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We study the problem of estimating the coefficients in linear ordinary differential equations (ODE's) with a diverging number of variables when the solutions are observed with noise. The solution trajectories are first smoothed with local…
This work develops Monte Carlo Euler adaptive time stepping methods for the weak approximation problem of jump diffusion driven stochastic differential equations. The main result is the derivation of a new expansion for the omputational…
We present locally stabilized, conforming space-time finite element methods for parabolic evolution equations on hexahedral decompositions of the space-time cylinder. Tensor-product decompositions allow for anisotropic a priori error…
By combining a standard symmetric, symplectic integrator with a new step size controller, we provide an integration scheme that is symmetric, reversible and conserves the values of the constants of motion. This new scheme is appropriate for…
In this paper, we propose a new test for the detection of a change in a non-linear (auto-)regressive time series as well as a corresponding estimator for the unknown time point of the change. To this end, we consider an at-most-one-change…
Explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable Hamiltonians. For the first time, the article proposes for arbitrary Hamiltonians similar integrators,…
The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schr\"odinger equation $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with a step-like…
We discuss local linear smooth backfitting for additive non-parametric models. This procedure is well known for achieving optimal convergence rates under appropriate smoothness conditions. In particular, it allows for the estimation of each…
We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schr\"odinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense of the underlying PDE. The main idea thereby lies in…
This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an…
Efficient simulation of the semiclassical Schr\"odinger equation has garnered significant attention in the numerical analysis community. While controlling the error in the unitary evolution or the wavefunction typically requires the time…
The paper is concerned with locally stabilized space-time IgA approximations to initial boundary value problems of the parabolic type. Originally, similar schemes (but weighted with a global mesh parameter) was presented and studied by U.…
We develop a novel asymptotic theory for local polynomial extremum estimators of time-varying parameters in a broad class of nonlinear time series models. We show the proposed estimators are consistent and follow normal distributions in…
The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential…
The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The…
We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several…
We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. Based on this fact is our method of group classification of potential symmetries of systems of…
We study statistical estimators computed using iterative optimization methods that are not run until completion. Classical results on maximum likelihood estimators (MLEs) assert that a one-step estimator (OSE), in which a single…
Establishing a fast rate of convergence for optimization methods is crucial to their applicability in practice. With the increasing popularity of deep learning over the past decade, stochastic gradient descent and its adaptive variants…
Parametric high-dimensional regression analysis requires the usage of regularization terms to get interpretable models. The respective estimators can be regarded as regularized M-functionals which are naturally highly nonlinear. We study…