156 0.01575 0.0159 0.01605 0.0162 0.01635 0.0165 0.01665 0.0168 0.01695 0.0171 0.01725 0.0174 0.01755 0.0177 0.01785 0.018 0.01815 0.0183 0.01845 0.0186 0.01875 0.0189 0.01905 0.0192 0.01935 0.0195 0.01965 0.0198 0.01995 0.0201 0.02025 0.0204 0.02055 0.0207 0.02085 0.021 0.02115 0.0213 0.02145 0.0216 0.02175 0.0219 0.02205 0.0222 0.02235 0.0225 0.02265 0.0228 0.02295 0.0231 0.02325 0.0234 0.02355 0.0237 0.02385 0.024 0.02415 0.0243 0.02445 0.0246 0.02475 0.0249 0.02505 0.0252 0.02535 0.0255 0.02565 0.0258 0.02595 0.0261 0.02625 0.0264 0.02655 0.0267 0.02685 0.027 0.02715 0.0273 0.02745 0.0276 0.02775 0.0279 0.02805 0.0282 0.02835 0.0285 0.02865 0.0288 0.02895 0.0291 0.02925 0.0294 0.02955 0.0297 0.02985 0.03 0.03015 0.0303 0.03045 0.0306 0.03075 0.0309 0.03105 0.0312 0.03135
+-0.00015 0 0.00015 0.0003 0.00045 0.0006 0.00075 0.0009 0.00105 0.0012 0.00135 0.0015 0.00165 0.0018 0.00195 0.0021 0.00225 0.0024 0.00255 0.0027 0.00285 0.003 0.00315 0.0033 0.00345 0.0036 0.00375 0.0039 0.00405 0.0042 0.00435
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diff --git a/input/thesis_part1/concrete3/large_concrete_s2_initial.csv b/input/thesis_part1/concrete3/large_concrete_s2_initial.csv
new file mode 100644
index 0000000000000000000000000000000000000000..1261a71478829e89d8641c60e7839a3de5067c65
--- /dev/null
+++ b/input/thesis_part1/concrete3/large_concrete_s2_initial.csv
@@ -0,0 +1,35 @@
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diff --git a/input/thesis_part1/concrete3/synthetics.txt b/input/thesis_part1/concrete3/synthetics.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d135a753f9596942da2df671ca91db71764c4689
--- /dev/null
+++ b/input/thesis_part1/concrete3/synthetics.txt
@@ -0,0 +1,10 @@
+name = synthetics
+#Configuration
+unknown = none
+m0_typec0 = file
+m0_pathc0 = ../input/thesis_part1/concrete3/large_concrete_s2
+#
+#Output
+write_data_fields = 0
+write_model_field0 = 1
+write_wave_fields = 0
diff --git a/input/thesis_part1/cross/common.txt b/input/thesis_part1/cross/common.txt
new file mode 100644
index 0000000000000000000000000000000000000000..103dc71780dc49f1eab38ac773e5856b43c9a373
--- /dev/null
+++ b/input/thesis_part1/cross/common.txt
@@ -0,0 +1,63 @@
+prename = cross_
+#Equation
+physic = acoustic
+equation = simplewave
+equation_boundary = 0
+#
+#Parametrization
+parametrization = permcond_bireal
+parametrization_frequency0 = 0.33
+#
+#Configuration
+configuration = circular_acquisition
+#
+rer = 5.
+rext = 7.
+np = 120
+emitter_offset = 0
+emitter_skip = 3
+receiver_offset = 0
+receiver_skip = 1
+receiver_on_emitter=0
+#
+Re(mbc0) = 4.0
+Im(mbc0) = 0.
+Re(mbc1) = 0.5393
+Im(mbc1) = 0.
+#
+n_freq = 7
+#Frequency
+frequency0 = 0.17
+frequency1 = 0.2
+frequency2 = 0.23
+frequency3 = 0.27
