Tikhonov Regularization

The projected Tikhonov regularization method developed and used in this investiga- tion to solve the Fredholm integral equations of the first kind is very simple and effective, owing to the fact that the dimension of the subspace of projection is very small (n = , ); moreover, the regularized solution remains stable for a strong noise (ε = /) and for regular data.

Recently, there has been a great deal of work developing super-resolution reconstruction (SRR) algorithms. While many such algorithms have been proposed, the almost SRR estimations are based on L1 or L2 statistical norm estimation, therefore these SRR algorithms are usually very sensitive to their assumed noise model that limits their utility. The real noise models that corrupt the measure sequence are unknown; consequently, SRR algorithm using L1 or L2 norm may degrade the image sequence rather than enhance it. Therefore, the robust norm applicable to several noise and data models is desired in SRR algorithms. This pa- per first comprehensively reviews the SRR algorithms in this last decade and addresses their shortcomings, and latter proposes a novel robust SRR algorithm that can be applied on several noise models. The proposed SRR algorithm is based on the stochas- tic regularization technique of Bayesian MAP estimation by minimizing a cost function. For removing outliers in the data, the Lorentzian error norm is used for measuring the difference between the projected estimate of the high-resolution image and each low-resolution image. Moreover, Tikhonov regularization and Lorentzian-Tikhonov regularization are used to remove artifacts from the final answer and improve the rate of convergence. The experimental results confirm the effectiveness of our method and demonstrate its superiority to other super-resolution methods based on L1 and L2 norms for several noise models such as noise- less, additive white Gaussian noise (AWGN), poisson noise, salt and pepper noise, and speckle noise.

A trace gas column retrieval represents a typical inversion problem of atmospheric remote sensing with limited verti- cal sensitivity. For measurements with a sensitivity limited to the total column abundance of a trace gas, an unregularized profile retrieval would infer a vertical distribution which is dominated by the contribution of measurement noise. Thus, the profile inversion problem is ill-posed and requires reg- ularization to obtain a stable solution. In practice, there are two common ways to regularize the least-squares solution. First, a vertical trace gas profile is retrieved using a Tikhonov regularization approach or optimal estimation to stabilize the inversion. Due to the regularization, the retrieved profile es- timates a smoothed version of the true profile, where the smoothing is described by the so-called profile averaging ker- nel. Subsequently, the retrieved profile and the profile averag- ing kernel are vertically integrated. By that, an analytical ex- pression is given for the so-called column averaging kernel. It describes the sensitivity of the retrieved column with respect to changes of the true vertical trace gas distribution as a func- tion of altitude and is defined by corresponding derivatives. In an ideal case, the column averaging kernel represents a geometrical integration of the true trace gas profile, whereas in practice it describes a vertically weighted height integra- tion. Here, the weights of the integration are mainly due to atmospheric scattering and the temperature dependence of atmospheric absorption. Hence, for the further usage of the retrieved trace gas columns (e.g. for assimilation or valida- tion) it is essential to account for the total column averaging kernel to prevent misinterpretations. Despite its theoretical advantages, only a few retrieval algorithms derive trace gas columns and the according total column averaging kernels via a profile retrieval.

In this paper, we consider the problem for determining an unknown source in the heat equation. The Tikhonov regularization method in Hilbert scales is presented to deal with ill-posedness of the problem and error estimates are obtained with a posteriori choice rule to find the regularization parameter. The smoothness parameter and the a priori bound of exact solution are not needed for the choice rule. Numerical tests show that the proposed method is effective and stable.

