scholarly journals Identification of linear response functions from arbitrary perturbation experiments in the presence of noise – Part 1: Method development and toy model demonstration

2021 ◽  
Vol 28 (4) ◽  
pp. 501-532
Author(s):  
Guilherme L. Torres Mendonça ◽  
Julia Pongratz ◽  
Christian H. Reick
Keyword(s):  
Response Function ◽  
Noise Level ◽  
Linear Response ◽  
Method Development ◽  
Control Experiment ◽  
Regularization Parameter ◽  
Response Functions ◽  
Perturbation Experiment ◽  
Ill Posed ◽  
Proper Estimation

Abstract. Existent methods to identify linear response functions from data require tailored perturbation experiments, e.g., impulse or step experiments, and if the system is noisy, these experiments need to be repeated several times to obtain good statistics. In contrast, for the method developed here, data from only a single perturbation experiment at arbitrary perturbation are sufficient if in addition data from an unperturbed (control) experiment are available. To identify the linear response function for this ill-posed problem, we invoke regularization theory. The main novelty of our method lies in the determination of the level of background noise needed for a proper estimation of the regularization parameter: this is achieved by comparing the frequency spectrum of the perturbation experiment with that of the additional control experiment. The resulting noise-level estimate can be further improved for linear response functions known to be monotonic. The robustness of our method and its advantages are investigated by means of a toy model. We discuss in detail the dependence of the identified response function on the quality of the data (signal-to-noise ratio) and on possible nonlinear contributions to the response. The method development presented here prepares in particular for the identification of carbon cycle response functions in Part 2 of this study (Torres Mendonça et al., 2021a). However, the core of our method, namely our new approach to obtaining the noise level for a proper estimation of the regularization parameter, may find applications in also solving other types of linear ill-posed problems.

scholarly journals Identification of linear response functions from arbitrary perturbation experiments in the presence of noise – Part I. Method development and toy model demonstration

2021 ◽  
Author(s):  
Guilherme L. Torres Mendonça ◽  
Julia Pongratz ◽  
Christian H. Reick
Keyword(s):  
Response Function ◽  
Noise Level ◽  
Linear Response ◽  
Method Development ◽  
Control Experiment ◽  
Regularization Parameter ◽  
Response Functions ◽  
Perturbation Experiment ◽  
Ill Posed ◽  
Proper Estimation

Abstract. Existent methods to identify linear response functions from data require tailored perturbation experiments, e.g. impulse or step experiments. And if the system is noisy, these experiments need to be repeated several times to obtain a good statistics. In contrast, for the method developed here, data from only a single perturbation experiment at arbitrary perturbation is sufficient if in addition data from an unperturbed (control) experiment is available. To identify the linear response function for this ill-posed problem we invoke regularization theory. The main novelty of our method lies in the determination of the level of background noise needed for a proper estimation of the regularization parameter: This is achieved by comparing the frequency spectrum of the perturbation experiment with that of the additional control experiment. The resulting noise level estimate can be further improved for linear response functions known to be monotonic. The robustness of our method and its advantages are investigated by means of a toy model. We discuss in detail the dependence of the identified response function on the quality of the data (signal-to-noise ratio) and on possible nonlinear contributions to the response. The method development presented here prepares in particular for the identification of carbon-cycle response functions in Part II of this study. But the core of our method, namely our new approach to obtain the noise level for a proper estimation of the regularization parameter, may find applications in solving also other types of linear ill-posed problems.

scholarly journals ON NUMERICAL REALIZATION OF QUASIOPTIMAL PARAMETER CHOICES IN (ITERATED) TIKHONOV AND LAVRENTIEV REGULARIZATION

2009 ◽  
Vol 14 (1) ◽  
pp. 99-108 ◽  
Author(s):  
Toomas Raus ◽  
Uno Hämarik
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Numerical Realization ◽  
A Posteriori ◽  
Parameter Choice ◽  
Linear Combinations ◽  
Choice Alternative ◽  
Lavrentiev Regularization ◽  
Ill Posed

