Suppose $A \in \mathbb{R}^{n \times m}$ is a rectangular matrix and $D_i \in \mathbb{R}^{m \times m}$ for $i = 1, \ldots, N$ are diagonal matrices.
I'm interested if there exists a fast way to compute the singular values of $B_i := A D_i$ for all $i$, without requiring to perform $N$ different singular value decompositions, since $N$ is really big in my case.
In particular I'm only interested in the computation of the singular values. Thank you very much in advance.
In general, no such formula will exist. To better see this, think about the case where your matrix is real and symmetric positive definite. Then the svd reduces to diagonalization, and it might be clearer that nothing can be said in general.