+frequency4 = 0.33
+frequency5 = 0.5
+frequency6 = 0.67
+#
+#Discretization
+h = 0.1
+integration_type = Gauss
+wave_FunctionSpaceType = HierarchicalH1
+#
+wave_FunctionSpaceDegree0 = 1
+equation_integration_degree_blk0 = 3
+equation_integration_degree_bnd0 = 4
+wave_FunctionSpaceDegree1 = 1
+equation_integration_degree_blk1 = 3
+equation_integration_degree_bnd1 = 4
+wave_FunctionSpaceDegree2 = 2
+equation_integration_degree_blk2 = 5
+equation_integration_degree_bnd2 = 6
+wave_FunctionSpaceDegree3 = 2
+equation_integration_degree_blk3 = 5
+equation_integration_degree_bnd3 = 6
+wave_FunctionSpaceDegree4 = 2
+equation_integration_degree_blk4 = 5
+equation_integration_degree_bnd4 = 6
+wave_FunctionSpaceDegree5 = 3
+equation_integration_degree_blk5 = 7
+equation_integration_degree_bnd5 = 8
+wave_FunctionSpaceDegree6 = 4
+equation_integration_degree_blk6 = 9
+equation_integration_degree_bnd6 = 10
diff --git a/input/thesis_part1/cross/gradient.txt b/input/thesis_part1/cross/gradient.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3e1103135b20d44eb1cc7ef9668bea7b3c73ba1b
--- /dev/null
+++ b/input/thesis_part1/cross/gradient.txt
@@ -0,0 +1,31 @@
+name = gradient
+#Configuration
+unknown = all
+#
+#Objective
+objective = l2distance
+#
+#
+#InnerProduct
+innerproduct_diag_preconditioner_scale = 0
+innerproduct_diag_preconditioner_smooth = 0
+innerproduct_preconditioner_ref = 1.
+innerproduct_integration_degree_blk = 7
+innerproduct_integration_degree_bnd = 8
+#
+#Scale
+scale0 = 0.75
+scaleN = 0.75
+interval = linear
+N = 0
+#
+#Frequency
+n_group=1
+#
+#Data
+data = cross_synthetics_data
+#
+#Discretization
+#Model
+model_FunctionSpaceType = HierarchicalH1
+model_FunctionSpaceDegree = 1
diff --git a/input/thesis_part1/cross/synthetics.txt b/input/thesis_part1/cross/synthetics.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8f980adb475a8f782132d7afe1b46ad496a13128
--- /dev/null
+++ b/input/thesis_part1/cross/synthetics.txt
@@ -0,0 +1,29 @@
+name = synthetics
+#Configuration
+unknown = none
+#
+inclusion_ni = 2
+inclusion_filled = 1
+#Inclusion 0
+inclusion0 = cross
+inclusion_x0i0 = -1.
+inclusion_y0i0 = -1.
+inclusion_ri0 = 1.0
+Re(mi0c0) = 1.
+Im(mi0c0) = 0.
+Re(mi0c1) = 0.0180
+Im(mi0c1) = 0.
+#Inclusion 1
+inclusion1 = cross
+inclusion_x0i1 = 1.25
+inclusion_y0i1 = 1.25
+inclusion_ri1 = 1.0
+Re(mi1c0) = 8.
+Im(mi1c0) = 0.
+Re(mi1c1) = 1.7975
+Im(mi1c1) = 0.
+#
+#Output
+write_data_fields = 0
+write_wave_fields = 0
+write_model_field0 = 1
diff --git a/input/thesis_part1/marmousi/common.txt b/input/thesis_part1/marmousi/common.txt
new file mode 100644
index 0000000000000000000000000000000000000000..913af4babdba49135628353813850df29c8fb69e
--- /dev/null
+++ b/input/thesis_part1/marmousi/common.txt
@@ -0,0 +1,54 @@
