A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This

We consider numerical approximations of ill-posed elliptic problems with conditional stability. The notion of optimal error estimates is defined including both convergence with respect to discretisation and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the approximation space. A proof is given

We study a finite-element based space-time discretisation for the 2D stochastic Navier–Stokes equations in a bounded domain supplemented with no-slip boundary conditions. We prove optimal convergence rates in the energy norm with respect to convergence in probability, that is convergence of order (almost) 1/2 in time and 1 in space. This was previously only known in the space-periodic case, where higher

In earlier work, the second author showed that a closed subset of a polynomial functor can always be defined by finitely many polynomial equations. In follow-up work on \({\text {GL}}_\infty \)-varieties, Bik–Draisma–Eggermont–Snowden showed, among other things, that in characteristic zero every such closed subset is the image of a morphism whose domain is the product of a finite-dimensional affine

In this paper, we study an online learning algorithm with a robust loss function \(\mathcal {L}_{\sigma }\) for regression over a reproducing kernel Hilbert space (RKHS). The loss function \(\mathcal {L}_{\sigma }\) involving a scaling parameter \(\sigma >0\) can cover a wide range of commonly used robust losses. The proposed algorithm is then a robust alternative for online least squares regression

Let A be an \(m\times n\) matrix with nonnegative real entries. The psd rank of A is the smallest k for which there exist two families \((P_1,\ldots ,P_m)\) and \((Q_1,\ldots ,Q_n)\) of positive semidefinite Hermitian \(k\times k\) matrices such that \(A(i|j)={\text {tr}}(P_i Q_j)\) for all i, j. Several questions on the algorithmic complexity of related matrix invariants were posed in recent literature: