Consider a Hilbert-Schmidt integral operator $T:L^2(\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n)$, whose explicit mapping is given by \begin{align} f(y)\mapsto \int k(x,y) f(y) dy. \end{align} Hopefully, the kernel of interest has bounded support and we can replace (if needed) $\mathbb{R}^n$ with $n$-dimensional intervals. I want to analyze a numerical method (which will be explained) for computing the singular values of $T$, for given kernel $k$.
When it comes to a numerical evaluation, the first method one can think of might be a discretization. In [R1], the singular values of $T$ is approximated by the singular values of (truncated) discrete representation of $T$—the matrix $A\in\mathbb{R}^{N\times N}$ whose $(i,j)$-entry is given by \begin{align} a_{i,j} = \langle\psi_i,T\varphi_j\rangle \end{align} where $\psi_i$ and $\varphi_j$ are orthonormal functions—and this method has a theoretical error bound. To numerically compute it, we need a specific choice of orthonormal functions (and we prefer computationally-friendly one). Constructing the grid on $\mathbb{R}^n$, in the numerical example in [R1], the orthonormal functions \begin{align} \psi_i = \frac{1}{\sqrt{\text{area of grid } i}}\boldsymbol{1}_{\text{grid } i}, \varphi_j = \frac{1}{\sqrt{\text{area of grid } j}}\boldsymbol{1}_{\text{grid } j} \end{align} are used.
Question. I want to further simplify this procedure and eliminate the use of integration. For sufficiently smooth $k$ and sufficiently fine grid, the Riemann sum approximation gives \begin{align} a_{i,j} \approx b_ {i,j}= \sqrt{\text{area of grid } i} \sqrt{\text{area of grid } j} \cdot k(\text{some point in grid } i, \text{some point in grid } j). \end{align} I count on that the theoretical error bound of this method can be readily obtained from combining the existing results. Precisely, the error bound between the singular values of $T$ and the singular values in $A$ is established in [R1]. Also, the error bound between the singular values of $A$ and that of $B$, where $B\in\mathbb{R}^{N\times N}$ is a matrix having its $(i,j)$-entry as $b_{i,j}$, can be straightforwardly followed from the perturbation bound on the spectrum for matrix, together with the error bound on the Riemann sum approximation. Invoking the triangle inequality, we can argue that singular values of $B$ can reasonably approximate the singular values of $T$ when grids are sufficiently fine. However, I reckon that it would be merely reinventing the wheel. Are there any publications related to it?
Any comments or suggestions are welcome.
[R1] P. C. Hansen, Computation of the singular value expansion, Computing, 40 (1988), pp. 185–199.