Added back-end API figure to tutorial

Author: sichinaga

tutorials/developers-tutorial1/PyDMD-structure.png

@@ -23,15 +23,47 @@
     "In this tutorial we will indeed create from scratch a new version of the original DMD; such extension will be totally useless from a scientific perspective, the only purpose is showing the suggested steps to implement new variants inside **PyDMD**."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## The Back-end API"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "The necessary bricks for building the new DMD version are:\n",
     "\n",
-    "- [`DMDBase`](https://mathlab.github.io/PyDMD/dmdbase.html), the actual backbone of all the different implemented versions;\n",
-    "- `DMDTimeDict`, the class that manages the time window;\n",
-    "- `DMDOperator`, the class that manages the so-called DMD operator;\n",
+    "- [`DMDBase`](https://pydmd.github.io/PyDMD/dmdbase.html), the actual backbone of all the different implemented versions;\n",
+    "- [`DMDTimeDict`](https://github.com/PyDMD/PyDMD/blob/7ac9f3fa855e9a5e7008daad9906eaa6e59ba80a/pydmd/dmdbase.py#L759), the class that manages the time window;\n",
+    "- [`DMDOperator`](https://pydmd.github.io/PyDMD/dmdoperator.html), the class that manages the so-called DMD operator;\n",
+    "\n",
+    "The following schematic outlines the general structure of every `DMDBase` module.\n",
+    "![](PyDMD-structure.png)\n",
+    "In general, all `PyDMD` modules should be capable of the following tasks:\n",
+    "1) Accepting and storing DMD algorithm parameters.\n",
+    "2) Performing DMD given snapshot data $\\mathbf{X}$ via the `fit` method, which often does the following:\n",
+    "    - prepares and stores the input data,\n",
+    "    - sets the `DMDTimeDict`s,\n",
+    "    - computes the DMD operator and its eigendecomposition, and\n",
+    "        - (this is done by invoking the `DMDOperator`'s `compute_operator` function)\n",
+    "    - computes the DMD amplitudes using the computed operator.\n",
+    "3) Fetching and utilizing DMD results, such as:\n",
+    "    - the reduced DMD operator $\\tilde{\\mathbf{A}}$\n",
+    "    - the spatial modes $\\mathbf{\\Phi}$ (the eigenvectors of $\\mathbf{A}$)\n",
+    "    - the temporal frequencies $\\mathbf{\\Lambda}$ (the eigenvalues of $\\mathbf{A}$)\n",
+    "    - the spatiotemporal mode amplitudes $\\mathbf{b}$\n",
+    "\n",
+    "Hence different modules implement different DMD variants, where any number of these steps may be performed differently."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Building a New PyDMD Module\n",
     "\n",
     "We start the new module by importing all these classes and the usual math environment (`matplotlib`+`numpy`)."
    ]
@@ -393,9 +425,9 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.6"
+   "version": "3.11.5"
   }
  },
  "nbformat": 4,
- "nbformat_minor": 2
+ "nbformat_minor": 4
 }