54 problems

Let $d\in\mathbb{N}$, and let an order-2 digital sequence mean a digital sequence over the finite field $\mathbb{F}_2$ with order parameter $2$. For an infinite sequence…

Let $g_1$ be a differentiable function, let $U$ be an orthogonal matrix, and define $g_2(\bm{y}):= g_1(U\bm{y})$ under the coordinate transformation $\bm{y}=U^T\bm{x}$. Let…

Let $N$ points in the relevant $d$-dimensional domain be chosen deterministically, and let the generalized discrepancy be measured in the $p=1$ case. Curse-of-dimensionality conjec…

Let $W_{\rho}^{\alpha,2}(\mathbb{R})$ be the one-dimensional $\rho$-weighted Sobolev space, and consider trapezoidal rules combined with the boundary-damping importance sampling me…

Let $\boldsymbol{x}_p$ and $\boldsymbol{x}_q$ be points of the Sobol' sequence, and let $q\ge 4$. A pair $(\boldsymbol{x}_p,\boldsymbol{x}_q)$ is said to update the minimum distanc…

Let $(\boldsymbol{x}_n)_{n\ge 0}$ be the $d$-dimensional Sobol' sequence, and let $Q_N=\{\boldsymbol{x}_0,\ldots,\boldsymbol{x}_{N-1}\}$. The separation radius of a finite point se…

Let $w^*(x)\in C_0^\infty([0,1])$ be a weighting function used in a weighted Birkhoff average, and let $N$ denote the number of samples. The preceding result gives a nearly linear-…

Independent-Bernoulli approximation conjecture. For large $m$, $\mathrm{SUM}'_{1,m}$ can be approximated by $\mathrm{SUM}”_{1,m}$, where the variables $Z'(\boldsymbol{k})$ are samp…

Higher-way rank-deficiency conjecture. One might conjecture that

Let $n$ be an odd prime. For $1\leq\kappa\leq n-1$, define … where $r(j,n)$ is the unique integer congruent to $j$ modulo $n$ in $\{-(n-1)/2,\ldots,(n-1)/2\}$. Kazashi and Sloan's…

Let $d=2$, and consider Fibonacci lattices as point sets in the two-dimensional torus. For a large class of potentials, Fibonacci lattice optimality conjecture. Fibonacci lattices…

Let $d$ be the dimension, let $P$ be a point set in $[0,1)^d$, and let $D_N^*$ denote its star-discrepancy when the point set has $N$ points. Niederreiter–Wills conjecture. $n=10d$…

Let $(x_n)_{n\geq 1}$ be a sequence in $[0,1)^d$, and let … where the supremum is over all intervals $B=[0,b_1)\times\cdots\times[0,b_d)$ with $0\leq b_i\leq 1$. Optimal-order conj…

Let $\prescript{\mathrm{Ext}}{}{\Phi}^{\ell}$ and $\prescript{\mathrm{Ext}}{}{\Psi}^{\ell}$ denote the extrapolated fine- and coarse-level quantities at level $\ell$. Extrapolated…

Let $\prescript{\mathrm{Ext}}{}{\Phi}^{\ell}$ and $\prescript{\mathrm{Ext}}{}{\Psi}^{\ell}$ denote the extrapolated fine- and coarse-level quantities at level $\ell$. Assume that…

Let $\mathcal{A}(M,P,N)$ be the single-level estimator, and let $\bm X_N^P$ denote the particle-system realization. For $M$ samples and $P$ particles, variance-decay conjecture. Th…

Let $\bm X^P$ be the continuous-time particle-system approximation and let $Z(t)$ be the true solution of the corresponding McKean–Vlasov equation. Let…

Complete quasi-equidistribution conjecture. Let $\gamma$ be the largest root of the polynomial $x^2-px-q$ for $1\leq q\leq p$. The van der Corput sequence in base $\gamma$ is c.q.e…

Uniform distribution conjecture. The $d$-dimensional interlaced Halton sequence in Algorithm is uniformly distributed.

Discrepancy-order conjecture. For $m e 0$,

Asymptotic discrepancy-ratio conjecture. Based on numerical evidence, it was conjectured that

QMC strength conjecture. It is conjectured that the strengths of Fekete points, equal area points, and Coulomb energy points are, respectively, $1.5$, $2$, and $2$.

Let $\mathbf{N}$ be a collocation point set generated by a sampling method with low-discrepancy or local-regularity properties, and let the computational domain be the domain on wh…

Woźniakowski's conjecture.

Star discrepancy lower-bound conjecture. For every dimension $d$ there exists a constant $c_d>0$ such that every $n$-point set $P\subseteq[0,1]^d$ satisfies