In this paper, we are concerned with an inverse source problem for the time-fractional diffusion equation with variable coefficients in a general bounded domain. The problem is mildly ill-posed. A ...new fractional Tikhonov method is proposed. We discuss the a-priori regularization parameter choice rule and the a-posteriori regularization parameter choice rule, and prove the corresponding convergence estimates. Numerical experiments are conducted for illustrating effectiveness 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
Aiming at achieving the in-orbit diagnostic of Hall drift current, this study focuses on estimation through the indirect measurement methodology using a magnetic sensor array. It elaborates ...on the application of a pseudo-seminorm defined for the Hall drift current solution to address the inverse magnetostatic problems, which are formulated with a two-dimensional Tikhonov regularization constraint, and thereby offering a systematic approach to select regularization parameters. Our investigation discusses factors influencing the formation of the L-curve and the accuracy of the resultant solution obtained via the L-curve criterion. The results reveal that the formation of the defined pseudo-seminorm of the Hall drift current solution in the semi-logarithmic coordinate system is independent of the number of calibrating current elements or the number of magnetic sensors. This effectively resolves the issue of failing to generate an L-curve during regularization parameter selection. Furthermore, the study indicates that expanding the number of calibrating current elements—essentially increasing the unknown variables in the inverse magnetostatic equations—contributes to a significant enhancement in the accuracy of Hall drift current solutions. It also has extensibility to be applied to other areas where the contactless current measuring is required.
Tikhonov regularization is one of the most popular methods for computing an approximate solution of linear discrete ill-posed problems with error-contaminated data. A regularization parameter λ>0 ...balances the influence of a fidelity term, which measures how well the data are approximated, and of a regularization term, which dampens the propagation of the data error into the computed approximate solution. The value of the regularization parameter is important for the quality of the computed solution: a too large value of λ>0 gives an over-smoothed solution that lacks details that the desired solution may have, while a too small value yields a computed solution that is unnecessarily, and possibly severely, contaminated by propagated error. When a fairly accurate estimate of the norm of the error in the data is known, a suitable value of λ often can be determined with the aid of the discrepancy principle. This paper is concerned with the situation when the discrepancy principle cannot be applied. It then can be quite difficult to determine a suitable value of λ. We consider the situation when the Tikhonov regularization problem is in general form, i.e., when the regularization term is determined by a regularization matrix different from the identity, and describe an extension of the COSE method for determining the regularization parameter λ in this situation. This method has previously been discussed for Tikhonov regularization in standard form, i.e., for the situation when the regularization matrix is the identity. It is well known that Tikhonov regularization in general form, with a suitably chosen regularization matrix, can give a computed solution of higher quality than Tikhonov regularization in standard form.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The projection immersed boundary method calculates boundary forces (i.e., surface stresses) from velocity constraints without introducing additional timestep restrictions. However, the projection ...method generally fails to produce boundary forces that converge to the actual physical surface stresses when the grid is refined. In most cases, the boundary forces show spurious oscillation. This spurious oscillation is particularly violent when the Lagrangian-to-Eulerian grid spacing ratio is small. As a result, the traditional projection method may encounter problems in fluid-structural interaction (FSI) simulations where accurate boundary forces are needed. To address these challenges, we propose a regularized projection method using (generalized) Tikhonov regularization techniques. The regularized projection method yields smoother boundary force solutions that closely approximate the actual boundary force distributions for a wider range of Lagrangian-to-Eulerian grid spacing ratios. The boundary force shows first order convergence with properly chosen parameters. We demonstrate the effectiveness and accuracy of our approach through various test cases.
