On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification

Melicher, Valdemar; Vrábel, Vladimír

doi:10.1088/0266-5611/29/11/115008

Cited by 6 publications

(3 citation statements)

References 41 publications

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“…The ASM is used in many different fields, such as control theory (Lions, 1971), data assimilation in meteorology (Lewis et al, 2006) or parameter identification (Melicher and Vrábel', 2013;Cimrák and Melicher, 2007). It is difficult to precisely trace its origin, since it is based on a general principle -the duality.…”

Section: Adjoint-state Methodsmentioning

confidence: 99%

Fast derivatives of likelihood functionals for ODE based models using adjoint-state method

2017

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We consider time series data modeled by ordinary differential equations (ODEs), widespread models in physics, chemistry, biology and science in general. The sensitivity analysis of such dynamical systems usually requires calculation of various derivatives with respect to the model parameters.We employ the adjoint state method (ASM) for efficient computation of the first and the second derivatives of likelihood functionals constrained by ODEs with respect to the parameters of the underlying ODE model. Essentially, the gradient can be computed with a cost (measured by model evaluations) that is independent of the number of the ODE model parameters and the Hessian with a linear cost in the number of the parameters instead of the quadratic one. The sensitivity analysis becomes feasible even if the parametric space is high-dimensional.The main contributions are derivation and rigorous analysis of the ASM in the statistical context, when the discrete data are coupled with the continuous ODE model. Further, we present a highly optimized implementation of the results and its benchmarks on a number of problems.The results are directly applicable in (e.g.) maximum-likelihood estimation or Bayesian sampling of ODE based statistical models, allowing for faster, more stable estimation of parameters of the underlying ODE model.

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Section: Adjoint-state Methodsmentioning

confidence: 99%

Fast derivatives of likelihood functionals for ODE based models using adjoint-state method

2017

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“…In some cases, small varieties in the right-hand side of (15) may lead to dramatic varieties in the solution [25][26][27][28]. In other words, they are very susceptible to error, in the inverse problem, this may directly lead to greater deviations of the identified results [29][30][31], and thus it is not realistic to solve (15) using traditional mathematical methods [32][33][34].…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Identification of Random Dynamic Force Using an Improved Maximum Entropy Regularization Combined with a Novel Conjugate Gradient

Ren

Wang

Liu

2017

Mathematical Problems in Engineering

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We propose a novel mathematical algorithm to offer a solution for the inverse random dynamic force identification in practical engineering. Dealing with the random dynamic force identification problem using the proposed algorithm, an improved maximum entropy (IME) regularization technique is transformed into an unconstrained optimization problem, and a novel conjugate gradient (NCG) method was applied to solve the objective function, which was abbreviated as IME-NCG algorithm. The result of IME-NCG algorithm is compared with that of ME, ME-CG, ME-NCG, and IME-CG algorithm; it is found that IME-NCG algorithm is available for identifying the random dynamic force due to smaller root mean-square-error (RMSE), lower restoration time, and fewer iterative steps. Example of engineering application shows that L-curve method is introduced which is better than Generalized Cross Validation (GCV) method and is applied to select regularization parameter; thus the proposed algorithm can be helpful to alleviate the ill-conditioned problem in identification of dynamic force and to acquire an optimal solution of inverse problem in practical engineering.

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“…J. Gao et al [12] utilized the standard TR method for identifying the unknown load source. V. Melicher et al [13] identified the piecewise constant parameter using the standard TR method. In addition, many works are mainly concerned with the identification of load position and load number in nonlinear system using the standard TR method [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

A Novel Fractional Tikhonov Regularization Coupled with an Improved Super-Memory Gradient Method and Application to Dynamic Force Identification Problems

Wang

Ren

Liu

2018

Mathematical Problems in Engineering

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This paper presents a novel inverse technique to provide a stable optimal solution for the ill-posed dynamic force identification problems. Due to ill-posedness of the inverse problems, conventional numerical approach for solutions results in arbitrarily large errors in solution. However, in the field of engineering mathematics, there are famous mathematical algorithms to tackle the illposed problem, which are known as regularization techniques. In the current study, a novel fractional Tikhonov regularization (NFTR) method is proposed to perform an effective inverse identification, then the smoothing functional of the ill-posed problem processed by the proposed method is regarded as an optimization problem, and finally a stable optimal solution is obtained by using an improved super-memory gradient (ISMG) method. The result of the present method is compared with that of the standard TR method and FTR method; new findings can be obtained; that is, the present method can improve accuracy and stability of the inverse identification problem, robustness is stronger, and the rate of convergence is faster. The applicability and efficiency of the present method in this paper are demonstrated through a mathematical example and an engineering example.

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On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification

Cited by 6 publications

References 41 publications

Fast derivatives of likelihood functionals for ODE based models using adjoint-state method

Fast derivatives of likelihood functionals for ODE based models using adjoint-state method

Identification of Random Dynamic Force Using an Improved Maximum Entropy Regularization Combined with a Novel Conjugate Gradient

A Novel Fractional Tikhonov Regularization Coupled with an Improved Super-Memory Gradient Method and Application to Dynamic Force Identification Problems

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