+prename = thesis_part1_marmousi_
+#Configuration
+configuration = surface_acquisition
+#
+#Parametrization
+parametrization = slowness2_real
+model_reference_frequency = 1.
+#
+xoffset = 0
+Ler = 8.712
+np = 243
+re_start = 0
+re_skip = 1
+em_start = 0
+em_skip = 2
+ReceiverOnEmitter=0
+ymax = 0.216
+ymin = 0.072
+L = 9.192
+H = 2.904
+#
+Re(m_superc0) = 0.44
+Im(m_superc0) = 0.
+Re(m_subc0) = 0.25
+Im(m_subc0) = 0.
+#
+#Equation
+physic = acoustic
+equation = simplewave
+#
+#Discretization
+h = 0.036
+integration_type = Gauss
+#Wave
+wave_FunctionSpaceType = HierarchicalH1
+equation_integration_degree_blk0 = 5
+equation_integration_degree_bnd0 = 6
+equation_integration_degree_blk1 = 7
+equation_integration_degree_bnd1 = 8
+equation_integration_degree_blk2 = 9
+equation_integration_degree_bnd2 = 10
+#Wave
+wave_FunctionSpaceDegree0 = 2
+wave_FunctionSpaceDegree1 = 3
+wave_FunctionSpaceDegree2 = 4
+#
+#Frequencies
+n_freq = 3
+#Frequency0
+frequency0 = 4.
+#Frequency1
+frequency1 = 6.
+#Frequency2
+frequency2 = 8.
diff --git a/input/thesis_part1/marmousi/gradient.txt b/input/thesis_part1/marmousi/gradient.txt
new file mode 100644
index 0000000000000000000000000000000000000000..79b5f04c78da120705fddb8223320562df9c0b20
--- /dev/null
+++ b/input/thesis_part1/marmousi/gradient.txt
@@ -0,0 +1,30 @@
+name = gradient
+#
+#Configuration
+unknown = subsurface
+m0_typec0 = file.pos
+m0_pathc0 = marmousi_initial
+#
+#Objective
+objective = l2distance
+#
+#InnerProduct
+innerproduct_diag_preconditioner_scale = 0
+innerproduct_diag_preconditioner_smooth = 0
+innerproduct_preconditioner_ref = 1.
+innerproduct_integration_degree_blk = 7
+innerproduct_integration_degree_bnd = 8
+#
+#Scale
+scale0 = 0.25
+scaleN = 0.25
+interval = linear
+N = 0
+#
+#Frequency
+n_group=1
+#Data
+data = thesis_part1_marmousi_synthetics_data
+#
+model_FunctionSpaceType = HierarchicalH1
+model_FunctionSpaceDegree = 1
diff --git a/input/thesis_part1/marmousi/synthetics.txt b/input/thesis_part1/marmousi/synthetics.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4f9b521a17a1a8c5c4c2812b005b6e31ebcf327c
--- /dev/null
+++ b/input/thesis_part1/marmousi/synthetics.txt
@@ -0,0 +1,15 @@
+name = synthetics
+#Configuration
+unknown = none
+m0_typec0 = file
+m0_pathc0 = ../input/paper2/marmousi_slowness2
+#
+#Output
+write_data_fields = 1
+write_wave_fields = 0
+write_model_field0 = 1
+#write_single_wave_fields = 4
+#write_single_wave_field0 = 0
+#write_single_wave_field1 = 1
+#write_single_wave_field2 = 2
+#write_single_wave_field3 = 121
diff --git a/plot_timewave.py b/plot_timewave.py
index 56d3999a885fc8a3826c60dd6c71143755f67ff0..ac4d3c4b448eaead683b06a8a76c7ad17282e996 100644
--- a/plot_timewave.py
+++ b/plot_timewave.py
@@ -40,6 +40,7 @@ time_samp = 4
 ΔT = 1.
 """
 
