12 problems

Let $d>1$ be the dimension and let $r:=1+\frac{1}{p}$, where $p$ is the exponent in the relevant $L_p$-norm. For the Lipschitz function class ${\cal L}$ considered in the paper, th…

Average-case complexity conjecture. The average case complexity of $\app$ in $\lstd$ is of the same order as the average case complexity of $\app$ in $\lall$.

Optimality conjecture for Krylov information. For the class $\Lambda$ of matrix-vector multiplications and every orthogonally invariant class $F$, Krylov information is optimal.

For randomized approximation of the embedding $\ell_1^m \hookrightarrow \ell_\infty^m$, let $m$ be the dimension and $n$ the complexity, and fix a small error threshold…

Let $h$ denote the discretization step size, and let $\beta$ be the parameter in the Hölder-type assumptions on the differential equation and its data. Under the assumptions of the…

Decay-threshold conjecture.

The asymptotic-constant conjecture. All these asymptotic constants exist, and possibly they do not depend on the shape of $\Omega$, only on its volume. This concerns the finer asym…

Uniform lower-bound conjecture. There exists a constant $C>0$, independent of $d$ and $D_d$, such that

Random-information conjecture. Random information is as good as optimal information, in order of convergence, provided that $p>q$; when $p\leq q$, the radius of random information…

Non-polynomial lower-bound conjecture. A similar negative result should hold for such approximations: there exist stochastic differential equations for which no adaptive approximat…

Let $(a_j)_{j\geq 1}$ and $(b_j)_{j\geq 1}$ be the sequences defining the Hermite space, and let EC-WT denote exponential-convergence weak tractability. Sufficient-condition conjec…

Let $F$ be a Banach space, let $S$ be a continuous operator on $F$, and let $e^{\rm all-wor}_n(S)$ denote its $n$-th minimal worst-case error using arbitrary continuous linear info…