SVD on Intersection Spaces

We are interested in applying SVD to more general spaces, the motivating example being the Sobolev space $H^1(\Omega)$of weakly differentiable functions over a domain$\Omega\subset\R^d$. Controlling the truncation error in the energy norm is particularly interesting for PDE applications. To this end, one can apply SVD to tensor products in $H^1(\Omega_1)\otimes H^1(\Omega_2)$with the induced tensor scalar product. However, the resulting space is not$H^1(\Omega_1\times\Omega_2)$but is instead the space$H^1_{\text{mix}}(\Omega_1\times\Omega_2)$of functions with mixed regularity. For large$d>2$this poses a restrictive regularity requirement on$u\in H^1(\Omega)$. On the other hand, the space $H^1(\Omega)$is not a tensor product Hilbert space, in particular$\|\cdot\|_{H^1}$is not a reasonable crossnorm. Thus, we can not identify$H^1(\Omega)$with the space of Hilbert Schmidt operators and apply SVD. However, it is known that$H^1(\Omega)$is isomorph (here written for$d=2$) to the Banach intersection space $$H^1(\Omega_1\times\Omega_2)=H^1(\Omega_1)\otimes L_2(\Omega_2)\cap L_2(\Omega_1)\otimes H^1(\Omega_2)$$ with equivalent norms. Each of the spaces in the intersection is a tensor product Hilbert space where SVD applies. We investigate several approaches to construct low-rank approximations for a function $u\in H^1(\Omega_1\times\Omega_2)$.