draft parametric problems tutorial

Author: shanemcq18

docs/source/tutorials/inputs.ipynb

@@ -422,7 +422,7 @@
    "source": [
     "We will use a {class}`opinf.basis.PODBasis` to reduce the dimension of the snapshot training data, which approximates the discretized state vector as $\\q(t) \\approx \\Vr\\qhat(t)$ for some $\\Vr\\in\\RR^{n\\times r}$ with orthonormal columns and $\\qhat(t)\\in\\RR^{r}$, with and $r\\ll n$.\n",
     "Input training data are *not* typically compressed with dimensionality reduction or subjected to other pre-processing routines.\n",
-    "Because the FOM {eq}`eq_inputs_fom` has the linear-time invariant form $\\ddt\\q(t) = \\A\\q(t) + \\B u(t)$, we seek a ROM with the structure, i.e.,\n",
+    "Because the FOM {eq}`eq_inputs_fom` has the linear-time invariant form $\\ddt\\q(t) = \\A\\q(t) + \\B u(t)$, we seek a ROM with the same structure, i.e.,\n",
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
@@ -432,7 +432,19 @@
     "\\end{aligned}\n",
     "$$\n",
     "\n",
-    "Data for the time derivative $\\ddt\\qhat(t)$ are estimated in this example with sixth-order finite differences using {class}`opinf.ddt.UniformFiniteDifferencer`."
+    "Data for the time derivative $\\ddt\\qhat(t)$ are estimated in this example with sixth-order finite differences using {class}`opinf.ddt.UniformFiniteDifferencer`.\n",
+    "The underlying least-squares problem to determine $\\Ahat$ and $\\Bhat$ is given by\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "    \\min_{\\Ahat,\\Bhat}\n",
+    "    \\sum_{j=0]^{k-1}\\left\\|\n",
+    "        \\Ahat\\qhat_{j} + \\Bhat\\u_j - \\dot{\\qhat}_j\n",
+    "    \\right\\|_{2}^{2},\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "where $\\qhat_j = \\qhat(t_j)\\in\\RR^{r}$ and $u_j = u(t_j)\\in\\RR$ are the state snapshots and input data, respectively, and $\\dot{\\qhat}_j \\approx \\ddt\\qhat(t)|_{t=t_j}\\in\\RR^{r}$ are the estimated time derivatives."
    ]
   },
   {