WAVELET-BASED ADAPTIVE COLLOCATION METHOD FOR THE RESOLUTION OF NONLINEAR PDEs

2007 ◽  
Vol 05 (06) ◽  
pp. 957-973 ◽  
Author(s):  
A. KARAMI ◽  
H. R. KARIMI ◽  
B. MOSHIRI ◽  
P. JABEDAR MARALANI
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Multiresolution Analysis ◽  
Chemical Engineering ◽  
Nonlinear Pdes ◽  
Dynamic Processes ◽  
Partial Differential ◽  
Numeric Solution ◽  
Interpolating Wavelets

Theoretical modeling of dynamic processes in chemical engineering often implies the numeric solution of one or more partial differential equations. The complexity of such problems is increased when the solutions exhibit sharp moving fronts. An efficient adaptive multiresolution numerical method is described for solving systems of partial differential equations. This method is based on multiresolution analysis and interpolating wavelets, that dynamically adapts the collocation grid so that higher resolution is automatically attributed to domain regions where sharp features are present. Space derivatives were computed in an irregular grid by cubic splines method. The effectiveness of the method is demonstrated with some relevant examples in a chemical engineering context.

scholarly journals Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
J. Nathan Kutz ◽  
J. L. Proctor ◽  
S. L. Brunton
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Data Driven ◽  
Nonlinear Pdes ◽  
Dynamic Mode ◽  
Partial Differential ◽  
Koopman Operator ◽  
Dimensional Approximation ◽  
Spatio Temporal ◽  
The Impact

We consider the application of Koopman theory to nonlinear partial differential equations and data-driven spatio-temporal systems. We demonstrate that the observables chosen for constructing the Koopman operator are critical for enabling an accurate approximation to the nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition (DMD) algorithm can be enacted to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues, and Koopman modes. We demonstrate simple rules of thumb for selecting a parsimonious set of observables that can greatly improve the approximation of the Koopman operator. Further, we show that the clear goal in selecting observables is to place the DMD eigenvalues on the imaginary axis, thus giving an objective function for observable selection. Judiciously chosen observables lead to physically interpretable spatio-temporal features of the complex system under consideration and provide a connection to manifold learning methods. Our method provides a valuable intermediate, yet interpretable, approximation to the Koopman operator that lies between the DMD method and the computationally intensive extended DMD (EDMD). We demonstrate the impact of observable selection, including kernel methods, and construction of the Koopman operator on several canonical nonlinear PDEs: Burgers’ equation, the nonlinear Schrödinger equation, the cubic-quintic Ginzburg-Landau equation, and a reaction-diffusion system. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.

scholarly journals Robust Low-Rank Discovery of Data-Driven Partial Differential Equations

2020 ◽  
Vol 34 (01) ◽  
pp. 767-774
Author(s):  
Jun Li ◽  
Gan Sun ◽  
Guoshuai Zhao ◽  
Li-wei H. Lehman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Minimization Problem ◽  
Real World ◽  
Low Rank ◽  
Dynamic Processes ◽  
Complex Data ◽  
Sparse Regression ◽  
Partial Differential ◽  
Regression Methods

Partial differential equations (PDEs) are essential foundations to model dynamic processes in natural sciences. Discovering the underlying PDEs of complex data collected from real world is key to understanding the dynamic processes of natural laws or behaviors. However, both the collected data and their partial derivatives are often corrupted by noise, especially from sparse outlying entries, due to measurement/process noise in the real-world applications. Our work is motivated by the observation that the underlying data modeled by PDEs are in fact often low rank. We thus develop a robust low-rank discovery framework to recover both the low-rank data and the sparse outlying entries by integrating double low-rank and sparse recoveries with a (group) sparse regression method, which is implemented as a minimization problem using mixed nuclear norms with ℓ1 and ℓ0 norms. We propose a low-rank sequential (grouped) threshold ridge regression algorithm to solve the minimization problem. Results from several experiments on seven canonical models (i.e., four PDEs and three parametric PDEs) verify that our framework outperforms the state-of-art sparse and group sparse regression methods. Code is available at https://github.com/junli2019/Robust-Discovery-of-PDEs

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