Tikhonov Regularization Technique Research Articles

Investigation of a new method for direct chemistry integration in Conditional Source-term Estimation

This paper examines a new Conditional Source-term Estimation (CSE) implementation in which an integral inversion is performed for species mass fractions without any chemistry tabulation. To tackle the numerical stiffness due to chemistry, a constant-pressure chemical reactor is considered in conditional space. The objective of this work is to investigate this new method in CSE by comparing its numerical performance (accuracy of predictions and computational efficiency) with previous CSE implementations using tabulated chemistry. For the first time, the constraints for mass and mixture fraction conservation in conditional space are included in the inversion process. The investigation is performed in a Reynolds Averaged Navier-Stokes (RANS) framework using a well-documented turbulent flame with detailed measurements in conditional and physical spaces for many reacting scalars. A chemical mechanism including 16 species and 35 reactions is incorporated. CSE predictions of conditional and Favre averaged temperature and different species mass fractions are analysed. CSE with full chemistry performs as well as CSE with tabulated chemistry, if not slightly better. Numerical errors are discussed. The new CSE implementation is made possible by the proposed numerical treatment of stiffness due to chemistry and first-order Tikhonov regularisation technique with additional physical constraints. The new approach is found to be more CPU intensive than CSE with a pre-tabulation technique, but as a preliminary analysis, the computational cost of CSE with direct integration of a chemical mechanism appears to be much smaller than the run time associated with CMC for a similar set-up. Further investigation is required to refine this initial conclusion. These results open new opportunities towards CSE simulations that involve fuel blends and more complex operating conditions for example with spray and multi-stream inlets with non-uniform compositions for which accurate chemistry tabulations are difficult to generate.

A Novel Mayfly Algorithm with Response Surface for Static Damage Identification Based on Multiple Indicators

This paper proposes a novel structural damage identification approach coupling the Mayfly algorithm (MA) with static displacement-based response surface (RS). Firstly, a hybrid optimal objective function is established that simultaneously considers the sensitivity-based residual errors of static damage identification equation and the static displacement residual. In the objective function, the static damage identification equation is addressed by the Tikhonov regularization technique. The MA is subsequently employed to conduct an optimal search and pinpoint the location and intensity of damages at the structural element level. To handle the inconformity of the static loading points and the measurement points of displacements, the model reduction and displacement extension techniques are implemented to reconstruct the static damage identification equation. Meanwhile, the static displacement-based RS is constructed to calculate the displacement residual in the hybrid objective function, thereby circumventing the time-consuming finite element calculations and improving computational efficiency. The identification results of the numerical box girder bridge demonstrate that the proposed method outperforms the particle swarm optimization, differential evolution, Jaya and whale optimization algorithms about both convergence rate in optimal searching and identification accuracy. The proposed method enables more accurate damage identification compared to methods solely based on the indicator of the residual of static damage identification equations or displacement residual. The results of identifying damage in the 21 element-truss structure and the static experiments on identifying damage in an aluminum alloy cantilever beam confirm the high efficiency of the proposed approach.

Determination of Timewise-Source Coefficient in Time-Fractional Reaction-Diffusion Equation from First Order Heat Moment

This article aims to determine the time-dependent heat coefficient together with the temperature solution for a type of semi-linear time-fractional inverse source problem by applying a method based on the finite difference scheme and Tikhonov regularization. An unconditionally stable implicit finite difference scheme is used as a direct (forward) solver. While by the MATLAB routine lsqnonlin from the optimization toolbox, the inverse problem is reformulated as nonlinear least square minimization and solved efficiently. Since the problem is generally incorrect or ill-posed that means any error inclusion in the input data will produce a large error in the output data. Therefore, the Tikhonov regularization technique is applied to obtain stable and accurate results. Finally, to demonstrate the accuracy and effectiveness of our scheme, two benchmark test problems have been considered, and its good working with different noise levels.

Application of Regularization Methods in the Sky Map Reconstruction of the Tianlai Cylinder Pathfinder Array

The Tianlai cylinder pathfinder is a radio interferometer array to test 21 cm intensity mapping techniques in the post-reionization era. It works in passive drift scan mode to survey the sky visible in the northern hemisphere. To deal with the large instantaneous field of view and the spherical sky, we decompose the drift scan data into m-modes, which are linearly related to the sky intensity. The sky map is reconstructed by solving the linear interferometer equations. Due to incomplete uv coverage of the interferometer baselines, this inverse problem is usually ill-posed, and regularization method is needed for its solution. In this paper, we use simulation to investigate two frequently used regularization methods, the Truncated Singular Value Decomposition (TSVD), and the Tikhonov regularization techniques. Choosing the regularization parameter is very important for its application. We employ the generalized cross validation method and the L-curve method to determine the optimal value. We compare the resulting maps obtained with the different regularization methods, and for the different parameters derived using the different criteria. While both methods can yield good maps for a range of regularization parameters, in the Tikhonov method the suppression of noisy modes are more gradually applied, produce more smooth maps which avoids some visual artefacts in the maps generated with the TSVD method.

