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 $u\in H^1(\Omega_1\times\Omega_2)$.