We consider linear ill‐posed problems in Hilbert spaces with noisy right hand side and given noise level. For approximation of the solution the Tikhonov method or the iterated variant of this method may be used. In self‐adjoint problems the Lavrentiev method or its iterated variant are used. For a posteriori choice of the regularization parameter often quasioptimal rules are used which require computing of additionally iterated approximations. In this paper we propose for parameter choice alternative numerical schemes, using instead of additional iterations linear combinations of approximations with different parameters.

Linear Response Functions of a Cumulus Ensemble to Temperature and Moisture Perturbations and Implications for the Dynamics of Convectively Coupled Waves

2010 ◽  
Vol 67 (4) ◽  
pp. 941-962 ◽  
Author(s):  
Zhiming Kuang
Keyword(s):  
Boundary Layer ◽  
Linear Response ◽  
Large Scale ◽  
Control Experiment ◽  
Convective Heating ◽  
Tropospheric Temperature ◽  
Response Functions ◽  
Convectively Coupled Waves ◽  
Toga Coare ◽  
Mean State

Abstract An approach is presented for the construction of linear response functions of a cumulus ensemble to large-scale temperature and moisture perturbations using a cloud system–resolving model (CSRM). A set of time-invariant, horizontally homogeneous, anomalous temperature and moisture tendencies is added, one at a time, to the forcing of the CSRM. By recording the departure of the equilibrium domain-averaged temperature and moisture profiles from those of a control experiment and through a matrix inversion, a sufficiently complete and accurate set of linear response functions is constructed for use as a parameterization of the cumulus ensemble around the reference mean state represented by the control experiment. This approach is applied to two different mean state conditions in which the CSRM, when coupled with 2D gravity waves, exhibits interestingly different behaviors. With a more strongly convecting mean state forced by the large-scale vertical velocity profile taken from the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE), spontaneous development of convectively coupled waves requires moisture variations above the boundary layer, whereas with a mean state of radiative–convective equilibrium (RCE) not forced by large-scale vertical advection, the development of convectively coupled waves is stronger and persists even when moisture variations above the boundary layer are removed. The linear response functions were able to reproduce these behaviors of the full CSRM with some quantitative accuracy. The linear response functions show that both temperature and moisture perturbations at a range of heights can regulate convective heating. The ability for convection to remove temperature anomalies, thus maintaining convective neutrality, decreases considerably from the lower troposphere to the middle and upper troposphere. It is also found that the response of convective heating to a lower tropospheric temperature anomaly is more top-heavy in the RCE case than in the TOGA COARE case. Comparing the linear response functions with the treatment of convection in an earlier simple model by the present author indicates general consistency, lending confidence that the instability mechanisms identified in that model provide the correct explanation to the instability seen in the CSRM simulations and the instability’s dependence on the mean state.

On The Analog Of The Monotone Error Rule For Parameter Choice In The (Iterated) Lavrentiev Regularization

2008 ◽  
Vol 8 (3) ◽  
pp. 237-252 ◽  
Author(s):  
U HAMARIK ◽  
R. PALM ◽  
T. RAUS
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Optimal Error Estimates ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Optimal Error ◽  
Numerical Examples ◽  
Lavrentiev Regularization ◽  
Ill Posed

AbstractWe consider linear ill-posed problems in Hilbert spaces with a noisy right hand side and a given noise level. To solve non-self-adjoint problems by the (it-erated) Tikhonov method, one effective rule for choosing the regularization parameter is the monotone error rule (Tautenhahn and Hamarik, Inverse Problems, 1999, 15, 1487– 1505). In this paper we consider the solution of self-adjoint problems by the (iterated) Lavrentiev method and propose for parameter choice an analog of the monotone error rule. We prove under certain mild assumptions the quasi-optimality of the proposed rule guaranteeing convergence and order optimal error estimates. Numerical examples show for the proposed rule and its modifications much better performance than for the modified discrepancy principle.

