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Approximate solutions of the 2D space-time fractional diffusion equation via a gradient-descent iterative algorithm with Grünwald-Letnikov approximation
(2022-01-01)
Kittisopaporn, Adisorn
;
Chansangiam, Pattrawut
We consider the two-dimensional space-time fractional differential equation with the Caputo’s time derivative and the Riemann-Liouville space derivatives on bounded domains. The equation is subjected to the zero Dirichlet boundary condition and the zero initial condition. We discretize the equation by finite difference schemes based on Grünwald-Letnikov approximation. Then we linearize the discretized equations into a sparse linear system. To solve such linear system, we propose a gradient-descent iterative algorithm with a sequence of optimal convergence factor aiming to minimize the error occurring at each iteration. The convergence analysis guarantees the capability of the algorithm as long as the coefficient matrix is invertible. In addition, the convergence rate and error estimates are provided. Numerical experiments demonstrate the efficiency, the accuracy and the performance of the proposed algorithm.
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Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm
(2021-12-01)
Kittisopaporn, Adisorn
;
Chansangiam, Pattrawut
This paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.
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Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations
(2020-12-01)
Kittisopaporn, Adisorn
;
Chansangiam, Pattrawut
In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form ∑t=1pAtXBt=C which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson’s equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.
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An enhanced learning algorithm with a particle filter-based gradient descent optimizer method
(2020-08-01)
Kamsing, Patcharin
;
Torteeka, Peerapong
;
Yooyen, Soemsak
This experiment integrates a particle filter concept with a gradient descent optimizer to reduce loss during iteration and obtains a particle filter-based gradient descent (PF-GD) optimizer that can determine the global minimum with excellent performance. Four functions are applied to test optimizer deployment to verify the PF-GD method. Additionally, the Modified National Institute of Standards and Technology (MNIST) database is used to test the PF-GD method by implementing a logistic regression learning algorithm. The experimental results obtained with the four functions illustrate that the PF-GD method performs much better than the conventional gradient descent optimizer, although it has some parameters that must be set before modeling. The results of implementing the MNIST dataset demonstrate that the cross-entropy of the PF-GD method exhibits a smaller decrease than that of the conventional gradient descent optimizer, resulting in higher accuracy of the PF-GD method. The PF-GD method provides the best accuracy for the training model, 97.00%, and the accuracy of evaluating the model with the test dataset is 90.37%, which is higher than the accuracy of 90.08% obtained with the conventional gradient descent optimizer.
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Traffic signal timings optimization based on genetic algorithm and gradient descent
(2020-05-01)
Yadav, Alok
;
Nuthong, Chaiwat
Traffic congestions are a recurring problem that results in significant losses both financially and environmentally. optimizing traffic signal timings is one of the most cost-effective ways to mitigate such effects. optimization of traffic signal timings capable of minimizing congestion is, however, computationally expensive. Research needs to be conducted to develop algorithms capable of better optimization using fewer computational resources. This paper presents a novel approach to traffic signal optimization that combines genetic algorithms and a gradient descent like algorithm to obtain optimized traffic signal timings. The genetic algorithm is used to arrive at a starting point for gradient descent; gradient descent is then used to obtain further improvement.
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Multiple linear regression using gradient descent: A case study on Thailand car sales
(2017-01-01)
Netisopakul, Ponrudee
;
Leenawong, Chartchai
Severe fluctuations in Thailand car sales had enormous impacts on the automobile and related industries. A reliable forecasting model is needed to accurately forecast the car sales for the next production batch. Using ten-year car sales data, this research proposes a machine learning approach using gradient descent (GD) to fitting multiple linear regression for Thailand car sales forecasts. The resulted forecasting accuracy is then compared with that of a normal equation method (NE) as well as that obtained from a statistical package (SP). First, two independent variables (2IVs): Thailand’s Gross Domestic Product and the 12-month Loan Rate are used in the proposed models. Then, dummy seasonal variables (Season) are added to the regression equations. Finally, dummy event flag variables (Event) are added. Totally, five sets of experiments are conducted. The experiment results show that NE produces the same regression equations as SP. Both GD and NE methods yield exactly the same results for 2IVs, but GD yields slightly less prediction accuracy than NE’s in Season and Event experiments. This research concludes that gradient descent has comparable forecasting accuracy to those from other methods. Nevertheless, when the regression contains dummy variables, caution is recommended.
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Estimation of the single GPS-receiver bias using the gradient descent algorithm
(2016-09-06)
Chiablaem, Athiwat
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Supnithi, Pornchai
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Klinngam, Somjai
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Panachart, Chaiwat
;
Saekow, Apithep
The ionospheric Total Electron Content (TEC) can be obtained from processing measurements of the dual-frequency Global Positioning System (GPS) receiver. The main sources of errors in the TEC calculation are satellite and receiver biases. In this paper, we apply the gradient descent algorithm on the receiver bias estimation. The TEC is derived from measurements at 12 dual-frequency GPS stations in Thailand. The criterion of receiver bias estimation is based on the minimum sum of the vertical TEC (VTEC) standard deviation method. The results show that the maximum receiver bias value is approximately 3.69 ns at UDON station, while the minimum value is -5.91 ns at SRTN station. The accuracy of the receiver biases from this algorithm is compared with the reference method. The maximum percentage deviation is about 7.5% at SRTN station. The percentage deviation of the minimum sum of the VTEC between the reference method and the proposed method from all stations are less than 0.05%. Thus, the proposed algorithm is a viable option to estimate the receiver bias.

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