The new procedure is applied to two different sets of problems in this paper: one based on synthetic data and the other on experimental data. For the first set, we assume that data collection is done by a set of receivers which are located on a circle around the object and that the object is illuminated by Transverse Magnetic (TM) plane- waves impinging on the object from different angles of incidence. The geometrical configuration is the same as that described in [11]. In the second set, we use measurement data collected by researchers at the Institut Fresnel for two different targets, namely FoamDielIntTM and FoamDielExtTM [33, 35]. Here, we use single frequency data, at 2 GHz, for reconstructing the contrast profiles of both synthetic and experimental data. We only show results for 2 GHz because the higher frequency data, provided by Institut Fresnel, is very difficult to invert using the BIM. For other inversion techniques, such as the Distorted Born Iterative Method (DBIM) [12], the NCP parameter- choice method for Tikhonov regularization which is proposed in this paper, is also applicable to the case of higher frequency and multi- frequency inversion. For the case of multi-frequency inversion the frequency hopping method can be used [34]. A review of alternative more robust inversion techniques, that have been used on the Fresnel data, such as the Multiplicative Regularized Contrast Source Inversion (MR-CSI) and the modified gradient methods, is available in [35].

l − . As for the measured data function g δ is only in L 2 ( ) 0, π , we cannot expect that it possess such a decay property. So some special regularization methods are required. In the following, we apply the Tikhonov regularization method in Hilbert scales to reconstruct a new function h δ from the perturbed data g δ and Th δ will be used as an approximation of f. It is well known that for any ill-posed problems an a priori bound assumption for the exact solution is needed and necessary. In this paper, we assume the following a priori bound holds:

In this work, we give another way for approaching the ill-posedness of an inverse source problem. We deliver a Tikhonov regularization method to consider the above Gaussian random model. The right-hand side is a function represented in the form of variable sep- aration. To determine the source term f (x), we require the following assumptions: The functions (g, F) are approximated by the noisy observation data (g ε , F ε ) such that

Tikhonov regularization can be used for both classification and regression tasks, but we refer to the function f as the regularized solution in all cases. We call the left-hand portion the regularization term, and the right-hand portion the loss term. We assume a loss function v(y, y) ˆ that is convex in its first argument and minimized at y = y (thereby ruling out, for example, the 0/1 “misclassification ˆ rate”). We call such a loss function valid. Aside from convexity, we will be unconcerned with the form of the loss function and often take the loss term in the optimization in (1) to be some convex function V : R n → R which is minimized by the vector ˆ y of ˆ y i ’s.

chine learning field. Classically, regularization [24] seeks to minimize the weighted sum of a risk functional and some measure of discriminant complexity although how to decide on the weighting (the so-called regularization constant) between the two terms usually involves cross-validation [4]. The parsimony principle [18] much- used in GP is an example of regularization. Minimum description length (MDL) ap- proaches [20] can also be viewed as regularization. Iba et al. [13] attempted to apply MDL to GP but failed to account for the not-necessarily-minimal form of the trees – logically, MDL can only be applied to trees which have been simplified to truly minimal algebraic form which, we suspect, is an NP-complete task. In Bayesian ap- proaches, the log prior can be interpreted as a regularization term [25].

The proposed method is essentially to a hybrid splitting method since it combines the parallel splitting method and the alternating direction method, which are two power tools for the co[r]

We consider an inverse heat conduction problem with convection term which appears in some applied subjects. This problem is ill posed in the sense that the solution (if it exists) does not depend continuously on the data. A generalized Tikhonov regularization method for this problem is given, which realizes the best possible accuracy.

To the best of the knowledge of the authors, the results available in the literature are mainly devoted to the IHCP with known initial-boundary value. However, in practical real-life problems we cannot know the initial condition because the heat process has al- ready started before we estimate the problem. A few works are developed for the IHCP without initial value [, ]. Ginsberg [] used a cutoff method for an IHCP with only boundary value and gave a Hölder type error estimate. Recently, Liu and Wei [] used a quasi-reversibility regularization method for solving an IHCP without initial data. Yang and Fu [] applied a simplified Tikhonov regularization method for determining the heat source. In this paper, we will use a modified Tikhonov regularization method to deal with the IHCP without initial value (.) and obtain an order optimal error estimate between the approximate solution and the exact solution.

If T is a positive and selfadjoint operator on a Hilbert space, then the sim- plified regularization introduced by Lavrentiev is better suited than Tikhonov regularization in terms of speed of convergence and condition number in the case of finite-dimensional approximations (cf. Schock [11]).

