This paper deals with a second order dynamical system with vanishing damping that contains a Tikhonov regularization term, in connection to the minimization problem of a convex Fréchet differentiable ...function g. We show that for appropriate Tikhonov regularization parameters the value of the objective function in a generated trajectory converges fast to the global minimum of the objective function and a trajectory generated by the dynamical system converges weakly to a minimizer of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Nevertheless, our main goal is to extend and improve some recent results obtained in 6 and 12 concerning the strong convergence of the generated trajectories to an element of minimal norm from the argmin set of the objective function g. Our analysis also reveals that the damping coefficient and the Tikhonov regularization coefficient are strongly correlated.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Excitation identification has received considerable attention because of its importance in safety assessment and structural design. This paper proposed a power spectral density (PSD) identification ...method for stationary stochastic excitations considering multi-source uncertainties in load fluctuations, material dispersions and measurement noises. Based on the traditional inverse pseudo-excitation method, a two-step weighting regularization strategy is creatively developed to reduce the amplification effects of the uncertainties in the transfer matrix and measurements on reconstructed results near natural frequencies. Especially, to enhance the generalizability of regularization operations, a weighting matrix is defined based on the interval-quantized deviation analysis of pseudo excitations and then an improved Tikhonov regularizing operator is defined given the features of the weighting transfer matrix and pseudo responses. Next, the response superposition-decomposition principle is performed to determine the boundaries of excitation PSD and two uncertainty propagation methods are developed. To guarantee the accuracy and efficiency of uncertainty analysis, the adaptive reduced-dimension Chebyshev model is adopted to characterize the nonlinear response-parameter relationships, and the first-order Taylor series approximation is used to describe the linear response-excitation relationships. Eventually, two numerical examples and one experimental example are discussed to demonstrate the feasibility of the developed approach. The results suggest its promising applications in complicated structures and loading conditions.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this paper we revisit the discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems in Hilbert spaces and provide some new and improved saturation results under less ...restrictive conditions, comparing with the existing results in the literature.
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Dynamic light scattering (DLS) from ultra-low concentration suspensions (the average number of particles in the scattering volume is less than ~100) gives rise to autocorrelation functions (ACFs) ...containing a non-Gaussian term due to particle number fluctuations. This term is difficult to characterize and account for and makes recovery of particle size distribution (PSD) information unreliable. We show that an initial analysis of the intensity ACF to determine parameters describing the amplitude and relaxation rate of the non-Gaussian term and then using these parameters to create a better theoretical non-Gaussian ACF model allows a more accurate recovery of the PSD. The modified model is consistent with the measured ACF data, and a reconstructed kernel matrix matching the measured data is obtained. When compared with the usual kernel function reconstruction (KFR) method, the proposed method gives significantly improved PSD recovery accuracy with experimental data. Furthermore, the PSDs obtained have no obvious differences to those obtained from measurements at normal particle concentrations.
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•Fitting the measured ACF data with a non-Gaussian ACF model of lower concentration.•Reducing focused beam waist value to obtain non-Gaussian model at lower concentration.•Providing a solution for nanoparticle size measurement at ultra-low concentration.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Abstract
Runaway electrons (REs) in experimental advanced superconducting tokamak (EAST) typically have an energy level of tens of MeV. Synchrotron radiation images of REs contain a wealth of ...information, making it crucial to handle this data properly to uncover hidden features. We observed distinct synchrotron radiation images emitted by the REs at different time points throughout the discharges and the focus of our study was an Ohmic discharge characterized by an evident synchrotron pattern and weaker background radiation. This paper aims to verify the feasibility of utilizing the green function and regularization method to obtain the distribution function information of synchrotron radiation images on EAST. We reconstruct the radial density profile of the REs beam by assuming the monoenergetic nature of REs at different time points. We employ the Tikhonov regularization method for this purpose. The results indicate that the radial density profile of the REs beam follows a roughly Gaussian distribution. Moreover, the peak density of the REs beam shifts inward to the magnetic axis as time progresses, a behavior also observed in the experimental image. However, a good fit between the simulated and experimental images is not achieved for all time points. This discrepancy is likely due to the intervention of
q
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magnetic islands and the increased complexity of the distribution function in both radial and momentum spaces.
•Nonlinear acoustic tomography is combined with particle swarm optimization as a means of reconstructing the two-dimensional thermal vortex field.•The proposed model considers acoustic ray bending to ...improve the accuracy of the co-reconstruction of temperature and flow fields.•The effectiveness of the proposed model for the cooperative reconstruction of low velocity flow field and non-uniform temperature distribution is experimentally verified.
