On Reduced Models For The Chemical Master Equation

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

Adiabatic Midpoint Rule for the dispersion-managed nonlinear Schrödinger Equation

The dispersion-managed nonlinear Schrödinger equation contains a rapidly changing discontinuous coefficient function. Approximating the solution numerically is a challenging task because typical solutions oscillate in time which imposes severe step-size restrictions for traditional methods. We present and analyze a tailor-made time integrator which attains the desired accuracy with a significantly larger step-size than traditional methods. The construction of this method is based on a favorable transformation to an equivalent problem and the explicit computation of certain integrals over highly oscillatory phases. The error analysis requires the thorough investigation of various cancellation effects which result in improved accuracy for special step-sizes

Improved error bounds for approximations of high-frequency wave propagation in nonlinear dispersive media

High-frequency wave propagation is often modelled by nonlinear Friedrichs systems where both the differential equation and the initial data contain the inverse of a small parameter $\varepsilon$, which causes oscillations with wavelengths proportional to $\varepsilon$ in time and space. A prominent example is the Maxwell--Lorentz system, which is a well-established model for the propagation of light in nonlinear media. In diffractive optics, such problems have to be solved on long time intervals with length proportional to $1/\varepsilon$. Approximating the solution of such a problem numerically with a standard method is hopeless, because traditional methods require an extremely fine resolution in time and space, which entails unacceptable computational costs. A possible alternative is to replace the original problem by a new system of PDEs which is more suitable for numerical computations but still yields a sufficiently accurate approximation. Such models are often based on the \emph{slowly varying envelope approximation} or generalizations thereof. Results in the literature state that the error of the slowly varying envelope approximation is of $\mathcal{O}(\varepsilon)$. In this work, however, we prove that the error is even proportional to $\varepsilon^2$, which is a substantial improvement, and which explains the error behavior observed in numerical experiments. For a higher-order generalization of the slowly varying envelope approximation we improve the error bound from $\mathcal{O}(\varepsilon^2)$ to $\mathcal{O}(\varepsilon^3)$. Both proofs are based on a careful analysis of the nonlinear interaction between oscillatory and non-oscillatory error terms, and on \textit{a priori} bounds for certain ``parts'' of the approximations which are defined by suitable projections

A multi-level stochastic collocation method for Schrödinger equations with a random potential

We propose and analyze a numerical method for time-dependent linear Schrödinger equations with uncertain parameters in both the potential and the initial data. The random parameters are discretized by stochastic collocation on a sparse grid, and the sample solutions in the nodes are approximated with the Strang splitting method. The computational work is reduced by a multi-level strategy, i.e. by combining information obtained from sample solutions computed on different refinement levels of the discretization. We prove new error bounds for the time discretization which take the finite regularity in the stochastic variable into account, and which are crucial to obtain convergence of the multi-level approach. The predicted cost savings of the multi-level stochastic collocation method are verified by numerical examples

Approximation of high-frequency wave propagation in dispersive media

We consider semilinear hyperbolic systems with a trilinear nonlinearity. Both the differential equation and the initial data contain the inverse of a small parameter $\varepsilon$, and typical solutions oscillate with frequency proportional to $1/\varepsilon$ in time and space. Moreover, solutions have to be computed on time intervals of length $1/\varepsilon$ in order to study nonlinear and diffractive effects. As a consequence, direct numerical simulations are extremely costly or even impossible. We propose an analytical approximation and prove that it approximates the exact solution up to an error of $\mathcal{O}(\varepsilon^2)$ on time intervals of length $1/\varepsilon$. This is a significant improvement over the classical nonlinear Schrödinger approximation, which only yields an accuracy of $\mathcal{O}(\varepsilon)$

On numerical methods for the semi-nonrelativistic system of the nonlinear Dirac equation

Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of $\mathcal{O}(\varepsilon^{-2})$, where $0 < ε \ll 1$ is inversely proportional to the speed of light. It was shown in [7], however, that such solutions can be approximated up to an error of $\mathcal{O}(\varepsilon^{-2})$ by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the exponential explicit midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy by a factor of six

Analysis of a Peaceman–Rachford ADI scheme for Maxwell equations in heterogenous media

The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than $H^2$-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equations in detail in an appropriate functional analytical framework. The theoretical findings are complemented by a numerical experiment indicating that the proven convergence rate is indeed observable and optimal

Analysis of a Peaceman-Rachford ADI scheme for Maxwell equations in heterogeneous media

The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than $H^2$-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equations in detail in an appropriate functional analytical framework. The theoretical findings are complemented by a numerical experiment indicating that the proven convergence rate is indeed observable and optimal