Space–time spectral collocation method for one-dimensional PDE constrained optimisation

Rezazadeh, Arezou; Mahmoudi, Mahmoud; Darehmiraki, Majid

doi:10.1080/00207179.2018.1501161

Cited by 13 publications

(5 citation statements)

References 19 publications

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“…Chen et al [4] follow the FOTD approach to solve a coupled system of PDEs, which is derived from the continuous first order optimality conditions. A. Rezazadeh et al [10] use the FOTD approach and the space-time spectral collocation method to solve the parabolic constrained optimal control problem. It is proved that the accuracy of numerical solution through this method is much higher than the classical numerical solutions.…”

Section: A Exiting Methodsmentioning

confidence: 99%

First-Optimize-Then-Discretize Strategy for the Parabolic PDE Constrained Optimization Problem With Application to the Reheating Furnace

2021

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Each slab entering to the reheating furnace has an unique and optimal reheating curve. The process of obtaining the optimal reheating curve is to solve the typical Partial differential equations (PDE) constrained optimization problem. Obviously, the solution of optimization problem is determined by both the precision of the mathematical PDE model and the numerical method. Firstly, the more accurate mathematical PDE model, in which some key parameters are reconsidered as temperature-dependent, is built for the reheating furnace. Secondly, the first-optimize-then-discretize approach is introduced to solve this PDE-constrained optimization problem. The analysis of the Fréchet gradient of the cost functional is given and we can prove the gradient is Lipschitz continuous. Then, an improved conjugate gradient method is proposed to solve this problem. Finally, numerical simulations and experiment examples are given and analyzed. The results can prove the effectiveness of the proposed strategy. INDEX TERMSReheating furnace; PDE-constrained optimization problem; first-optimize-thendiscretize; Lipschitz continuous; improved conjugate gradient algorithm;

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Section: A Exiting Methodsmentioning

confidence: 99%

First-Optimize-Then-Discretize Strategy for the Parabolic PDE Constrained Optimization Problem With Application to the Reheating Furnace

2021

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“…This is why in practical calculations often small artificial damping is added to the model. (3) Spatial-temporal semianalytical basis function methods [37][38][39] employ the spatial-temporal semi-analytical basis function a priori to satisfy the transient wave equation and then solve it directly. Among these three time-discretization schemes, the first two have been widely used for transient wave propagation analysis; the last one has not been widely used because the time-dependent semi-analytical basis functions are not easy to construct, in particular, the transient wave equation, including the source excitations.…”

Section: Introductionmentioning

confidence: 99%

A Novel Spatial–Temporal Radial Trefftz Collocation Method for 3D Transient Wave Propagation Analysis with Specified Sound Source Excitation

Chen

2022

Mathematics

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In this paper, a novel semi-analytical collocation solver, the spatial–temporal radial Trefftz collocation method (STRTCM) is proposed to solve 3D transient wave equations with specified sound source excitations. Unlike the traditional time discretization strategies, the proposed numerical scheme introduces the spatial–temporal radial Trefftz functions (STRTFs) as the basis functions for the spatial and temporal discretization of the transient wave equations. The STRTFs are constructed in the spatial–temporal domain, which is a combination of 3D Euclidean space and time into a 4D manifold. Moreover, since the initial and boundary conditions are imposed on the spatial–temporal domain boundaries, the original transient wave propagation problem can be converted to an inverse boundary value problem. To deal with the specified time-dependent sound source excitations, the composite multiple reciprocity technique is extended from the spatial domain to the spatial–temporal domain, which transforms the original problem with a source term into a high-order problem without a source term. By deriving the related STRTFs for the considered high-order problem, the proposed scheme only requires the node discretization on the spatial–temporal domain boundaries. The efficiency of the proposed method is numerically verified by four benchmark examples under 3D transient wave equations with specified time-dependent sound source excitation.

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“…In the indirect method, the optimality conditions are derived. This method leads to a multiple-point boundary-value (BVP) problem that is solved to determine candidate optimal trajectories called extremals using the existing methods such as spectral methods [19,25,26,29], simple shooting method [8,18] and finite element method [2,16,32]. Each of the computed extremals is then examined to see if it is a local minimum, maximum, or a saddle point and then the particular extremal with the lowest cost is chosen.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of optimal control of atherosclerosis using direct and indirect methods with shooting/collocation approach

Nasresfahani¹,

Eslahchi²

2022

Preprint

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We present a direct numerical method for the solution of an optimal control problem controlling the growth of LDL, HDL and plaque. The optimal control problem is constrained with a system of coupled nonlinear free and mixed boundary partial differential equations consisting of three parabolics one elliptic and one ordinary differential equations. In the first step, the original problem is transformed from a free boundary problem into a fixed one and from the mixed boundary condition to a Neumann one. Then, employing a fixed point-collocation method, we solve the optimal control problem. In each step of the fixed point iteration, the problem is changed to a linear one and then, the equations are solved using the collocation method bringing about an NLP which is solved using sequential quadratic programming. Then, the obtained solution is verified using indirect methods originating from the firstorder optimality conditions. Numerical results are considered to illustrate the efficiency of methods.

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Space–time spectral collocation method for one-dimensional PDE constrained optimisation

Cited by 13 publications

References 19 publications

First-Optimize-Then-Discretize Strategy for the Parabolic PDE Constrained Optimization Problem With Application to the Reheating Furnace

First-Optimize-Then-Discretize Strategy for the Parabolic PDE Constrained Optimization Problem With Application to the Reheating Furnace

A Novel Spatial–Temporal Radial Trefftz Collocation Method for 3D Transient Wave Propagation Analysis with Specified Sound Source Excitation

Numerical solution of optimal control of atherosclerosis using direct and indirect methods with shooting/collocation approach

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