+"""
 # Concrete 3
 
 path = "concrete3_data/"
@@ -59,6 +60,7 @@ Te=100 #[ms]
 time_samp = 4
 ΔNr = 50
 ΔT = 25 #[ms]
+"""
 
 # Cross
 """
@@ -82,7 +84,6 @@ time_samp = 20
 ΔT = 10
 """
 
-"""
 # S-Cube
 path = "win_data/"
 
@@ -98,7 +99,6 @@ Te=6.98
 time_samp = 1
 ΔNr = 80
 ΔT = 4.
-"""
 
 #
 # Time domain
@@ -124,6 +124,6 @@ plt.pcolormesh(np.arange(0,Nr+1,1)-0.5,time-Δt*time_samp/2,data[:,:],vmin=-ampl
 plt.xticks(np.arange(0,Nr,ΔNr));
 plt.yticks(np.arange(0,Te*1.1,ΔT));
 plt.xlabel(r'receiver index [-]')
-#plt.ylabel(r'time [s]')
-plt.ylabel(r'time [ms]')
+plt.ylabel(r'time [s]')
+#plt.ylabel(r'time [ms]')
 #plt.ylabel(r'$c_{0}$ $\times$ time [m]')
\ No newline at end of file
diff --git a/specific/model/innerproduct/sobolev.cpp b/specific/model/innerproduct/sobolev.cpp
index 7bcfbc9e5de5797417be94178a31a2c8436a2f36..73fbc784a16546079fbabf2e47609dbb592fb485 100644
--- a/specific/model/innerproduct/sobolev.cpp
+++ b/specific/model/innerproduct/sobolev.cpp
@@ -380,7 +380,14 @@ namespace sobolev_df
         std::complex<double> sum = 0.;
         for (unsigned int c = 0; c < _m_size; c++)
         {
-            sum += _weight[c] * gmshfem::post::integrate( grad(m1[c]) * _D * grad(m2[c]), _config->model_unknown(Support::BLK), integrationType(6*degree-2) ) / _beta[c] / _beta[c];
+            if(_structureIsUpToDate)
+            {
+                sum += _weight[c] * gmshfem::post::integrate( grad(m1[c]) * _D * grad(m2[c]), _config->model_unknown(Support::BLK), integrationType(6*degree-2) ) / _beta[c] / _beta[c];
+            }
+            else
+            {
+                msg::warning << "Penalization requested while structure is not up to date." << msg::endl;
+            }
         }
         return sum;
     };
diff --git a/specific/optimization/localminimumsearch/classic.cpp b/specific/optimization/localminimumsearch/classic.cpp
index a401e69d478677118968485442a71fc7cefa37a3..3c345564968a2a42a365d96ab1ad4540d6cf5404 100644
--- a/specific/optimization/localminimumsearch/classic.cpp
+++ b/specific/optimization/localminimumsearch/classic.cpp
@@ -4,9 +4,6 @@
 //GmshFWI Library
 #include "classic.h"
 
-//to be removed or implemented properly
-//#include "../../../common/statefunctional.h"
-
 using namespace gmshfem;
 using namespace gmshfem::common;
 
@@ -91,9 +88,6 @@ namespace classic
         double mmn = 0.;
 
         double j0 = functional->performance();
-        //to be removed or implemented properly
-        //StateFunctional<Physic::acoustic>* const statefunctional = dynamic_cast<StateFunctional<Physic::acoustic>* const>(functional);
-        //statefunctional->write_data(Type::FS,_name+"d"+std::to_string(0));
 
         double jn = 0.;
         double jn_1 = 0.;
@@ -146,12 +140,7 @@ namespace classic
                 n++;
                 descentsearch->update(functional);
                 functional->iteration(n);
-                if((n % _writeInterval) == 0){
-                    m->write(_name+"m"+std::to_string(n));
-                    //to be removed or implemented properly
-                    //StateFunctional<Physic::acoustic>* const statefunctional = dynamic_cast<StateFunctional<Physic::acoustic>* const>(functional);
-                    //statefunctional->write_data(Type::FS,_name+"d"+std::to_string(n));
-                }
+                if((n % _writeInterval) == 0){m->write(_name+"m"+std::to_string(n));}
                 msg::unindent();
             }
         }
diff --git a/specific/wave/correlation.cpp b/specific/wave/correlation.cpp
index a15a693ce4bcb4de440f9d8e187f7364d0f4d07f..05dc7ac2b0a0eb0efbe8946b71be5fe3ba7607b5 100644
--- a/specific/wave/correlation.cpp
+++ b/specific/wave/correlation.cpp
@@ -34,7 +34,7 @@ namespace gmshfem
     namespace function
     {
         /* acoustic */
-        void Correlate<Physic::acoustic>::operator()(const WaveMultiField<Physic::acoustic> *fieldA, const WaveMultiField<Physic::acoustic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const
+        void Correlate<Physic::acoustic>::operator()(const WaveMultiField<Physic::acoustic> *fieldA, const WaveMultiField<Physic::acoustic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const
         {
             const unsigned int nbrOfElements = points.size() / gaussPoints.size();
             const unsigned int numThreads = omp::getNumThreads();
@@ -103,7 +103,7 @@ namespace gmshfem
         }
 
