The Experts below are selected from a list of 52392 Experts worldwide ranked by ideXlab platform
X Marchandise - One of the best experts on this subject based on the ideXlab platform.
-
treatment planning optimization by Conjugate Gradients and simulated annealing methods in stereotactic radiosurgery
International Journal of Radiation Oncology Biology Physics, 1995Co-Authors: David Gibon, J Rousseau, B Castelain, S Blond, Christian Vasseur, X MarchandiseAbstract:This paper presents a new optimization method of treatment planning in linac stereotactic radiosurgery. On a workstation integrating x-rays, computed tomography (CT), magnetic resonance imaging (MRI), and digital subtracted angiography (DSA) images, we first determine the outlines of the target volume and surrounding healthy tissues to spare. To achieve complete optimization of the treatment plans, this method decomposes the optimization process in two steps. The position of the isocenters and the diameter of the collimators are first deduced by a Conjugate Gradients method, from the position and size of ellipsoids or spheres modeling the target volume. The other irradiation parameters, such as the isocenter dose, the aperture, and the weight of each irradiation place and of their irradiation sectors are finally deduced by a simulated annealing optimization algorithm. The system can perform multitarget/multisector treatment plans that are automatically obtained in a satisfactory time (as a rule, 20 min for a two-target irradiation), much faster than the time needed for a manual treatment planning. We present the results in two cases: the simulation of a single-target treatment and a two-target real treatment with constraints. In these two cases, we can control the dose received by target and sensitive volumes. This methodmore » achieves an excellent conformation of the estimated isodose curves with the outlines of the target volume, which allows us to avoid the surrounding healthy tissues, thanks to the different weighting factors given on each volume concerned according to the importance we grant to each of them. 34 refs., 4 figs., 3 tabs.« less
-
treatment planning optimization by Conjugate Gradients and simulated annealing methods in stereotactic radiosurgery
International Stereotactic Radiosurgery Society. Congress, 1995Co-Authors: David Gibon, J Rousseau, B Castelain, S Blond, Christian Vasseur, X MarchandiseAbstract:Purpose : This paper presents a new optimization method of treatment planning in linac stereotactic radio-surgery. Methods and Materials : On a workstation integrating x-rays, computed tomography (CT), magnetic resonance imaging (MRI), and digital subtracted angiography (DSA) images, we first determine the outlines of the target volume and surrounding healthy tissues to spare. To achieve complete optimization of the treatment plans, this method decomposes the optimization process in two steps. The position of the isocenters and the diameter of the collimators are first deduced by a Conjugate Gradients method, from the position and size of ellipsoids or spheres modeling the target volume. The other irradiation parameters, such as the isocenter dose, the aperture, and the weight of each irradiation plane and of their irradiation sectors are finally deduced by a simulated annealing optimization algorithm. Results : The system can perform multitarget/multisector treatment plans that are automatically obtained in a satisfactory time (as a rule, 20 min for a two-target irradiation), much faster than the time needed for a manual treatment planning. We present the results in two cases : the simulation of a single-target treatment and a two-target real treatment with constraints. In these two cases, we can control the dose received by target and sensitive volumes. Conclusion : This method achieves an excellent conformation of the estimated isodose curves with the outlines of the target volume, which allows us to avoid the surrounding healthy tissues, thanks to the different weighting factors given on each volume concerned according to the importance we grant to each of them.
Serge Gratton - One of the best experts on this subject based on the ideXlab platform.
-
Minimizing convex quadratics with variable precision Conjugate Gradients
Numerical Linear Algebra with Applications, 2021Co-Authors: Serge Gratton, Ehouarn Simon, David Titley-peloquin, Philippe TointAbstract:We investigate the method of Conjugate Gradients, exploiting inac-curate matrix-vector products, for the solution of convex quadratic op-timization problems. Theoretical performance bounds are derived, andthe necessary quantities occurring in the theoretical bounds estimated,leading to a practical algorithm. Numerical experiments suggest thatthis approach has significant potential, including in the steadily moreimportant context of multi-precision computations.
-
differentiating the method of Conjugate Gradients
SIAM Journal on Matrix Analysis and Applications, 2014Co-Authors: Serge Gratton, David Titleypeloquin, Philippe L Toint, Jean Tshimanga IlungaAbstract:The method of Conjugate Gradients (CG) is widely used for the iterative solution of large sparse systems of equations $Ax=b$, where $A\in\Re^{n\times n}$ is symmetric positive definite. Let $x_k$ denote the $k$th iterate of CG. This is a nonlinear differentiable function of $b$. In this paper we obtain expressions for $J_k$, the Jacobian matrix of $x_k$ with respect to $b$. We use these expressions to obtain bounds on $\|J_k\|_2$, the spectral norm condition number of $x_k$, and discuss algorithms to compute or estimate $J_kv$ and $J_k^Tv$ for a given vector $v$.
