Regularization of Inverse Problems

Engl, Heinz W.; Hanke, Martin; Neubauer, A.

doi:10.1007/978-94-009-1740-8

Cited by 4,154 publications

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“…In practice, this means that small errors in the measured BOLD signal can induce large errors in the estimate of the parameters. To solve IPP (7), we propose a solution methodology that is based on the Tikhonov regularized Newton method (TNM) [32,33], since regularized iterative methods appear to be the primary candidates for solving nonlinear and ill-posed problems (see, e.g., [34], and the references therein). The Newton algorithm addresses the nonlinear aspect of IPP (7), whereas the Tikhonov regularization procedure is incorporated to address its ill-posed nature [35,36].…”

Section: Parameter Estimation: the Solution Methodologymentioning

confidence: 99%

A novel approach to calibrate the hemodynamic model using functional Magnetic Resonance Imaging (fMRI) measurements

Khoram

Zayane

Djellouli

et al. 2016

Journal of Neuroscience Methods

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Page 1 of 42A c c e p t e d M a n u s c r i p t• An efficient solution methodology for estimating the parameters of the brain response model is proposed.• This method distinguishes itself from existing calibrating techniques by employing intelligently (a) the Newton algorithm, (b) a Tikhonov regularization approach, and (c) a Kalman filtering procedure.• Both synthetic and real fMRI measurements were used to assess the performance of this method.• The fast convergence, the accuracy, and the robustness to the noise effect of this method are clearly demonstrated by the reported numerical results.• This methods outperforms existing methods. Highlights (for review)Page 2 of 42 A c c e p t e d M a n u s c r i p t Abstract BackgroundThe calibration of the hemodynamic model that describes changes in blood flow and blood oxygenation during brain activation is a crucial step for successfully monitoring and possibly predicting brain activity. This in turn has the potential to provide diagnosis and treatment of brain diseases in early stages. New MethodWe propose an efficient numerical procedure for calibrating the hemodynamic model using some fMRI measurements. The proposed solution methodology is a regularized iterative method equipped with a Kalman filtering-type procedure. The Newton component of the proposed method addresses the nonlinear aspect of the problem. The regularization feature is used to ensure the stability of the algorithm. The Kalman filter procedure is incorporated here to address the noise in the data. Results Numerical results obtained with synthetic data as well as with real fMRI measurements are presented to illustrate the accuracy, robustness to the noise, and the cost-effectiveness of the proposed method.

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Section: Parameter Estimation: the Solution Methodologymentioning

confidence: 99%

A novel approach to calibrate the hemodynamic model using functional Magnetic Resonance Imaging (fMRI) measurements

Khoram

Zayane

Djellouli

et al. 2016

Journal of Neuroscience Methods

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“…On the other hand, such finite difference or finite element computations only work efficiently in low dimension, since the problem of finding the control of the ordinary differential equation (1.1) in R d has been transformed to solving a partial differential equation in R d × [0, T ], while the Pontryagin approach based on the solution of the two ordinary differential equations (1.15) with (1.8), forX and the adjoint variableλ, is computationally feasible even in very high dimension, d 1, with application to distributed control of partial differential equations cf. [17,18], optimal shape [21] and inverse problems [13,29], see Section 2. This work uses 1 Citation from chapter one in [24] "This equation of Bellman's yields an approach to the solution of the optimal control problem which is closely connected with, but different from, the approach described in this book (see Chap.…”

Section: U(x T) := Inf X(t)=x α∈A G X(t ) +mentioning

confidence: 99%

“…by the discrepancy principle, cf. [13,29]. The Newton method described in Section 3 works well to solve the discrete equations for d = 10.…”

Section: Ds(t) = µS(t)dt + σ T S(t) S(t)dw (T) (21)mentioning

confidence: 99%

Convergence rates of symplectic Pontryagin approximations in optimal control theory

Sandberg¹,

Szepessy²

2006

ESAIM: M2AN

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Abstract. Many inverse problems for differential equations can be formulated as optimal control problems. It is well known that inverse problems often need to be regularized to obtain good approximations. This work presents a systematic method to regularize and to establish error estimates for approximations to some control problems in high dimension, based on symplectic approximation of the Hamiltonian system for the control problem. In particular the work derives error estimates and constructs regularizations for numerical approximations to optimally controlled ordinary differential equations in R d , with non smooth control. Though optimal controls in general become non smooth, viscosity solutions to the corresponding Hamilton-Jacobi-Bellman equation provide good theoretical foundation, but poor computational efficiency in high dimensions. The computational method here uses the adjoint variable and works efficiently also for high dimensional problems with d 1. Controls can be discontinuous due to a lack of regularity in the Hamiltonian or due to colliding backward paths, i.e. shocks. The error analysis, for both these cases, is based on consistency with the Hamilton-Jacobi-Bellman equation, in the viscosity solution sense, and a discrete Pontryagin principle: the bi-characteristic Hamiltonian ODE system is solved with a C 2 approximate Hamiltonian. The error analysis leads to estimates useful also in high dimensions since the bounds depend on the Lipschitz norms of the Hamiltonian and the gradient of the value function but not on d explicitly. Applications to inverse implied volatility estimation, in mathematical finance, and to a topology optimization problem are presented. An advantage with the Pontryagin based method is that the Newton method can be applied to efficiently solve the discrete nonlinear Hamiltonian system, with a sparse Jacobian that can be calculated explicitly.Mathematics Subject Classification. 49M29, 49L25.

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“…So it was not our goal in this paper to compare Conjugate Gradient with those methods, but to show that our approximation of GCV is applicable as a stopping rule to the Conjugate Gradient method for large scale ill-posed problems, similar to Positron Emission Tomography (very large scale, not severely ill-posed and relatively large noise in the data). Some authors (see, for example, [15,9]) have suggested the use of other approximations to GCV. Our experimental results show (Fig.…”

Section: Positron Emission Tomographymentioning

confidence: 99%

The effect of the nonlinearity on GCV applied to conjugate gradients in computerized tomography

Santos¹,

Pierro²

2006

Mat. apl. comput.

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Abstract.We study the effect of the nonlinear dependence of the iterate x k of Conjugate Gradients method (CG) from the data b in the GCV procedure to stop the iterations. We compare two versions of using GCV to stop CG. In one version we compute the GCV function with the iterate x k depending linearly from the data b and the other one depending nonlinearly. We have tested the two versions in a large scale problem: positron emission tomography (PET). Our results suggest the necessity of considering the nonlinearity for the GCV function to obtain a reasonable stopping criterion.Mathematical subject classification: 62G05, 92C55, 65F10, 65C05.

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Regularization of Inverse Problems

Cited by 4,154 publications

References 0 publications

A novel approach to calibrate the hemodynamic model using functional Magnetic Resonance Imaging (fMRI) measurements

A novel approach to calibrate the hemodynamic model using functional Magnetic Resonance Imaging (fMRI) measurements

Convergence rates of symplectic Pontryagin approximations in optimal control theory

The effect of the nonlinearity on GCV applied to conjugate gradients in computerized tomography

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