minor corrections

docs/notebooks/burger.ipynb

@@ -57,24 +57,25 @@
     "where $v \\in V$ is an arbitrary test function. The Jacobian of $F(u^{n+1}, u^n, v)$ is obtained through the Gateaux derivative. \n",
     "\n",
     "The time sequence of the adjoint system for the functional (Eq. (1)) is given by:\n",
-    "\n",
-    "\\begin{bmatrix}\n",
-    "    \\frac{\\partial F_{N + 1}}{\\partial u^{N + 1}} \\lambda^{N + 1} + \\frac{\\partial I}{\\partial u^{N + 1}} = 0 \\\\\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "    \\frac{\\partial F_{N + 1}}{\\partial u^{N + 1}} \\lambda^{N + 1} &= \\frac{\\partial I}{\\partial u^{N + 1}} \\\\\n",
     "    \\\\\n",
-    "    \\frac{\\partial F_{N}}{\\partial u^{N}} \\lambda^{N} + \\frac{\\partial F_{N + 1}}{\\partial u^{N}} \\lambda^{N + 1} = 0 \\\\\n",
+    "    \\frac{\\partial F_{N}}{\\partial u^{N}} \\lambda^{N} + \\frac{\\partial F_{N + 1}}{\\partial u^{N}} \\lambda^{N + 1} &= 0 \\\\\n",
     "    \\\\\n",
     "    \\vdots \\\\\n",
     "    \\\\\n",
-    "    \\frac{\\partial F_{1}}{\\partial u^{1}} \\lambda^{1} + \\frac{\\partial F_{2}}{\\partial u^{1}} \\lambda^{2} = 0 \\\\\n",
+    "    \\frac{\\partial F_{1}}{\\partial u^{1}} \\lambda^{1} + \\frac{\\partial F_{2}}{\\partial u^{1}} \\lambda^{2} &= 0 \\\\\n",
     "    \\\\\n",
-    "    \\frac{\\partial F_{0}}{\\partial u^{0}} \\lambda^{0} + \\frac{\\partial F_{1}}{\\partial u^{0}} \\lambda^{1} = 0\n",
-    "\\end{bmatrix} \\tag{5}\n",
-    "\n",
+    "    \\frac{\\partial F_{0}}{\\partial u^{0}} \\lambda^{0} + \\frac{\\partial F_{1}}{\\partial u^{0}} \\lambda^{1} &= 0\n",
+    "\\end{align*}\n",
+    "\\tag{5}\n",
+    "$$\n",
     "where $\\lambda^{n}$ is the adjoint variable at time step $n$.\n",
     "\n",
     "The adjoint system is solved backward in time, we can verify this by taking the adjoint of Eq. (5), where at the time step zero we have two unknowns, $\\lambda^{0}$ and $\\lambda^{1}$, and at the time step $N$ we have only one unknown, $\\lambda^{N}$.\n",
     "\n",
-    "For additonal details on the adjoint-based gradient computation, you can refer to the following references [1, 2].\n"
+    "For additional details on the discrete adjoint-based gradient computation, you may refer to the following references [1, 2].\n"
    ]
   },
   {
@@ -87,18 +88,18 @@
     "The Burger's equation is discretised using the Finite Element Method (FEM). We use continuous first-order Lagrange basis functions to discretise the spatial domain.\n",
     "\n",
     "After discretising, the residual of the forward system is given by:\n",
-    "$$F = M \\mathbf{U}^{n + 1} - M \\mathbf{U}^{n} + \\Delta t \\left(C(\\mathbf{U}^{n + 1}) \\mathbf{U}^{n + 1} - K \\mathbf{U}^{n + 1}\\right) = 0, \\tag{6}$$\n",
+    "$$\\mathbf{F} = \\mathbf{M} \\mathbf{U}^{n + 1} - \\mathbf{M} \\mathbf{U}^{n} + \\Delta t \\left(\\mathbf{C}(\\mathbf{U}^{n + 1}) \\mathbf{U}^{n + 1} - \\mathbf{K} \\mathbf{U}^{n + 1}\\right) = 0, \\tag{6}$$\n",
     "where $M$ and $K$ are the mass and stiffness, respectively. $U$ represents the solution vector. The matrices $C$ is obtained from the advection term. The superscript $n$ denotes the time step, and $\\Delta t$ is the time step size. The solution vector $\\mathbf{U}^{n + 1}$ is obtained by solving the non-linear forward system using the Newton method. The Jacobian of the forward system has the form:\n",
     "\n",
-    "Let us consider the residual of the forward system as $R(\\mathbf{U}^{n + 1}, \\mathbf{U}^{n})$. The Jacobian of the forward system is given by:\n",
-    "$$J_{n, n} = \\frac{\\partial F^{n}}{\\partial U^{n}} = M + \\Delta t \\left(J_C(\\mathbf{U}^{n}) - K \\right), \\tag{7}$$\n",
-    "$$J_{n + 1, n} = \\frac{\\partial F^{n + 1}}{\\partial U^{n}} = - M, \\tag{8}$$\n",