Simultaneous identification of the solely time‐dependent potential and source control terms from known moisture moments

Pseudo‐parabolic equations are commonly used as mathematical models in mechanics, biology, and physics to address various applied problems. One particular application involves describing moisture transfer dynamics in subsoil layers using pseudo‐parabolic equations. This manuscript examines the inverse problem (IP) of identifying the moisture transfer function, along with the time‐varying potential and source control terms, in a linear pseudo‐parabolic equation from known moisture moments. We prove the existence of a unique solution to the inverse problem for sufficiently small times by employing the contraction principle. The inverse problem is reformulated as a nonlinear least‐squares minimization problem, with the unknowns subjected to simple bounds. To guarantee stability, the Tikhonov regularization technique is utilized. For the numerical discretization, we develop the Crank–Nicolson finite‐difference method to solve the direct problem. To solve the nonlinear least‐squares minimization problem, we utilize the built‐in subroutine lsqnonlin from the MATLAB optimization toolbox. We present and thoroughly discuss numerical outcomes for a benchmark test example, providing insights into the performance and effectiveness of the proposed methodology.

Output-only structural damage identification based on Q-learning hybrid evolutionary algorithm and response reconstruction technique

Various structural damage identification methods have been developed and employed, while the absence of input excitation measurements may pose a huge challenge in their application since the input forces such as seismic load, traffic load, wind load, are not directly measurable. To address this issue, in the present paper, a novel output-only structural damage detection approach based on Q-learning hybrid evolutionary algorithm (QHEA) and response reconstruction technique is presented. On the one hand, the external excitation and structural acceleration responses are reconstructed with the aid of response reconstruction and Tikhonov regularization techniques in time domain. Structural damage identification can be formulated as an optimization-based inverse problem. On the other hand, a new optimization framework QHEA integrating Jaya algorithm, differential evolution, Q-learning algorithm is developed as search tool. The unknown structural parameters and unmeasured input force are iteratively updated by using two different measurement sets until the reconstructed acceleration responses agree well with the measured responses. Numerical examples involving a cantilever beam structure under single excitation and a simply-supported 51-bar truss structure under multiple excitations, as well as an experimental five-floor steel frame structure in the laboratory are carried out to validate the effectiveness of the proposed method. The final results demonstrate that the proposed method can accurately detect damage locations and quantify damage extents without the information of input excitation. In addition, the superiority of QHEA over other heuristic algorithms, the uncertainties of measurement noise and modeling errors on damage identification results are further examined.

A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique

In this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the Moreau envelope of the objective function and its properties as well as Tikhonov regularization properties, which we extend to a nonsmooth case. We introduce the setting, which at the same time guarantees the fast convergence of the function (and Moreau envelope) values and strong convergence of the trajectories of the system to a minimal norm solution—the element of the minimal norm of all the minimizers of the objective. Moreover, we deduce the precise rates of convergence of the values for the particular choice of parameters. Various numerical examples are also included as an illustration of the theoretical results.

A regularized projection immersed boundary method for smooth boundary forces

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.

Estimation of soils thermophysical characteristics in a nonlinear inverse heat transfer problem

It is well known that knowledge of thermophysical parameters is a leading strategy to research effects of energy transfer in soils. The present article proposes an inverse analysis for numerical solving of nonlinear heat transfer problem to determine the thermophysical properties of two different soil types: sand and chernozem. First, estimation of thermophysical parameters is performed using temperature data from experimental set-up, which is two-chambered container for two soil types. Second, numerical algorithm is based on implicit Euler scheme for discretization, Newton method to solve nonlinear system of equations and Levenberg-Marquardt method to minimize nonlinear estimator with Tikhonov's regularization technique. Simulations have been efficiently carried out for two different soil types, showing that the reliability of the model is satisfying with a discrepancy between numerical predictions and experimental observations remaining within the measurement error.

