Sequential function approximation for the solution of differential equations

Meade, Andrew; Kokkolaras, Michael; Zeldin, B. A.

doi:10.1002/(sici)1099-0887(199712)13:12<977::aid-cnm113>3.0.co;2-x

Cited by 7 publications

(3 citation statements)

References 16 publications

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“…The method of weighted residuals is a possible alternative, as discussed in Meade et al (1997). For example, minimizing R, R , where R is the residual of a differential equation and • is some appropriate inner product, corresponds to the well-known least-squares weighted residual method.…”

Section: Challenges and Opportunitiesmentioning

confidence: 99%

“…Motivated by similarities observed in artificial neural network algorithms and adaptive grid optimization in computational mechanics, Meade et al (1997) formulated the concept of sequential function approximation (SFA) for the solution of differential equations. Using the closely related methods of weighted residuals and variational principles, appropriate optimization problems were formulated, and incrementally built solutions were obtained for one-and two-dimensional boundary value problems involving linear self-adjoint differential operators associated with homogeneous Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Concurrent Implementation of the Optimal Incremental Approximation Method for the Adaptive and Meshless Solution of Differential Equations

Kokkolaras

Meade

Zeldin

2003

Optimization and Engineering

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The optimal incremental function approximation method is implemented for the adaptive and meshless solution of differential equations. The basis functions and associated coefficients of a series expansion representing the solution are selected optimally at each step of the algorithm according to appropriate error minimization criteria. Thus, the solution is built incrementally. In this manner, the computational technique is adaptive in nature, although a grid is neither built nor adapted in the traditional sense using a posteriori error estimates. Since the basis functions are associated with the nodes only, the method can be viewed as a meshless method. Variational principles are utilized for the definition of the objective function to be extremized in the associated optimization problems. Complicated data structures, expensive remeshing algorithms, and systems solvers are avoided. Computational efficiency is increased by using low-order local basis functions and the parallel direct search (PDS) optimization algorithm. Numerical results are reported for both a linear and a nonlinear problem associated with fluid dynamics. Challenges and opportunities regarding the use of this method are discussed.

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Section: Challenges and Opportunitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Concurrent Implementation of the Optimal Incremental Approximation Method for the Adaptive and Meshless Solution of Differential Equations

Kokkolaras

Meade

Zeldin

2003

Optimization and Engineering

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“…Generally, function approximation is performed on the final function with methods such as GLOMAP [3], Sequential Function Approximation (SFA) [4], [5], least-squares, etc. Other methods have been developed recently that actually learn using a least-squares approximation.…”

Section: Introductionmentioning

confidence: 99%

Multiresolution state-space discretization method for Q-learning with function approximation and policy iteration

Lampton

Valasek

2009

2009 IEEE International Conference on Systems, Man and Cybernetics

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A multiresolution state-space discretization method is developed for the episodic unsupervised learning method of Q-Learning. In addition, a genetic algorithm is used periodically during learning to approximate the action-value function. Policy iteration is added as a stopping criterion for the algorithm. For large scale problems Q-Learning often suffers from the Curse of Dimensionality due to large numbers of possible stateaction pairs. This paper develops a method whereby a statespace is adaptively discretized by progressively finer grids around the areas of interest within the state or learning space. Policy iteration is added to prevent unnecessary episodes at each level of discretization once the learning has converged. Utility of the method is demonstrated with application to the problem of a morphing airfoil with two morphing parameters (two state variables). By setting the multiresolution method to define the area of interest by the goal the agent seeks, it is shown that this method can learn a specific goal within ±0.002, while reducing the total number episodes needed to converge by 85% from the allotted total possible episodes. It is also shown that a good approximation of the action-value function is produced with 80% agreement between the tabulated and approximated policy, though empirically the approximated policy appears to be superior.

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Solution of the radiative transfer equation in discrete ordinate form by sequential function approximation

Thomson

Meade

Bayazıtoğlu

2001

International Journal of Thermal Sciences

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Sequential function approximation for the solution of differential equations

Cited by 7 publications

References 16 publications

Concurrent Implementation of the Optimal Incremental Approximation Method for the Adaptive and Meshless Solution of Differential Equations

Concurrent Implementation of the Optimal Incremental Approximation Method for the Adaptive and Meshless Solution of Differential Equations

Multiresolution state-space discretization method for Q-learning with function approximation and policy iteration

Solution of the radiative transfer equation in discrete ordinate form by sequential function approximation

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