-    "where $J_C$ is the Jacobian of the advection term. Considering the adjoint as shownd in Eq. (5), and the Jacobian of the forward system (Eq. (7) a (8)), we arrive at the following expression for the adjoint system.\n",
-    "$$J^{T}_{n, n} \\Lambda^{n} + J^{T}_{n+1, n} \\Lambda^{n + 1} = 0, \\tag{9}$$\n",
+    "Let us consider the residual of the forward system as $\\mathbf{R}(\\mathbf{U}^{n + 1}, \\mathbf{U}^{n})$. The Jacobian of the forward system is given by:\n",
+    "$$\\mathbf{J}_{n, n} = \\frac{\\partial \\mathbf{F}^{n}}{\\partial U^{n}} = \\mathbf{M} + \\Delta t \\left(\\mathbf{J_C}(\\mathbf{U}^{n}) - \\mathbf{K} \\right), \\tag{7}$$\n",
+    "$$\\mathbf{J_{n + 1, n}} = \\frac{\\partial \\mathbf{F}^{n + 1}}{\\partial \\mathbf{U}^{n}} = - \\mathbf{M}, \\tag{8}$$\n",
+    "where $\\mathbf{J_C}$ is the Jacobian of the advection term. Considering the adjoint as shownd in Eq. (5), and the Jacobian of the forward system (Eq. (7) a (8)), we arrive at the following expression for the adjoint system.\n",
+    "$$\\mathbf{J}^{T}_{n, n} \\Lambda^{n} + \\mathbf{J}^{T}_{n+1, n} \\boldsymbol{\\Lambda}^{n + 1} = 0, \\tag{9}$$\n",
     "for $n = N - 1, \\ldots, 0$. $\\Lambda$ is the adjoint solution vector. \n",
     "\n",
     "At the time $n = N$, the adjoint is initialized by solving the following system:\n",
-    "$$J^{T}_{N,N} \\Lambda^{N} + \\frac{\\partial I}{\\partial U^{N}} = 0. \\tag{10}$$\n"
+    "$$\\mathbf{J}^{T}_{N,N} \\boldsymbol{\\Lambda}^{N} + \\frac{\\partial \\mathbf{I}}{\\partial \\mathbf{U}^{N}} = 0. \\tag{10}$$\n"
    ]
   },
   {
@@ -171,7 +172,7 @@
     "            if write_adj_deps:\n",
     "                self._store_data(u, step, storage, write_adj_deps, write_ics)\n",
     "            def _residual(u_new):\n",
-    "                C = self._convection_matrix(u_new)\n",
+    "                C = self._advection_matrix(u_new)\n",
     "                residual = (M + self.model[\"dt\"] * (- K + C)).dot(u_new) - M.dot(u)\n",
     "                residual[0] = residual[self.model[\"nx\"] - 1] = 0\n",
     "                return residual\n",
@@ -218,12 +219,6 @@
     "            step -= 1\n",
     "        self.adjoint_work_memory[StorageType.WORK][step] = copy.deepcopy(u)\n",
     "\n",
-    "    def _initialize_adjoint(self):\n",
-    "        J = self._jacobian(self.forward_final_solution, self.model[\"max_n\"], self.model[\"max_n\"])\n",
-    "        M = self._mass_matrix()\n",
-    "        self.adjoint_work_memory[StorageType.WORK][self.model[\"max_n\"]] = spsolve(J.T, - self.forward_final_solution)\n",
-    "\n",
-    "\n",
     "    def gradient(self):\n",
     "        \"\"\"Compute the adjoint-based gradient.\n",
     "\n",
@@ -276,6 +271,11 @@
     "                if step in self.restart_forward[from_storage]:\n",
     "                    del self.restart_forward[from_storage][step]\n",
     "\n",
+    "    def _initialize_adjoint(self):\n",
+    "        J = self._jacobian(self.forward_final_solution, self.model[\"max_n\"], self.model[\"max_n\"])\n",
+    "        M = self._mass_matrix()\n",
+    "        self.adjoint_work_memory[StorageType.WORK][self.model[\"max_n\"]] = spsolve(J.T, - self.forward_final_solution)\n",
+    "\n",
     "    def _jacobian(self, u, i, j):\n",
     "        M = self._mass_matrix()\n",
     "        if i == j:\n",
@@ -327,7 +327,7 @@
     "            K[i, i] = - 2 * b\n",
     "        return K\n",
     "    \n",
-    "    def _convection_matrix(self, u):\n",
+    "    def _advection_matrix(self, u):\n",
     "        num_nodes = self.model[\"nx\"]\n",
     "        # 1D mesh is uniform. Thus, the mesh spacing is constant.\n",
     "        h = self.model[\"lx\"] / self.model[\"nx\"] \n",