-
Conjugate Gradients versus multigrid solvers for diffusion based correlation models in data assimilation
Quarterly Journal of the Royal Meteorological Society, 2013Co-Authors: Serge Gratton, Philippe L Toint, J TshimangaAbstract:This article provides a theoretical and experimental comparison between Conjugate Gradients and multigrid, two iterative schemes for solving linear systems, in the context of applying diffusion-based correlation models in data assimilation. In this context, a large number of such systems has to be (approximately) solved if the implicit mode is chosen for integrating the involved diffusion equation over pseudo-time, thereby making their efficient handling crucial for practical performance. It is shown that the multigrid approach has a significant advantage, especially for larger correlation lengths and/or large problem sizes.
-
Preconditioning and globalizing Conjugate Gradients in dual space for quadratically penalized nonlinear-least squares problems
Computational Optimization and Applications, 2013Co-Authors: Serge Gratton, Selime Gürol, Philippe L TointAbstract:When solving nonlinear least-squares problems, it is often useful to regularize the problem using a quadratic term, a practice which is especially common in applications arising in inverse calculations. A solution method derived from a trust-region Gauss-Newton algorithm is analyzed for such applications, where, contrary to the standard algorithm, the least-squares subproblem solved at each iteration of the method is rewritten as a quadratic minimization subject to linear equality constraints. This allows the exploitation of duality properties of the associated linearized problems. This paper considers a recent Conjugate-gradient-like method which performs the quadratic minimization in the dual space and produces, in exact arithmetic, the same iterates as those produced by a standard Conjugate-Gradients method in the primal space. This dual algorithm is computationally interesting whenever the dimension of the dual space is significantly smaller than that of the primal space, yielding gains in terms of both memory usage and computational cost. The relation between this dual space solver and PSAS (Physical-space Statistical Analysis System), another well-known dual space technique used in data assimilation problems, is explained. The use of an effective preconditioning technique is proposed and refined convergence bounds derived, which results in a practical solution method. Finally, stopping rules adequate for a trust-region solver are proposed in the dual space, providing iterates that are equivalent to those obtained with a Steihaug-Toint truncated Conjugate-gradient method in the primal space.
David Gibon - One of the best experts on this subject based on the ideXlab platform.
-
treatment planning optimization by Conjugate Gradients and simulated annealing methods in stereotactic radiosurgery
International Journal of Radiation Oncology Biology Physics, 1995Co-Authors: David Gibon, J Rousseau, B Castelain, S Blond, Christian Vasseur, X MarchandiseAbstract:This paper presents a new optimization method of treatment planning in linac stereotactic radiosurgery. On a workstation integrating x-rays, computed tomography (CT), magnetic resonance imaging (MRI), and digital subtracted angiography (DSA) images, we first determine the outlines of the target volume and surrounding healthy tissues to spare. To achieve complete optimization of the treatment plans, this method decomposes the optimization process in two steps. The position of the isocenters and the diameter of the collimators are first deduced by a Conjugate Gradients method, from the position and size of ellipsoids or spheres modeling the target volume. The other irradiation parameters, such as the isocenter dose, the aperture, and the weight of each irradiation place and of their irradiation sectors are finally deduced by a simulated annealing optimization algorithm. The system can perform multitarget/multisector treatment plans that are automatically obtained in a satisfactory time (as a rule, 20 min for a two-target irradiation), much faster than the time needed for a manual treatment planning. We present the results in two cases: the simulation of a single-target treatment and a two-target real treatment with constraints. In these two cases, we can control the dose received by target and sensitive volumes. This methodmore » achieves an excellent conformation of the estimated isodose curves with the outlines of the target volume, which allows us to avoid the surrounding healthy tissues, thanks to the different weighting factors given on each volume concerned according to the importance we grant to each of them. 34 refs., 4 figs., 3 tabs.« less
-
treatment planning optimization by Conjugate Gradients and simulated annealing methods in stereotactic radiosurgery
International Stereotactic Radiosurgery Society. Congress, 1995Co-Authors: David Gibon, J Rousseau, B Castelain, S Blond, Christian Vasseur, X MarchandiseAbstract:Purpose : This paper presents a new optimization method of treatment planning in linac stereotactic radio-surgery. Methods and Materials : On a workstation integrating x-rays, computed tomography (CT), magnetic resonance imaging (MRI), and digital subtracted angiography (DSA) images, we first determine the outlines of the target volume and surrounding healthy tissues to spare. To achieve complete optimization of the treatment plans, this method decomposes the optimization process in two steps. The position of the isocenters and the diameter of the collimators are first deduced by a Conjugate Gradients method, from the position and size of ellipsoids or spheres modeling the target volume. The other irradiation parameters, such as the isocenter dose, the aperture, and the weight of each irradiation plane and of their irradiation sectors are finally deduced by a simulated annealing optimization algorithm. Results : The system can perform multitarget/multisector treatment plans that are automatically obtained in a satisfactory time (as a rule, 20 min for a two-target irradiation), much faster than the time needed for a manual treatment planning. We present the results in two cases : the simulation of a single-target treatment and a two-target real treatment with constraints. In these two cases, we can control the dose received by target and sensitive volumes. Conclusion : This method achieves an excellent conformation of the estimated isodose curves with the outlines of the target volume, which allows us to avoid the surrounding healthy tissues, thanks to the different weighting factors given on each volume concerned according to the importance we grant to each of them.
