Related papers: Long-time relative error analysis for linear ordin…
We are interested in the relative conditioning of the problem $y_0\mapsto \mathrm{e}^{tA}y_0$, i.e., the relative conditioning of the action of the matrix exponential $\mathrm{e}% ^{tA}$ on a vector with respect to perturbations of this…
The paper \cite{M0} studied, for a \emph{complex} linear ordinary differential equation $y^\prime(t)=Ay(t)$, the long-time propagation to the solution $y(t)$ of a perturbation of the initial value. By measuring the perturbations with…
We obtain rigorous a priori upper and lower bounds to the exact period of the celebrated Rayleigh stretched string differential equation. We use them to show that Rayleigh's approximative period overestimates the true period and that the…
In this paper, we consider the initial value problem for some nonlinear second-order ODEs of Duffing type. We study the large time behavior of the solutions to this problem, from both the perspectives of mathematical and numerical analysis.…
Large deviation estimates for the following linear parabolic equation are studied: \[ \frac{\partial u}{\partial t}=\tr\Big(a(x)D^2u\Big) + b(x)\cdot D u + \int_{\R^N} \Big\{(u(x+y)-u(x)-(D u(x)\cdot y)\ind{|y|<1}(y)\Big\}\d\mu(y), \] where…
The propagation of primary discontinuities in initial value problems for linear delay differential-algebraic equations (DDAEs) is discussed. Based on the (quasi-) Weierstra{\ss} form for regular matrix pencil, a complete characterization of…
In this paper we consider the rate of convergence of solutions of a scalar ordinary differential equation which is a perturbed version of an autonomous equation with a globally stable equilibrium. Under weak assumptions on the nonlinear…
Computable estimates for the error of finite element discretisations of parabolic problems in the $L^\infty(0,T; L^2)$ norm are developed, which exhibit constant effectivities (the ratio of the estimated error to the true error) with…
We initiate the study of extended excitations in the long-range O(N) model. We focus on line and surface defects and we discuss the challenges of a naive generalization of the simplest defects in the short-range model. To face these…
We develop a technique to construct analytical solutions of the linear perturbations of inflation with a nonlinear dispersion relation, due to quantum effects of the early universe. Error bounds are given and studied in detail. The…
We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…
We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretisation errors…
Estimating long-term causal effects by combining long-term observational and short-term experimental data is a crucial but challenging problem in many real-world scenarios. In existing methods, several ideal assumptions, e.g. latent…
This paper focuses on propagation phenomena in reaction-diffusion equations with a weaklymonostable nonlinearity. The reaction term can be seen as an intermediate between the classicallogistic one (or Fisher-KPP) and the standard weak Allee…
There are many scenarios where short- and long-term causal effects of an intervention are different. For example, low-quality ads may increase short-term ad clicks but decrease the long-term revenue via reduced clicks. This work, therefore,…
As is known, the problems for the differential equations with continuously changing order of the derivatives are not considered completely. In this paper we consider the initial and boundary value problems for this type of linear ordinary…
The analysis of data arising from environmental health studies which collect a large number of measures of exposure can benefit from using latent variable models to summarize exposure information. However, difficulties with estimation of…
For a nonlinear ordinary differential equation with time delay, the differentiation of the solution with respect to the delay is investigated. Special emphasis is laid on the second-order derivative. The results are applied to an associated…
Temporal-Difference learning (TD) [Sutton, 1988] with function approximation can converge to solutions that are worse than those obtained by Monte-Carlo regression, even in the simple case of on-policy evaluation. To increase our…
Nonlinear ordinary differential equations (ODEs) are powerful tools for modeling real-world dynamical systems. However, propagating initial state uncertainty through nonlinear dynamics, especially when the ODE is unknown and learned from…