•Proposed a Tikhonov regularized projection immersed boundary method.•Accurately computed body forces, eliminating spurious oscillations.•Demonstrated convergence of body forces with test problems.•Validated applicability to FSI problems with Turek-Hron problems.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
•Revealed the mechanism and influence of the shape factor on DRT deconvolution;•Evaluated the optimisation methods of the shape factor by the synthetic EIS data;•Discussed the coupling effect of the ...shape and penalty factors on DRT deconvolution;•Proposed a collaborative optimisation strategy to improve the accuracy of DRT result;
Electrochemical impedance spectroscopy (EIS) is a powerful diagnosis tool for the performance of electrochemical energy storage and conversion devices. One challenge for EIS-based diagnosis is the difficulty in separating highly overlapped impedance spectra. To overcoming this challenge, the distribution of relaxation times (DRT) method based on Tikhonov regularization algorithm can be employed. However, the accuracy and stability of the DRT deconvolution method not only depend on the penalty factor (the actual regularization tool), but also on the rarely studied shape factor (a potential regularization tool). In this study, a comprehensive investigation on the effect of the shape factor on both the accuracy and the stability of the DRT deconvolution method was conducted. First, the influence of the shape factor on the DRT deconvolution method was theoretically derived and numerically simulated. Second, the optimization methods for the shape factor and the selection of regularization matrices for DRT deconvolution were quantitatively evaluated using synthetic impedance data. Third, the coupling effect of the shape and penalty factors on the DRT deconvolution method was analyzed quantitatively. Finally, the collaborative optimization strategy was proposed and evaluated. The method presented in this paper can be used to improve the accuracy and reliability of the DRT to decode highly overlapped impedance spectra.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
A reconstruction of an unknown source function is considered for hyperbolic partial differential equations with interior degeneracy. We identify the spatial element of the source term of a degenerate ...wave equation using the final observation data. The existence and uniqueness of the direct problem with interior degeneracy within the spatial domain are stated and proved. The inverse problem can be formulated as a nonlinear optimization problem and the unknown source term can be characterized as the solution to a minimization problem. The Tikhonov regularization technique is employed to accomplish the inclusion of noise in the input data, based on the insertion of the regularization term into the cost functional. The conjugate gradient algorithm in conjunction with Morozov's discrepancy principle as a stopping criterion is then utilized to develop an iterative reconstruction procedure. Finally, some numerical simulation results are provided to show the performance of the proposed scheme in one and two dimensions.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Extreme learning machine (ELM) is a single-hidden-layer feed-forward neural network in which the input weights linking the input layer to the hidden layer are randomly chosen. The output weights ...which link the hidden layer to the output layer are analytically determined by solving a linear system of equations and hence is one of the fastest learning algorithms. The Moore–Penrose (MP) generalized inverse is normally employed to obtain the output weights of the neural network. Although the random weight parameters between input and hidden layers are need not be tuned, ELM provides good generalization performance with fast learning speed. In general, the data sets from real-world problems tend to make the linear system of ELM ill-conditioned due to the presence of inconsistent noise levels in the input data which leads to unreliable solutions and over-fitting problems. The regularization techniques are developed to address such issues in ELM and it involves estimation of additional variables termed as a regularization parameter. In this context, the proper selection of the regularization parameter is a crucial task as it is going to decide the quality of the solution obtained from the linear system. Further, the popular choice is the Tikhonov regularization technique which penalizes the ℓ2-norm of the model parameters. In ELM, such inclusion results are giving equal weight to singular values of the matrix irrespective of the noise level present in the data. In the presented work, a fractional framework is introduced in the Tikhonov regularized ELM to weigh the singular values with respect to a fractional parameter to reduce the effect of different noise levels. Moreover, an automated golden-section method is applied to choose the optimal fractional parameter. Finally, the generalized cross-validation method is applied for obtaining the suitable value of the regularization parameter. The proposed strategy of applying fractional Tikhonov regularization to ELM results in improvement of performance when compared with the conventional methods with respect to the performance measures. Finally, the results obtained from the proposed fractional regularization is also shown to be statistically significant.
•A fractional framework is introduced in the Tikhonov regularized ELM to weigh the singular values with respect to a fractional parameter to reduce the effect of different noise levels.•The proposed strategy of applying fractional Tikhonov regularization to ELM results in improvement of performance when compared with the conventional methods with respect to all the performance measures.•The results obtained from the proposed fractional regularization is also shown to be statistically significant.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
This paper is devoted to solve the backward problem for a time-fractional diffusion-wave equation in a bounded domain. Based on the series expression of the solution for the direct problem, the ...backward problem for searching the initial data is converted into solving the Fredholm integral equation of the first kind. The existence, uniqueness and conditional stability for the backward problem are investigated. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of the regularized solution for the backward problem. Furthermore, the convergence rate for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical results for five examples in one-dimensional case and two-dimensional case show that the proposed method is efficient and stable.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
A learning approach for selecting regularization parameters in multi-penalty Tikhonov regularization is investigated. It leads to a bilevel optimization problem, where the lower level problem is a ...Tikhonov regularized problem parameterized in the regularization parameters. Conditions which ensure the existence of solutions to the bilevel optimization problem are derived, and these conditions are verified for two relevant examples. Difficulties arising from the possible lack of convexity of the lower level problems are discussed. Optimality conditions are given provided that a reasonable constraint qualification holds. Finally, results from numerical experiments used to test the developed theory are presented.
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