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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 $i=\overline{1,n-1}$.

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

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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

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

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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.

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  • $\begingroup$ 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. $\endgroup$
    – sound wave
    Mar 11, 2020 at 16:41
  • $\begingroup$ 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? $\endgroup$
    – sound wave
    Mar 11, 2020 at 17:52
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    $\begingroup$ @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 $t = (\ell + \frac12) h$... $\endgroup$
    – EditPiAf
    Mar 11, 2020 at 19:18

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