Fractional PDE constrained optimization: An optimize-then-discretize approach with L-BFGS and approximate inverse preconditioning

Cipolla, Stefano; Durastante, Fabio

doi:10.1016/j.apnum.2017.09.001

Cited by 11 publications

(10 citation statements)

References 24 publications

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“…S. Güttel et al [11] use the FOTD approach to for solve the time-dependent PDE-constrained optimization problems. S. Cipolla et al [12] use the FOTD approach to address the numerical solution for two Fraction PDE constrained optimization problems. Both theoretical and experimental analysis of the problem are carried out.…”

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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“…In many applications of interest, the preconditioner Dn,m has a diagonal times a multilevel banded Toeplitz structure (e.g. [17,53]). The application of the preconditioner in (3.13) consists in a single (LU or, if applicable, Cholesky) factorization of Dn,m at the beginning of the optimization, and subsequently two backward solves for every ADMM iteration.…”

Section: − −− → {Smentioning

confidence: 99%

“…Various specialized solution methods have been proposed in the literature, aiming at lowering the computational and storage cost of solving such problems (see for instance [17,23,24,40,53,77]). One popular and effective approach is to employ tensor product solvers.…”

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confidence: 99%

Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

Pougkakiotis¹,

Pearson²,

Leveque³

et al. 2020

SIAM J. Matrix Anal. Appl.

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In this paper, we present numerical methods suitable for solving convex quadratic Fractional Differential Equation (FDE) constrained optimization problems, with box constraints on the state and/or control variables. We develop an Alternating Direction Method of Multipliers (ADMM) framework, which uses preconditioned Krylov subspace solvers for the resulting subproblems. The latter allows us to tackle a range of Partial Differential Equation (PDE) optimization problems with box constraints, posed on space-time domains, that were previously out of the reach of state-of-the-art preconditioners. In particular, by making use of the powerful Generalized Locally Toeplitz (GLT) sequences theory, we show that any existing GLT structure present in the problem matrices is preserved by ADMM, and we propose some preconditioning methodologies that could be used within the solver, to demonstrate the generality of the approach. Focussing on convex quadratic programs with time-dependent 2-dimensional FDE constraints, we derive multilevel circulant preconditioners, which may be embedded within Krylov subspace methods, for solving the ADMM sub-problems. Discretized versions of FDEs involve large dense linear systems. In order to overcome this difficulty, we design a recursive linear algebra, which is based on the Fast Fourier Transform (FFT). We manage to keep the storage requirements linear, with respect to the grid size N , while ensuring an order N log N computational complexity per iteration of the Krylov solver. We implement the proposed method, and demonstrate its scalability, generality, and efficiency, through a series of experiments over different setups of the FDE optimization problem.

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“…We note at this point that FDE-constrained optimization problems, and preconditioners for the resulting matrix systems, have been considered previously (see for instance [14,17,29,54]): in the majority of previous work the cost functional J L 2 (y, u) is considered, and no additional algebraic constraints on the state and control variables are imposed. Recently, preconditioners for FDE-constrained optimization were considered for problems with algebraic and sparsity constraints in the time-independent setting [19].…”

Section: The Fde-constrained Optimization Modelmentioning

confidence: 99%

Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

Pougkakiotis,

Pearson,

Leveque

et al. 2019

Preprint

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In this paper, we present numerical methods suitable for solving convex Fractional Differential Equation (FDE) optimization problems, with potential box constraints on the state and control variables. First we derive powerful multilevel circulant preconditioners, which may be embedded within Krylov subspace methods, for solving equality constrained FDE optimization problems. We then develop an Alternating Direction Method of Multipliers (ADMM) framework, which uses preconditioned Krylov solvers for the resulting subproblems. The latter allows us to tackle FDE optimization problems with box constraints, posed on space-time domains, that were previously out of the reach of state-of-the-art preconditioners. Discretized versions of FDEs involve large dense linear systems. In order to overcome this difficulty, we design a recursive linear algebra, which is based on the fast Fourier transform (FFT), and our proposed multilevel circulant preconditioners are suitable for approximating discretized FDEs of arbitrary dimension. Focusing on timedependent 2-dimensional FDEs, we manage to keep the storage requirements linear, with respect to the grid size N , while ensuring an order N log N computational complexity per iteration of the Krylov solver. We implement the proposed method, and demonstrate its scalability, generality, and efficiency, through a series of experiments over different setups of the FDE optimization problem.

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Fractional PDE constrained optimization: An optimize-then-discretize approach with L-BFGS and approximate inverse preconditioning

Cited by 11 publications

References 24 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

Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

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