The high-temperature thermal vortex flows generated by the tangential combustion of large-scale coal-fired power plant boilers represent a complex medium that causes acoustic ray bending. In this paper, nonlinear acoustic tomography is combined with particle swarm optimization as a means of reconstructing the two-dimensional thermal vortex field. The nonuniform and large-gradient temperature distribution is the main cause of acoustic ray bending. We establish curved paths of acoustic propagation that are closely related to the temperature distribution. Based on these curved paths, a set of radial basis functions combined with Tikhonov regularization is derived to achieve more accurate reconstruction of the temperature field. The nonuniform temperature field affects the reconstruction of the flow field information. Our model considers a variation in the angle between the direction vector of the acoustic curved paths and the horizontal and vertical components of the flow velocity, and incorporates a priori information and a six-parameter hypersurface into the flow field reconstruction scheme. Particle swarm optimization is applied to minimize the objective function and obtain the vortex velocity field. Numerical experiments show that the proposed mathematical model of acoustic ray bending improves the reconstruction accuracy of the temperature distribution and vortex flow field. The acoustic time-of-flight is experimentally measured through the generalized cross-correlation with different window functions. The temperature and vortex fields reconstructed based on the actual time-of-flight are compared with thermocouple and anemometer measurements to demonstrate the validity and robustness of the proposed reconstruction model.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We investigate the method of nonstationary asymptotical regularization for solving linear ill-posed problems in Hilbert spaces. This method introduces the convex constraints that are proper lower ...semicontinuous and allowed to be non-smooth, therefore can be used for sparsity and discontinuity reconstruction. Under some suitable conditions , the convergence and regularity of the proposed method are established. Under the discretion of Runge–Kutta method, different iteration modes can be deduced for numerical implementation. The numerical results of iteration modes under one-stage explicit Euler, one-stage implicit Euler and two-stage explicit Runge–Kutta are presented to illustrate the efficiency of the proposed method.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Abstract
In this work we analyze the inverse problem of recovering a space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel ...(weighted) conditional stability estimates under very mild conditions on the problem data. Then we provide an error analysis of a standard reconstruction scheme based on the standard output least-squares formulation with Tikhonov regularization (by an
H
1
-seminorm penalty), which is then discretized by the Galerkin finite element method with continuous piecewise linear finite elements in space (and also backward Euler method in time for parabolic problems). We present a detailed error analysis of the discrete scheme, and provide convergence rates in a weighted
L
2
(
Ω
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for discrete approximations with respect to the exact potential. The error bounds explicitly depend on the noise level, regularization parameter and discretization parameter(s). Under suitable conditions, we also derive error estimates in the standard
L
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Ω
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and interior
L
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norms. The analysis employs sharp
a priori
error estimates and nonstandard test functions. Several numerical experiments are given to complement the theoretical analysis.
A classifier that couples nearest-subspace classification with a distance-weighted Tikhonov regularization is proposed for hyperspectral imagery. The resulting nearest-regularized-subspace classifier ...seeks an approximation of each testing sample via a linear combination of training samples within each class. The class label is then derived according to the class which best approximates the test sample. The distance-weighted Tikhonov regularization is then modified by measuring distance within a locality-preserving lower-dimensional subspace. Furthermore, a competitive process among the classes is proposed to simplify parameter tuning. Classification results for several hyperspectral image data sets demonstrate superior performance of the proposed approach when compared to other, more traditional classification techniques.
The Empirical Interpolation Method (EIM), and its generalized version (GEIM), are non-intrusive, reduced-basis model order reduction methods hereby adopted and modified to address the problem of ...optimal placement of sensors and real-time estimation in thermo-hydraulics systems. These techniques have been used to extract the characteristic spatial modes of the system and select a set of points (or functionals) corresponding to the optimal locations for the sensors. Collecting experimental measurements in the available points allows the construction of an empirical interpolation of the fields employed to estimate the variable of interest. However, when these data are affected by noise, the (G)EIM loses its good convergence properties. In this context, stabilization techniques allow good field reconstruction even with noisy data. This work provides an alternative and effective solution to the problem of reconstructing the system state in the presence of experimental data affected by random noise by using the Tikhonov regularization technique. The developed methods have been tested on a simple thermo-fluid dynamics problem known as “two-sided lid-driven differentially heated square cavity”.
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