# Error of approximation $\Phi\Phi^T \simeq \text{diag}(\Phi)$ with $\Phi$ vector of hat functions

Consider the partition of the time interval $$[0,T]$$ in $$n$$ equispaced subintervals of length $$h=T/n$$. The family of $$n+1$$ hat functions on $$[0, T]$$ is defined as $$\phi_0(t) = \begin{cases} \frac{h-t}{h}, &0\le t\le h \\ 0, &\text{ otherwise} \end{cases}\quad \phi_n(t) = \begin{cases} \frac{t-(T-h)}{h}, &T-h\le t\le T \\ 0, &\text{ otherwise} \end{cases}$$ $$\phi_i(t) = \begin{cases} \frac{t-(i-1)h}{h}, &(i-1)h\le t\le ih \\ \frac{(i+1)h-t}{h}, &ih\le t\le (i+1)h \\ 0, &\text{otherwise} \end{cases}$$ with $$i=\overline{1,n-1}$$.

Consider the vector $$\Phi(t) = [\phi_0(t),...,\phi_n(t)]^T$$, in many papers (such as this one) I'm seeing the approximation $$\Phi(t)\Phi(t)^T \simeq \text{diag}(\Phi(t))$$

What does it mean "expanding the entries"?

Moreover, since $$\phi_i(t)\phi_{i+1}(t) \ne 0$$ and $$\phi_i^2(t) \ne \phi_i(t)$$, what is the error when approximating the matrix $$\Phi(t)\Phi(t)^T$$ with $$\text{diag}(\Phi(t))$$?

MH Heydari, MR Hooshmandasl, FM Maleek Ghaini, C. Cattani: "A computational method for solving stochastic Itô–Volterra integral equations based on stochastic operational matrix for generalized hat basis functions", J. Comput. Phys. 270, 402-415. doi:10.1016/j.jcp.2014.03.064

• @Harry49 thanks I added tag and reference, can I add the sci-hub.tw link or I have to add the sciencedirect.com link? Mar 10, 2020 at 16:23

The entries of the matrix are of the form [\Phi \Phi^T]_{ij} = \left\lbrace \begin{aligned} &\phi_i\phi_j, & & |j-i|<2 ,\\ &0, & & \text{otherwise} . \end{aligned} \right. Let us expand $$[\Phi \Phi^T]_{ij}$$ in terms of hat functions $$\phi_k$$. To do so, introduce the representation $$[\Phi \Phi^T]_{ij} \simeq \Phi^T A = \sum_{k=0}^n a_k \phi_k .$$ By evaluating those matrix entries at the grid nodes $$t = \ell h$$, we find [\Phi(\ell h) \Phi(\ell h)^T]_{ij} = \left\lbrace \begin{aligned} &1, & & i=j=\ell ,\\ &0, & & \text{otherwise} , \end{aligned} \right. \qquad\qquad \sum_{k=0}^n a_k \phi_k(\ell h) = a_\ell Equating both sides, the expression $$a_\ell = \delta_{i\ell}\delta_{j\ell}$$ of the coefficients is obtained, where $$\delta$$ is the Kronecker symbol. Finally, the proposed pointwise approximation $$\Phi \Phi^T \simeq \text{diag}\,\Phi$$ is exact at the grid nodes, but error is made elsewhere.
• Thank you very much. However, to me, it is not clear how you found that expression for $a_k$...I see that you said "by evaluating those entries at the grid node $t=\ell h$", but I'm struggling with its meaning. Could you elaborate more? Thanks. Moreover, could you explain the difference between $\simeq$ and $\approx$? Because I see that you edited the question, thanks again. Mar 11, 2020 at 16:41
• Many thanks for the further explanation. I think I almost got it. So basically $\Phi\Phi^T$ can be approximated by $\text{diag}\Phi$ only when the nodes that we consider are all and only the grid nodes? Mar 11, 2020 at 17:52
• @soundwave yes, the formula is exact at the grid nodes $t = \ell h$ by construction, but it is not exact elsewhere. Consider for instance the abscissas $t = (\ell + \frac12) h$... Mar 11, 2020 at 19:18