Reconstruction of a time-dependent wave source and the initial condition in a hyperbolic equation

This article is concerned with determination of a time-dependent wave source together with the initial condition in a hyperbolic equation. Based on the given data including the flux tensions of a string at end points and a nonlocal integral observation along with the wave displacement at two distinct instants of time, we prove the unique solvability of the inverse problem. Moreover, we estimate the modulus of continuity of the inverse problem under special conditions. The numerical solution of the problem is also presented by means of Bernstein spectral method, which converts the problem to a linear system of algebraic equations and then Tikhonov regularization technique is employed to derive reliable solutions. Regarding the perturbed boundary conditions, we use a regularized numerical technique to obtain stable derivatives. Numerical simulations are provided to show the effectiveness of the proposed scheme.

An inversion for asymmetric hydraulic fracture growth and fracture opening distribution from tilt measurements

The eigenstrain tensor is defined in terms of the unit vector normal to the fracture plane and the dislocation vector to characterize the fracture geometry and to determine the fracture induced deformation field. Compared to the conventional nonlinear fracture model, the forward fracture model provides a linear relationship between fracture geometry parameters (eigenstrain) and the induced deformation. Therefore, a linear inverse problem needs to be tackled in mapping the fracture geometry by inversion of tilt observations. The fracture orientation and opening distribution of a tensile-dominated hydraulic fracture can be mapped iteratively in a coupled way by using the forward model defined in terms of the dislocation vector. The proposed analysis method is applied to field tilt data collected for monitoring the growth of a hydraulic fracture placed in the preconditioning of a roof rock over a coal longwall panel at a test site. Inversion of the tilt data produces a low fracture dip angle of 6.8°, which agrees with a nearly horizontal fracture orientation confirmed by intersection data collected in offset monitoring boreholes. Tikhonov regularization technique is applied to stabilize the discrete ill-posed inverse problem. The regularized solution provides information on the fracture opening distribution and asymmetric fracture growth. The inversion shows that the fracture grows uniformly and roughly in a radial pattern at early injection times, but grows asymmetrically, preferentially, and consistently to the northwest during later times. The predicted fracture growth agrees with the temperature logging to measure fracture intersections with the offset monitoring boreholes.

Automated Regularized Deconvolution for Eliminating Extra-Column Effects in Fast High-Efficiency Separations.

With the introduction of ultrahigh efficiency columns and fast separations, the need to eliminate peak deformation contributed by the instrument must be effectively solved. Herein, we develop a robust framework to automate deconvolution and minimize its artifacts, such as negative dips, wild noise oscillations, and ringing, by combining regularized deconvolution and Perona-Malik (PM) anisotropic diffusion methods. A asymmetric generalized normal (AGN) function is proposed to model the instrumental response for the first time. With no-column data at various flow rates, the interior point optimization algorithm extracts the parameters describing instrumental distortion. The column-only chromatogram was reconstructed using the Tikhonov regularization technique with minimal instrumental distortion. For illustration, four different chromatography systems are used in fast chiral and achiral separations with 2.1 and 4.6 mm i.d. columns. Ordinary HPLC data can approach highly optimized UHPLC data. Similarly, in fast HPLC-circular dichroism (CD) detection, 8000 plates were gained for a fast chiral separation. Moment analysis of deconvolved peaks confirms correction of the center of mass, variance, skew, and kurtosis. This approach can be easily integrated and used with virtually any separation and detection system to provide enhanced analytical data.

Identification of an unknown spatial source function in a multidimensional hyperbolic partial differential equation with interior degeneracy

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.

Microwave Spectrum Detection at Microscopic Scale Based on Nitrogen Vacancy Center in Diamond

A machine learning‐based microwave spectrum detection method based on the nitrogen vacancy (NV) color centers in diamonds is proposed. The functional relationship between the fluorescence spectrum and standard microwave spectrum is established. The response matrix is calculated using the Tikhonov regularization technique, and an unknown microwave spectrum is reconstructed. Diamond particles with a size of only 5 × 5 μm2 are placed in the microfluidic structures. Consequently, the frequency detection range of the microwave spectrum is from 2.892 to 6.214 GHz with a resolution of 22 kHz. The proposed research opens new paths for microwave spectrum detection and imaging at the microscopic scale.

Numerical Solution of Backward Problem of a Multi-term Time-space Fractional Diffusion Equation

There have been many studies from a variety of perspectives about the fractional diffusion equation for the backward problem. This work looks at the backward problem in a multi-term time-space fractional diffusion equation (TSFDE), which uses the observational final data. This paper proposes a numerical solution based on a Tikhonov regularization technique combined with an optimal perturbation algorithm (OPA) to take care of this poorly posed problem. An implicit finite difference technology is introduced by the matrix transfer technique as a numerical approach to solving the problem of the initial boundary value. It is both highly effective and relatively stable, as our suggested numerical method’s numerical results illustrate.