         /* electromagnetic */
-        void Correlate<Physic::electromagnetic>::operator()(const WaveMultiField<Physic::electromagnetic> *fieldA, const WaveMultiField<Physic::electromagnetic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const
+        void Correlate<Physic::electromagnetic>::operator()(const WaveMultiField<Physic::electromagnetic> *fieldA, const WaveMultiField<Physic::electromagnetic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const
         {
             const unsigned int nbrOfElements = points.size() / gaussPoints.size();
             const unsigned int numThreads = omp::getNumThreads();
@@ -144,7 +144,7 @@ namespace gmshfem
                 scalar::move(_determinants, _gmshDeterminants);
             }
 
-            std::vector< typename MathObject< std::complex<double>, Degree::Degree1 >::Object > normal_values;
+            OutputVector< std::complex<double>, Degree::Degree1 > normal_values;
             if(_trace!=CorrelationTrace::IDENTITY)
             {
                 normal_values.resize(values.size());
@@ -246,7 +246,7 @@ namespace gmshfem
         }
 
         /* elastodynamic */
-        void Correlate<Physic::elastodynamic>::operator()(const WaveMultiField<Physic::elastodynamic> *fieldA, const WaveMultiField<Physic::elastodynamic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const
+        void Correlate<Physic::elastodynamic>::operator()(const WaveMultiField<Physic::elastodynamic> *fieldA, const WaveMultiField<Physic::elastodynamic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const
         {
             const unsigned int nbrOfElements = points.size() / gaussPoints.size();
             const unsigned int numThreads = omp::getNumThreads();
@@ -287,7 +287,7 @@ namespace gmshfem
                 scalar::move(_determinants, _gmshDeterminants);
             }
 
-            std::vector< typename MathObject< std::complex<double>, Degree::Degree1 >::Object > normal_values;
+            OutputVector< std::complex<double>, Degree::Degree1 > normal_values;
             if(_trace!=CorrelationTrace::IDENTITY)
             {
                 normal_values.resize(values.size());
@@ -409,7 +409,7 @@ namespace gmshfem
     namespace function
     {
         /* acoustic */
-        void DerivativeCorrelate<Physic::acoustic>::operator()(const WaveMultiField<Physic::acoustic> *fieldA, const WaveMultiField<Physic::acoustic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const
+        void DerivativeCorrelate<Physic::acoustic>::operator()(const WaveMultiField<Physic::acoustic> *fieldA, const WaveMultiField<Physic::acoustic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const
         {
             const unsigned int nbrOfElements = points.size() / gaussPoints.size();
             const unsigned int numThreads = omp::getNumThreads();
@@ -485,7 +485,7 @@ namespace gmshfem
         }
 
         /* elastodynamic */
-        void DerivativeCorrelate<Physic::elastodynamic>::operator()(const WaveMultiField<Physic::elastodynamic> *fieldA, const WaveMultiField<Physic::elastodynamic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const
+        void DerivativeCorrelate<Physic::elastodynamic>::operator()(const WaveMultiField<Physic::elastodynamic> *fieldA, const WaveMultiField<Physic::elastodynamic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const
         {
             const unsigned int nbrOfElements = points.size() / gaussPoints.size();
             const unsigned int numThreads = omp::getNumThreads();
diff --git a/specific/wave/correlation.h b/specific/wave/correlation.h
index a5893884ecabb41824aadd8182ff405ead85d572..f169ea34bddd749a48003919edbf12887f930ce1 100644
--- a/specific/wave/correlation.h
+++ b/specific/wave/correlation.h
@@ -56,7 +56,7 @@ namespace gmshfem
                 return "Correlate";
             }
 