A fast multilevel iteration method for solving linear ill-posed integral equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hongqi Yang ◽  
Rong Zhang
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Integral Equations ◽  
Tikhonov Regularization ◽  
Iteration Method ◽  
Noise Level ◽  
Regularization Parameter ◽  
Ill Posed ◽  
Linear Integral ◽  
Linear Integral Equations

Abstract We propose a new concept of noise level: R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level for ill-posed linear integral equations in Tikhonov regularization, which extends the range of regularization parameter. This noise level allows us to choose a more suitable regularization parameter. Moreover, we also analyze error estimates of the approximate solution with respect to this noise level. For ill-posed integral equations, finding fast and effective numerical methods is a challenging problem. For this, we formulate a matrix truncated strategy based on multiscale Galerkin method to generate the linear system of Tikhonov regularization for ill-posed linear integral equations, which greatly reduce the computational complexity. To further reduce the computational cost, a fast multilevel iteration method for solving the linear system is established. At the same time, we also prove convergence rates of the approximate solution obtained by this fast method with respect to the R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level under the balance principle. By numerical results, we show that R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level is very useful and the proposed method is a fast and effective method, respectively.

scholarly journals NEW RULE FOR CHOICE OF THE REGULARIZATION PARAMETER IN (ITERATED) TIKHONOV METHOD

2009 ◽  
Vol 14 (2) ◽  
pp. 187-198 ◽  
Author(s):  
Toomas Raus ◽  
Uno Hämarik
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Optimal Error Estimates ◽  
A Posteriori ◽  
Optimal Error ◽  
Numerical Examples ◽  
Tikhonov Method ◽  
Mild Assumption ◽  
Ill Posed

We propose a new a posteriori rule for choosing the regularization parameter α in (iterated) Tikhonov method for solving linear ill‐posed problems in Hilbert spaces. We assume that data are noisy but noise level δ is given. We prove that (iterated) Tikhonov approximation with proposed choice of α converges to the solution as δ → 0 and has order optimal error estimates. Under certain mild assumption the quasioptimality of proposed rule is also proved. Numerical examples show the advantage of the new rule over the monotone error rule, especially in case of rough δ.

scholarly journals Penalty-based smoothness conditions in convex variational regularization

2019 ◽  
Vol 27 (2) ◽  
pp. 283-300 ◽  
Author(s):  
Bernd Hofmann ◽  
Stefan Kindermann ◽  
Peter Mathé
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Tikhonov Regularization ◽  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
The Family ◽  
Ill Posed ◽  
General Convex ◽  
Variational Regularization

Abstract The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization in Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.

scholarly journals EXTRAPOLATION OF TIKHONOV REGULARIZATION METHOD

2010 ◽  
Vol 15 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Uno Hämarik ◽  
Reimo Palm ◽  
Toomas Raus
Keyword(s):  
Approximate Solution ◽  
Tikhonov Regularization ◽  
Noise Level ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Noisy Data ◽  
Tikhonov Regularization Method ◽  
Guaranteed Accuracy ◽  
Numerical Examples ◽  
Ill Posed

We consider regularization of linear ill‐posed problem Au = f with noisy data fδ, ¦fδ - f¦≤ δ . The approximate solution is computed as the extrapolated Tikhonov approximation, which is a linear combination of n ≥ 2 Tikhonov approximations with different parameters. If the solution u* belongs to R((A*A) n ), then the maximal guaranteed accuracy of Tikhonov approximation is O(δ 2/3) versus accuracy O(δ 2n/(2n+1)) of corresponding extrapolated approximation. We propose several rules for choice of the regularization parameter, some of these are also good in case of moderate over‐ and underestimation of the noise level. Numerical examples are given.

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