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A Multilevel Algorithm for Simultaneously Denoising and Deblurring Images
"... In this paper, we develop a fast multilevel algorithm for simultaneously denoising and deblurring images under the totalvariation regularization. Although much effort has been devoted to developing fast algorithms for the numerical solution and the denoising problem was satisfactorily solved, fast ..."
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In this paper, we develop a fast multilevel algorithm for simultaneously denoising and deblurring images under the totalvariation regularization. Although much effort has been devoted to developing fast algorithms for the numerical solution and the denoising problem was satisfactorily solved, fast algorithms for the combined denoising and deblurring model remain to be a challenge. Recently several successful studies of approximating this model and subsequently finding fast algorithms were conducted which have partially solved this problem. The aim of this paper is to generalize a fast multilevel denoising method to solving the minimization model for simultaneously denoising and deblurring. Our new idea is to overcome the complexity issue by a detailed study of the structured matrices that are associated with the blurring operator. A fast algorithm can then be obtained for directly solving the variational model. Supporting numerical experiments on gray scale images are presented. AMS classifications. 68U10, 65F10, 65K10.
A twolevel approach to large mixedinteger programs with application to cogeneration in energyefficient buildings
, 2015
"... We study a twostage mixedinteger linear program (MILP) with more than 1 million binary variables in the second stage. We develop a twolevel approach by constructing a semicoarse model (coarsened with respect to variables) and a coarse model (coarsened with respect to both variables and constrain ..."
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We study a twostage mixedinteger linear program (MILP) with more than 1 million binary variables in the second stage. We develop a twolevel approach by constructing a semicoarse model (coarsened with respect to variables) and a coarse model (coarsened with respect to both variables and constraints). We coarsen binary variables by selecting a small number of prespecified daily on/off profiles. We aggregate constraints by partitioning them into groups and summing over each group. With an appropriate choice of coarsened profiles, the semicoarse model is guaranteed to find a feasible solution of the original problem and hence provides an upper bound on the optimal solution. We show that solving a sequence of coarse models converges to the same upper bound with proven finite steps. This is achieved by adding violated constraints to coarse models until all constraints in the semicoarse model are satisfied. We demonstrate the effectiveness of our approach in cogeneration for buildings. The coarsened models allow us to obtain good approximate solutions at a fraction of the time required by solving the original problem. Extensive numerical experiments show that the twolevel approach scales to large problems that are beyond the capacity of stateoftheart commercial MILP solvers.
Reduced order models in PIDE constrained optimization
, 2010
"... Mathematical models for option pricing often result in partial differential equations originally starting with the BlackScholes model. In this context, recent enhancements are models driven by Levy processes, which lead to a partial differential equation with an additional integral term. If one s ..."
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Mathematical models for option pricing often result in partial differential equations originally starting with the BlackScholes model. In this context, recent enhancements are models driven by Levy processes, which lead to a partial differential equation with an additional integral term. If one solves the problems mentioned last numerically, this yields large linear systems of equations with dense matrices. However, by using the special structure and an iterative solver the problem can be handled efficiently. To further reduce the computational cost in the calibration phase we implement a reduced order model, like proper orthogonal decomposition (POD), which proves to be very efficient. In this paper we use a special multilevel trust region POD algorithm to calibrate the option pricing model and give numerical results supporting the gain in efficiency.
Aircraft fuselage sizing with multilevel optimization
, 2013
"... LMS Samtech, Angleur (Belgium), in the framework of the project “Méthodes de résolution de problèmes d’optimisation de grande taille pour les structures en matériaux composites ” (Acronym LARGO “LARgescale Optimization problems”). LARGO was granted by the Walloon Region and LMS Samtech in the c ..."
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LMS Samtech, Angleur (Belgium), in the framework of the project “Méthodes de résolution de problèmes d’optimisation de grande taille pour les structures en matériaux composites ” (Acronym LARGO “LARgescale Optimization problems”). LARGO was granted by the Walloon Region and LMS Samtech in the context of the First Program (convention number 916981). 1 Overview of the LARGO project We address the optimization problem of sizing an aircraft fuselage and we focus on the problem of computing the dimensions of the different elements constituting a fuselage minimizing the total mass subject to some constraints. These constraints are mechanical stability constraints criteria (e.g. damage tolerance, buckling and post buckling) and they are formulated using Reserve Factors (RF): usually a structure is validated provided all its RFs are greater than one. These functions can be evaluated analytically by dedicated software but practically they have to be considered as blackbox functions, i.e. as unknown functions whose corresponding outputs can be obtained from a given list of inputs without knowing its expression or internal structure. Moreover these functions are computationallyexpensive since their evaluation is rather costly and plays the major role in the solution of the optimization problem.
Math. Program., Ser. A DOI 10.1007/s1010701206179 FULL LENGTH PAPER
, 2011
"... On the complexity of finding firstorder critical points in constrained nonlinear optimization ..."
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On the complexity of finding firstorder critical points in constrained nonlinear optimization
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, 2010
"... Solving structured nonlinear leastsquares and nonlinear feasibility problems ..."
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Solving structured nonlinear leastsquares and nonlinear feasibility problems
TRUST REGION METHODS WITH HIERARCHICAL FINITE ELEMENT MODELS FOR PDECONSTRAINED OPTIMIZATION
, 2011
"... Abstract. In this paper, a Hierarchical Trust Region Algorithm for solving PDEconstrained optimization problems is developed. A hierarchy of finite element meshes is used to define a hierarchy of quadratic models for the approximation of the discrete reduced cost functional on the finest mesh. The ..."
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Abstract. In this paper, a Hierarchical Trust Region Algorithm for solving PDEconstrained optimization problems is developed. A hierarchy of finite element meshes is used to define a hierarchy of quadratic models for the approximation of the discrete reduced cost functional on the finest mesh. The proposed algorithm simultaneously controls the choice of the model and the size of the trust region radius. Application of the trust region convergence theory allows for proving that every accumulation point of the sequence produced by the algorithm is a stationary point of the discretized problem. Numerical examples illustrate the behavior of the method and show a considerable reduction of computation time compared to the standard Newton trust region scheme.
FULL LENGTH PAPER Adaptive cubic regularisation methods for unconstrained optimization. Part II: worstcase
"... function and derivativeevaluation complexity ..."
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Journal of Complexity ( ) – Contents lists available at ScienceDirect Journal of Complexity
"... journal homepage: www.elsevier.com/locate/jco Complexity bounds for secondorder optimality in unconstrained optimization ..."
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journal homepage: www.elsevier.com/locate/jco Complexity bounds for secondorder optimality in unconstrained optimization
ADAPTIVE MULTILEVEL INEXACT SQP–METHODS FOR PDE–CONSTRAINED OPTIMIZATION
"... Abstract. We present a class of inexact adaptive multilevel trustregion SQPmethods for the efficient solution of optimization problems governed by nonlinear partial differential equations. The algorithm starts with a coarse discretization of the underlying optimization problem and provides during ..."
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Abstract. We present a class of inexact adaptive multilevel trustregion SQPmethods for the efficient solution of optimization problems governed by nonlinear partial differential equations. The algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process 1) implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and 2) implementable accuracy requirements for iterative solvers of the linearized PDE and adjoint PDE on the current grid. We prove global convergence to a stationary point of the infinite–dimensional problem. Moreover, we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equation. Numerical results are presented. Key words. Optimal control, adaptive mesh adaptation, PDE constraints, finite elements, a posteriori error estimator, trustregion methods, inexact linear system solvers. 1. Introduction. In this paper