Mathematically, inverse problems belong to the class of ill-posed problems. The matrix A is singular and ill-posed, thus the estimate of C 0 by (4.1) will be unstable so that the Tikhonov regularization method must be used to control this singularity. In our computations, we adapt the Tikhonov regularization method to solve the matrix system of equations (3.5) and (4.1). The Tikhonov regularized solutions to the systems of linear algebraic equations (3.5) and (4.1) are given by

Background: Local field potentials (LFPs) evoked by sensory stimulation are particularly useful in electrophysiological research. For instance, spike timing and current transmembrane current flow estimated from LFPs recorded in the barrel cortex in rats and mice are exploited to investigate how the brain represents sensory stimuli. Recent improvements in microelectrodes technology enable neuroscientists to acquire a great amount of LFPs during the same experimental session, calling for algorithms for their quantitative automatic analysis. Several computer tools were proposed for LFP analysis, but many of them incorporate algorithms that are not open to inspection or modification/personalization. We present a MATLAB software to automatically detect some important LFP features (latency, amplitude, time-derivative value in the inflection-point) for a quantitative analysis. The software features can be customized by the user according to his/her personal research needs. The incorporated algorithm is based on Phillips-Tikhonov regularization to deal with noise amplification due to ill-conditioning. In particular, its accuracy in the estimation of the features of interest is assessed in a Monte Carlo simulation mimicking the acquisition of LFPs in different SNR (signal-to-noise-ratio) conditions. Then, the algorithm is tested by analyzing a real set of 2500 LFPs recorded in rat after whisker stimulation at different depths in the primary somatosensory (S1) cortex, i.e., the region involved in the cortical representation of touch in mammals.

The multilevel iteration method have been developed to solve the integral equations since 1990s. Chen, Xu, and Yang [16-17] were the first scholars to utilize the multilevel iteration method to solve Fredholm integral equations of the first kind. By combining the Tikhonov regularization and the Multilevel Galerkin method, they truncated the dense matrix to be a sparse matrix so that a fast algorithm can be generated to solve Fredholm equations of the first kind. Fan-chun Li et al. [18-20] adopted various multilevel iteration methods to solve Fredholm integral equations of the first kind with disturbed initial data, proposed a fast algorithm with selection of regularized parameters. The complexity and convergence rate of the fast algorithm was analyzed and the approximation solution was proven to be optimized.

The LBP algorithm has been extensively applied in ECT image reconstruction because of its high reconstruction speed but it introduces large error in image reconstruction. The standard Tikhonov regularization method is one of the best method to solve the ill-posed problem, but in ECT it tends to generate a smooth approximation solution, in that case it can lead the lost detailed information which in turn result low image spatial resolution.

application of the theory of reproducing kernels to the Tikhonov regularization, which gives the approximate solutions for bounded linear operator equation on the Fock spaces F q,αc. Esp[r]

The Fourier method was used in [–]. The mollification method and projection regular- ization based on the Laplace and Fourier transforms are applied respectively in [] and []. For axisymmetric problems, we should mention recent articles. In [, ], the authors consider an axisymmetric IHCP of determining the surface temperature from a fixed lo- cation inside a cylinder. In [, ], the authors investigated the case of identifying a source from the final data. Xiong [] studied the problem of identifying a boundary condition by the method of quasi-reversibility. A modified Tikhonov regularization method was ap- plied for an axisymmetric backward heat equation in []. Lesnic et al. [] applied the method of fundamental solutions (MFS) (with a Tikhonov regularization) to the radially symmetric inverse heat conduction problem (IHCP) analogous to our problem. Inverse problems for fractional diffusion equations are studied by many authors; for example, we mention the recent article [].

The Matrix Λ is ill-conditioned. On the other hand, as ϕ(y, t) is affected by measurement errors, the estimate of Θ by (1.23) will be unstable so that the Tikhonov regularization method must be used to control this measurement errors. The Tikhonov regularized solution [13, 19, 30, 31] to the system of linear algebraic equation (1.23) is given by