-            virtual void operator()(const WaveMultiField<Physic::acoustic> *fieldA, const WaveMultiField<Physic::acoustic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override;
+            virtual void operator()(const WaveMultiField<Physic::acoustic> *fieldA, const WaveMultiField<Physic::acoustic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override;
         };
 
         /* electromagnetic */
@@ -86,7 +86,7 @@ namespace gmshfem
                 return "Correlate "+trace_to_string(_trace);
             }
 
-            void operator()(const WaveMultiField<Physic::electromagnetic> *fieldA, const WaveMultiField<Physic::electromagnetic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override;
+            void operator()(const WaveMultiField<Physic::electromagnetic> *fieldA, const WaveMultiField<Physic::electromagnetic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override;
         };
 
         /* elastodynamic */
@@ -119,7 +119,7 @@ namespace gmshfem
                 return "Correlate "+trace_to_string(_trace);
             }
 
-            void operator()(const WaveMultiField<Physic::elastodynamic> *fieldA, const WaveMultiField<Physic::elastodynamic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override;
+            void operator()(const WaveMultiField<Physic::elastodynamic> *fieldA, const WaveMultiField<Physic::elastodynamic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override;
         };
     }
 };
@@ -189,7 +189,7 @@ namespace gmshfem
                 return "Correlate "+trace_to_string(_trace);
             }
 
-            virtual void operator()(const WaveMultiField<Physic::acoustic> *fieldA, const WaveMultiField<Physic::acoustic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override;
+            virtual void operator()(const WaveMultiField<Physic::acoustic> *fieldA, const WaveMultiField<Physic::acoustic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override;
         };
 
         /* elastodynamic */
@@ -222,7 +222,7 @@ namespace gmshfem
                 return "Correlate "+trace_to_string(_trace);
             }
 
-            virtual void operator()(const WaveMultiField<Physic::elastodynamic> *fieldA, const WaveMultiField<Physic::elastodynamic> *fieldB, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override;
+            virtual void operator()(const WaveMultiField<Physic::elastodynamic> *fieldA, const WaveMultiField<Physic::elastodynamic> *fieldB, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override;
         };
     }
 };
@@ -277,7 +277,7 @@ namespace gmshfem
                 return "Auto-correlate";
             }
 
-            void operator()(const WaveMultiField<T_Physic>* field, std::vector< typename MathObject< std::complex<double>, Degree::Degree0 >::Object > &values, const std::vector< scalar::Precision< std::complex<double> > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override
+            void operator()(const WaveMultiField<T_Physic>* field, OutputVector< std::complex<double>, Degree::Degree0 > &values, const std::vector< scalar::Precision< std::complex<double> >, numa::allocator< scalar::Precision< std::complex<double> > > > &points, const std::vector< scalar::Precision< std::complex<double> > > &gaussPoints, const int elementType, const std::pair< int, int > &entity) const override
             {
                 const unsigned int nbrOfElements = points.size() / gaussPoints.size();
                 const unsigned int numThreads = omp::getNumThreads();
diff --git a/specific/wave/discretization.h b/specific/wave/discretization.h
index 307e1b6fe218dc0336bb6a8403480e324f332701..93d6511e3e913b861d30db1123e8ad5148df072f 100644
--- a/specific/wave/discretization.h
+++ b/specific/wave/discretization.h
@@ -6,7 +6,7 @@
 
 //GmshFEM library
 #include "GmshFem.h"
-#include "FunctionSpaceForm0.h"
+#include "FunctionSpace.h"
 #include "Message.h"
 #include "FieldObject.h"