5 problems

Let $H_d^s:=H_1^s\otimes\cdots\otimes H_1^s$ be the Sobolev space of dominating mixed smoothness, and let $B_d^s$ denote its unit ball. Smolyak's algorithm is a sparse-grid approxi…

Let $d\geq 3$, let $h_{x_i}=h_0\cdot 2^{-l_i}$ for $i=1,\ldots,d$, and let $\Delta=\Delta_1\otimes\cdots\otimes\Delta_d$ be the tensor product of the first-order difference operato…

Let $d\geq 3$, and let $v$ solve the $d$-dimensional Zakai SPDE. For a spatial grid with spacings $h_{x_i}>0$, let $V_{i_1,\ldots,i_d}^N$ denote the implicit finite-difference appr…

Let $Y_\ell$ be the sparse combination estimator formed from the multilevel samples $Y_{\boldsymbol \ell}$ with $\|{\boldsymbol \ell}\|_1=\ell$. Although substantial cancellation i…

Higher-dimensional calculations up to $D=7$ are needed to discretize phase space, for example in radiation transport in astrophysical scenarios. The conjecture. The limited use of…