Tove Odland - One of the best experts on this subject based on the ideXlab platform.
-
on the equivalence of the method of Conjugate Gradients and quasi newton methods on quadratic problems
arXiv: Optimization and Control, 2015Co-Authors: Anders Forsgren, Tove OdlandAbstract:In this paper we state necessary and sufficient conditions for equivalence of the method of Conjugate Gradients and quasi-Newton methods on a quadratic problem. We show that the set of quasi-Newton schemes that generate parallel search directions to those of the method of Conjugate Gradients is strictly larger than the one-parameter Broyden family. In addition, we show that this set contains an infinite number of symmetric rank-one update schemes.
-
on the equivalence of the method of Conjugate Gradients and quasi newton methods on quadratic problems
Computational Optimization and Applications, 2015Co-Authors: Anders Forsgren, Tove OdlandAbstract:In this thesis we present research on mathematical properties of methods for solv- ing symmetric systems of linear equations that arise in various optimization problem formulations and in methods for solving such problems.In the first and third paper (Paper A and Paper C), we consider the connection be- tween the method of Conjugate Gradients and quasi-Newton methods on strictly convex quadratic optimization problems or equivalently on a symmetric system of linear equa- tions with a positive definite matrix. We state conditions on the quasi-Newton matrix and the update matrix such that the search directions generated by the corresponding quasi-Newton method and the method of Conjugate Gradients respectively are parallel.In paper A, we derive such conditions on the update matrix based on a sufficient condition to obtain mutually Conjugate search directions. These conditions are shown to be equivalent to the one-parameter Broyden family. Further, we derive a one-to-one correspondence between the Broyden parameter and the scaling between the search directions from the method of Conjugate Gradients and a quasi-Newton method em- ploying some well-defined update scheme in the one-parameter Broyden family.In paper C, we give necessary and sufficient conditions on the quasi-Newton ma- trix and on the update matrix such that equivalence with the method of Conjugate gra- dients hold for the corresponding quasi-Newton method. We show that the set of quasi- Newton schemes admitted by these necessary and sufficient conditions is strictly larger than the one-parameter Broyden family. In addition, we show that this set of quasi- Newton schemes includes an infinite number of symmetric rank-one update schemes.In the second paper (Paper B), we utilize an unnormalized Krylov subspace frame- work for solving symmetric systems of linear equations. These systems may be incom- patible and the matrix may be indefinite/singular. Such systems of symmetric linear equations arise in constrained optimization. In the case of an incompatible symmetric system of linear equations we give a certificate of incompatibility based on a projection on the null space of the symmetric matrix and characterize a minimum-residual solu- tion. Further we derive a minimum-residual method, give explicit recursions for the minimum-residual iterates and characterize a minimum-residual solution of minimum Euclidean norm.
Edwin A H Vollebregt - One of the best experts on this subject based on the ideXlab platform.
-
a new solver for the elastic normal contact problem using Conjugate Gradients deflation and an fft based preconditioner
Journal of Computational Physics, 2014Co-Authors: Edwin A H VollebregtAbstract:This paper presents our new solver BCCG+FAI for solving elastic normal contact problems. This is a comprehensible approach that is based on the Conjugate Gradients (CG) algorithm and that uses FFTs. A first novel aspect is the definition of the "FFT-based Approximate Inverse" preconditioner. The underlying idea is that the inverse matrix can be approximated well using a Toeplitz or block-Toeplitz form, which can be computed using the FFT of the original matrix elements. This preconditioner makes the total number of CG iterations effectively constant in 2D and very slowly increasing in 3D problems. A second novelty is how we deal with a prescribed total force. This uses a deflation technique in such a way that CGs convergence and finite termination properties are maintained. Numerical results show that this solver is more effective than existing CG-based strategies, such that it can compete with Multi-Grid strategies over a much larger problem range. In our opinion it could be the new method of choice because of its simple structure and elegant theory, and because robust performance is achieved independently of any problem specific parameters. We present a new CG- and FFT-based solver for the elastic normal contact problem.Using FFTs we construct an effective preconditioner with (block) Toeplitz form.O(1) iterations are needed in 2D problems as predicted by theory.A prescribed total force is dealt with by "ignoring" the average, using deflation.The performance of the resulting algorithm is competitive with Multi-Grid strategies.