arXiv subjects

Daniel Eckhardt

Publications and source records attributed to Daniel Eckhardt.

4 recordsLinked to original sources

arXiv2024

Error analysis of an implicit-explicit time discretization scheme for semilinear wave equations with application to multiscale problems

We present an implicit-explicit (IMEX) scheme for semilinear wave equations with strong damping. By treating the nonlinear, nonstiff term explicitly and the linear, stiff part implicitly, we obtain a method which is not only unconditionally stable but also highly efficient. Our main results are error bounds of the full discretization in space and time for the IMEX scheme combined with a general abstract space discretization. As an application, we consider the heterogeneous multiscale method for wave equations with highly oscillating coefficients in space for which we show spatial and temporal convergence rates by using the abstract result.

math.NA↗

arXiv2022

Fully discrete Heterogeneous Multiscale Method for parabolic problems with multiple spatial and temporal scales

The aim of this work is the numerical homogenization of a parabolic problem with several time and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and the cell problem, method. Numerical experiments illustrate the theoretical convergence rates.

math.NA↗

arXiv2022

Parameter Inference of Time Series by Delay Embeddings and Learning Differentiable Operators

We provide a method to identify system parameters of dynamical systems, called ID-ODE -- Inference by Differentiation and Observing Delay Embeddings. In this setting, we are given a dataset of trajectories from a dynamical system with system parameter labels. Our goal is to identify system parameters of new trajectories. The given trajectories may or may not encompass the full state of the system, and we may only observe a one-dimensional time series. In the latter case, we reconstruct the full state by using delay embeddings, and under sufficient conditions, Taken's Embedding Theorem assures us the reconstruction is diffeomorphic to the original. This allows our method to work on time series. Our method works by first learning the velocity operator (as given or reconstructed) with a neural network having both state and system parameters as variable inputs. Then on new trajectories we backpropagate prediction errors to the system parameter inputs giving us a gradient. We then use gradient descent to infer the correct system parameter. We demonstrate the efficacy of our approach on many numerical examples: the Lorenz system, Lorenz96, Lotka-Volterra Predator-Prey, and the Compound Double Pendulum. We also apply our algorithm on a real-world dataset: propulsion of the Hall-effect Thruster (HET).

cs.LG↗

arXiv2019

Impact of Embedding View on Cross Mapping Convergence

Convergent cross mapping (CCM) provides a powerful technique for exploring causal relationships in nonlinear coupled systems. The method relies on Takens' theorem exploiting that time delay embeddings of infinite length general observations of smooth nonlinear coupled systems are diffeomorphic to the original coupled state space attractor. However, for finite length data that is corrupted by quantization and noise, not every view of the embedding is equally useful at identifying the system dynamics. Classically, the heuristic of the first minimum of mutual information (MI) has been proposed as a means to select appropriate lags for time delay embedding methods. This criteria is sensitive to additive noise and known to fail for systems with monotonically decreasing MI. In this work, alternative heuristics on MI for identifying useful embedding views are explored. The impact of coordinate system and noise level on the identification of a useful time lag representative of the structure in chaotic data is studied. The impact of selecting the first minimum of MI relative to several alternate heuristics for the appropriate time lag in the context of of CCM method is then presented. Both simple dynamical systems as well as experimental data derived from observations of a Hall Effect Thruster plasma propulsion device are used. It is found that the shorter of the two global maxima of MI from signal pairs in a stretched discrete Legendre orthonormal coordinate system is a more robust option relative to the alternatives studied for selecting embedding lag for use with the CCM method. The choice identifies time lags commensurate with the brute force maximum bidirectional CCM correlation results generated. This enhanced performance results from a decreased sensitivity to noise and fluctuations in the estimates of MI when compared to lags derived from local criteria on MI.

nlin.CD↗