Distributed Dynamic Load Identification of Beam Structures Using a Bayesian Method

The distributed dynamic load is difficult to obtain due to the complexity of loads in practical engineering, such as the aerodynamic loads of aircraft and the distributed dynamic loads of sea-crossing bridges. Thus, distributed dynamic load identification is important to deal with these difficulties, which is generally an ill-posed problem considering the inversion of the infinite dynamic loads. The traditional Tikhonov regularization technique is limited on the optimal regularization parameters selection. Consequently, in this paper, we develop a novel distributed dynamic load identification algorithm in combination with the orthogonal polynomials and the Bayesian framework. Thus, the orthogonal polynomial coefficients in the load identification model are regarded as the prior probability distribution of unknown variables in the Bayesian inference. Simultaneously, the posterior probability distribution of the orthogonal polynomial coefficients is derived based on the Bayesian formula and the likelihood function. The regularization parameters and the standard deviation of the response error are also treated as random variables to obtain the corresponding prior distribution in the multi-level Bayesian model. Moreover, the maximum posterior estimate is applied aiming at determining the regularization parameters, as well as the orthogonal polynomial coefficients to reconstruct the distributed dynamic loads. Compared with the Tikhonov regularization, a series of numerical simulations are studied to verify the effectiveness and high accuracy, as well as the noise resistance, and the results illustrate that this approach is effective to reconstruct the distributed dynamic loads.

The price of Anarchy as a classifier for mechanism design in a Pareto-Bayesian-Nash context

The complex interaction of fixed incentives for multiple agents can lead to inefficient results. The Price of Anarchy (PoA) is a concept employed in economics and game theory to quantify how, in the worst-case scenario, a system's efficiency diminishes due to the selfish behavior of its actors. It evaluates the outcomes obtained by perfect coordination to that obtained by self-interested players in a Nash equilibrium.This paper investigates the PoA in mechanism design for Pareto-Bayesian-Markov (PBN) games in which the players privately know their information. Our approach is restricted to the development of algorithmic mechanisms. We consider an ergodic finite Markov chain in which players' transitions are independent and decisions are made across a finite number of periods. As a key outcome, we completely characterize the Pareto frontier and calculate an efficient, ex-post incentive-compatible technique. We employ Tikhonov's regularization technique, for assuring that the original problem's solution converges to a unique solution. The ratio equilibria of the PoA is found employing a projection-gradient method, which is our main result. We compute the PoA using an auxiliary variable and present the formulas for retrieving the variables of interest. We show that our theoretical conclusions ensure convergence and we provide a numerical example to demonstrate the method's utility.

Localization of magnetocardiographic sources for myocardial infarction cases using deterministic and Bayesian approaches

In this paper, the inverse problems of cardiac sources using analytical and probabilistic methods are solved and discussed. The standard Tikhonov regularization technique is solved initially to estimate the under-determined heart surface potentials from Magnetocardiographic (MCG) signals. The results of the deterministic method subjected to noise in the measurements are discussed and compared with the probabilistic models. Hierarchical Bayesian modeling with fixed Gaussian prior is employed to quantify the uncertainties in source reconstructions. A novel application of Variational Bayesian inference approach has been presented to estimate the heart sources. The reconstruction results of Variational Bayesian model with non-stationary priors are compared with solutions of simplistic Bayesian approach; and the performances are evaluated using Root Mean Square Error (RMSE) and correlation co-efficient metrics. The Bayesian solutions in the study are also extended to localize the MCG sources for two types of Myocardial infarction cases.

Simultaneous determination of the time‐dependent potential and force terms in a fourth‐order Rayleigh–Love equation

This paper considers an inverse coefficient problem of simultaneous determination of solely time‐dependent potential and force terms with the unknown longitudinal displacement from a Rayleigh–Love equation from two integral overdetermination measurements. Unique solvability of this theorem is investigated theoretically by using contraction principle. Although, the aforesaid inverse identification problem is ill‐posed but has a unique solution. We use the cubic B‐spline (CBS) collocation and Tikhonov regularization techniques to discretize the direct problem and to obtain stable as well as accurate solutions, respectively. The stability, for the discretized system of the direct problem, is also carried out by means of the von Neumann method. The acquired results demonstrate that accurate as well as stable solutions for the and are accessed for , when for both smooth and discontinuous potential and force terms. The stability analysis shows that the discretized system of the direct problem is unconditionally stable. Since the noisy data are introduced, the investigation and analysis model real circumstances where the practical quantities are naturally infested with noise. The technique establishes that accurate, as well as stable, solutions are obtained.

Stabilization of Generalized Empirical Interpolation Method (GEIM) in presence of noise: A novel approach based on Tikhonov regularization

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”.