From: Wolfgang Bangerth <bangerth@colostate.edu>
Date: Wed, 24 Jun 2020 17:20:25 +0000 (-0600)
Subject: Augment documentation of project_onto_orthogonal_tensors().
X-Git-Tag: v9.3.0-rc1~1364^2
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F10595%2Fhead;p=dealii.git

Augment documentation of project_onto_orthogonal_tensors().
---

diff --git a/include/deal.II/base/tensor.h b/include/deal.II/base/tensor.h
index 956bc7e674..713e08a5d0 100644
--- a/include/deal.II/base/tensor.h
+++ b/include/deal.II/base/tensor.h
@@ -2640,16 +2640,66 @@ cofactor(const Tensor<2, dim, Number> &t)
 
 
 /**
- * Return the nearest orthogonal matrix by
- * combining the products of the SVD decomposition: $\mathbf U \mathbf{V}^T$,
- * where $\mathbf U$ and $\mathbf V$ are computed from the SVD decomposition:
- * $\mathbf U  \mathbf S \mathbf V^T$,
- * effectively replacing $\mathbf S$ with the identity matrix.
- * @param A The tensor which to find the closest orthogonal tensor to.
- * @pre @p Number must be either `float` or `double`.
+ * Return the nearest orthogonal matrix
+ * $\hat {\mathbf A}=\mathbf U \mathbf{V}^T$ by
+ * combining the products of the singular value decomposition (SVD)
+ * ${\mathbf A}=\mathbf U  \mathbf S \mathbf V^T$ for a given input
+ * ${\mathbf A}$, effectively replacing $\mathbf S$ with the identity matrix.
+ *
+ * This is a (nonlinear)
+ * [projection
+ * operation](https://en.wikipedia.org/wiki/Projection_(mathematics)) since when
+ * applied twice, we have $\hat{\hat{\mathbf A}}=\hat{\mathbf A}$ as is easy to
+ * see. (That is because the SVD of $\hat {\mathbf A}$ is simply
+ * $\mathbf U \mathbf I \mathbf{V}^T$.) Furthermore, $\hat {\mathbf A}$ is
+ * really an orthogonal matrix because orthogonal matrices have to satisfy
+ * ${\hat {\mathbf A}}^T \hat {\mathbf A}={\mathbf I}$, which here implies
+ * that
+ * @f{align*}{
+ *   {\hat {\mathbf A}}^T \hat {\mathbf A}
+ *   &=
+ *   \left(\mathbf U \mathbf{V}^T\right)^T\left(\mathbf U \mathbf{V}^T\right)
+ *   \\
+ *   &=
+ *   \mathbf V \mathbf{U}^T
+ *   \mathbf U \mathbf{V}^T
+ *   \\
+ *   &=
+ *   \mathbf V \left(\mathbf{U}^T
+ *   \mathbf U\right) \mathbf{V}^T
+ *   \\
+ *   &=
+ *   \mathbf V \mathbf I \mathbf{V}^T
+ *   \\
+ *   &=
+ *   \mathbf V \mathbf{V}^T
+ *   \\
+ *   &=
+ *   \mathbf I
+ * @f}
+ * due to the fact that the $\mathbf U$ and $\mathbf V$ factors that come out
+ * of the SVD are themselves orthogonal matrices.
+ *
+ * @param A The tensor for which to find the closest orthogonal tensor.
+ * @tparam Number The type used to store the entries of the tensor.
+ *   Must be either `float` or `double`.
  * @pre In order to use this function, this program must be linked with the
- *      LAPACK library.
- * @pre @p A must not be singular.
+ *   LAPACK library.
+ * @pre @p A must not be singular. This is because, conceptually, the problem
+ *   to be solved here is trying to find a matrix $\hat{\mathbf A}$ that
+ *   minimizes some kind of distance from $\mathbf A$ while satisfying the
+ *   quadratic constraint
+ *   ${\hat {\mathbf A}}^T \hat {\mathbf A}={\mathbf I}$. This is not so
+ *   dissimilar to the kind of problem where one wants to find a vector
+ *   $\hat{\mathbf x}\in{\mathbb R}^n$ that minimizes the quadratic objective
+ *   function $\|\hat {\mathbf x} - \mathbf x\|^2$ for a given $\mathbf x$
+ *   subject to the constraint $\|\mathbf x\|^2=1$ -- in other
+ *   words, we are seeking the point $\hat{\mathbf x}$ on the unit sphere
+ *   that is closest to $\mathbf x$. This problem has a solution for all
+ *   $\mathbf x$ except if $\mathbf x=0$. The corresponding condition
+ *   for the problem we are considering here is that $\mathbf A$ must not
+ *   have a zero eigenvalue.
+ *
  * @relatesalso Tensor
  */
 template <int dim, typename Number>