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Convergence rate for a Radau hp collocation method applied to constrained optimal control
For control problems with control constraints, a local convergence rate is established for an hp-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the sup-norm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the hp-scheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory.
Journal Article
A gradient reproducing kernel collocation method for high order differential equations
The High order Gradient Reproducing Kernel in conjunction with the Collocation Method (HGRKCM) is introduced for solutions of 2nd- and 4th-order PDEs. All the derivative approximations appearing in PDEs are constructed using the gradient reproducing kernels. Consequently, the computational cost for construction of derivative approximations reduces tremendously, basis functions for derivative approximations are smooth, and the accumulated error arising from calculating derivative approximations are controlled in comparison to the direct derivative counterparts. Furthermore, it is theoretically estimated and numerically tested that the same number of collocation points as the source points can be used to obtain the optimal solution in the HGRKCM. Overall, the HGRKCM is roughly 10–25 times faster than the conventional reproducing kernel collocation method. The convergence of the present method is estimated using the least squares functional equivalence. Numerical results are verified and compared with other strong-form-based and Galerkin-based methods.
Journal Article
Convergence Rate for a Gauss Collocation Method Applied to Unconstrained Optimal Control
A local convergence rate is established for an orthogonal collocation method based on Gauss quadrature applied to an unconstrained optimal control problem. If the continuous problem has a smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as the number of collocation points increases, the discrete solution converges to the continuous solution at the collocation points, exponentially fast in the sup-norm. Numerical examples illustrating the convergence theory are provided.
Journal Article
A barycentric rational interpolation collocation method for tri-harmonic equations
This essay mainly uses the barycentric rational interpolation collocation method to study two-dimensional tri-harmonic equations. By using this method to discretize the tri-harmonic equation, its linear equation is transformed into the corresponding matrix form, and its approximate solution is obtained. The effectiveness of the method compared to other methods is also verified through numerical examples.
Journal Article
Integration of Three Standardized Drought Indices utilizing Modified Triple Collocation and Scaled Triple Collocation relative to Triple Collocation
Droughts have a detrimental effect on plenty of social and economic endeavors along with surface and groundwater resources. Therefore, drought must be adequately considered when planning and regulating the water supply. This study will look at the latest developments in merging techniques to lessen the inconsistent drought monitoring and characterization attributed to the global standard shortage. The current research considers the framework of three distinct standardized indicators, SPI, SPTI, and SPEI, of six metrological stations in Pakistan from 1971 to 2017, intending to analyze drought using integrating techniques. Two merging techniques, Modified Triple Collocation (MTC) and proposed Scaled Triple Collocation (STC), are examined relative to Triple Collocation (TC). Correlation, Sen’s Slope, Taylor diagram, Kling Gupta Efficiency (KGE), and error variance analyses were used to evaluate their performance. The correlation study reveals that individual series have a comparable relationship with Merged Drought Index (MDI) model from MTC, STC, and TC. However, individual indices SPI and SPTI are strongly associated with MTC and STC-based MDI compared to SPEI. Sen's slope shows the same trend across all approaches with minimal amplitude divergence. KGE was assessed using an average of one hundred thousand simulated values, and STC and TC demonstrated higher efficiency than MTC. But MTC has a lower error variance in contrast to STC and TC. Overall, the current study's findings validate that Merged Drought Index (MDI) based on MTC and proposed STC provides a better quantitative way to merge three separate drought indices into a single index. So, MDI successfully captured recorded drought episodes throughout the research locations, indicating that the merging method can be a workable option to identify drought accurately.
Journal Article
A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate
In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff plate bending problems, and the boundary/initial conditions are minimised at those collocation points. A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters. In Kirchhoff plate bending problems, the C1 continuity requirement poses significant difficulties in traditional mesh-based methods. This can be solved by the proposed DCM, which uses a deep neural network to approximate the continuous transversal deflection, and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.
Journal Article
Enhancing PINNs for solving PDEs via adaptive collocation point movement and adaptive loss weighting
Physics-informed neural networks (PINNs) are an emerging method for solving partial differential equations (PDEs) and have been widely applied in the field of scientific computing. In this paper, we introduce a novel adaptive PINN model for solving PDEs. The model draws on the idea of traditional adaptive methods and incorporates the adaptive collocation point movement method into the PINNs model. It can use residual information from the PDE or characteristics of the solution function itself to guide the movement of collocation points, giving an appropriate distribution of collocation points for specific problems, improving the predictive accuracy of the model, and avoiding overfitting. Additionally, the model introduces an adaptive loss weighting strategy, which updates adaptive weights continuously by minimizing negative log-likelihood estimation to achieve adaptive weighting of the loss function, thereby improving the convergence rate and accuracy of the model. Finally, we conduct extensive experiments, including the one-dimensional Poisson equation, two-dimensional Poisson equation, Burgers equation, Klein–Gordon equation, Helmholtz equation, and Lid-Driven problem, to demonstrate the effectiveness and accuracy of the proposed model. The experimental results show that the model can significantly improve predictive accuracy and generalization ability. The data and code can be found at
https://github.com/hsbhc/AMAW-PINN
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Journal Article