"Real-Time Optical Laboratory Linear Algebra Solution Of Partial Differential Equations"

1986 ◽  
Author(s):  
David Casasent ◽  
James Jackson
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Time ◽  
Linear Algebra ◽  
Partial Differential

scholarly journals Infinite-dimensional linear algebra and solvability of partial differential equations

2021 ◽  
Vol 13 ◽  
Author(s):  
Todor D. Todorov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Linear Algebra ◽  
Differential Operators ◽  
Test Functions ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Partial Differential Operators ◽  
Linear Partial Differential Operators ◽  
Regular Operators

  We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of test-functions $\mathcal D(\Omega)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(\Omega)$. We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution $\mathcal D^\prime(\Omega)$, where these operators are often non-surjective. 

scholarly journals Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

2001 ◽  
Vol 124 (1) ◽  
pp. 70-80 ◽  
Author(s):  
C. Prud’homme ◽  
D. V. Rovas ◽  
K. Veroy ◽  
L. Machiels ◽  
Y. Maday ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Time ◽  
Galerkin Projection ◽  
Real Time Control ◽  
Reduced Basis ◽  
Partial Differential ◽  
Approximation Process ◽  
Essential Components ◽  
On Line

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations—Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation—relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

Advanced Image Analysis for Learning Underlying Partial Differential Equations for Anomaly Identification

2020 ◽  
Vol 64 (2) ◽  
pp. 20510-1-20510-10
Author(s):  
Andrew Miller ◽  
Jan Petrich ◽  
Shashi Phoha
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Time ◽  
Heat Diffusion ◽  
Series Data ◽  
Model Parameters ◽  
Model Structure ◽  
Partial Differential ◽  
Individual Parameter ◽  
Heat Diffusion Equation

Abstract In this article, the authors adapt and utilize data-driven advanced image processing and machine learning techniques to identify the underlying dynamics and the model parameters for dynamic processes driven by partial differential equations (PDEs). Potential applications include non-destructive inspection for material crack detection using thermal imaging as well as real-time anomaly detection for process monitoring of three-dimensional printing applications. A neural network (NN) architecture is established that offers sufficient flexibility for spatial and temporal derivatives to capture the physical dependencies inherent in the process. Predictive capabilities are then established by propagating the process forward in time using the acquired model structure as well as individual parameter values. Moreover, deviations in the predicted values can be monitored in real time to detect potential process anomalies or perturbations. For concept development and validation, this article utilizes well-understood PDEs such as the homogeneous heat diffusion equation. Time series data governed by the heat equation representing a parabolic PDE is generated using high-fidelity simulations in order to construct the heat profile. Model structure and parameter identification are realized through a shallow residual convolutional NN. The learned model structure and associated parameters resemble a spatial convolution filter, which can be applied to the current heat profile to predict the diffusion behavior forward in time.

Spatio-Temporal Estimation of Wildfire Growth

2013 ◽  
Author(s):  
Balaji R. Sharma ◽  
Manish Kumar ◽  
Kelly Cohen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Time ◽  
Growth Model ◽  
Wildland Fire ◽  
Time Estimation ◽  
Fire Growth ◽  
Growth Data ◽  
Partial Differential ◽  
Spatio Temporal

This work presents a methodology for real-time estimation of wildland fire growth, utilizing afire growth model based on a set of partial differential equations for prediction, and harnessing concepts of space-time Kalman filtering and Proper Orthogonal Decomposition techniques towards low dimensional estimation of potentially large spatio-temporal states. The estimation framework is discussed in its criticality towards potential applications such as forest fire surveillance with unmanned systems equipped with onboard sensor suites. The effectiveness of the estimation process is evaluated numerically over fire growth data simulated using a well-established fire growth model described by coupled partial differential equations. The methodology is shown to be fairly accurate in estimating spatio-temporal process states through noise-ridden measurements for real-time deploy ability.

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