• Structural Monitoring and Maintenance
• /
• v.3 no.3
• /
• pp.195-214
• /
• 2016

Monitoring of impact loads is a very important technique in the field of structural health monitoring (SHM). However, in most cases it is not possible to measure impact events directly, so they need to be reconstructed. Impact load reconstruction refers to the problem of estimating an input to a dynamic system when the system output and the impulse response function are usually known. Generally this leads to a so called ill-posed inverse problem. It is reasonable to use prior knowledge of the force in order to develop more suitable reconstruction strategies and to increase accuracy. An impact event is characterized by a short time duration and a spatial concentration. Moreover the force time history of an impact has a specific shape, which also can be taken into account. In this contribution these properties of the external force are employed to create a sample-force-dictionary and thus to transform the ill-posed problem into a sparse recovery task. The sparse solution is acquired by solving a minimization problem known as basis pursuit denoising (BPDN). The reconstruction approach shown here is capable to estimate simultaneously the magnitude of the impact and the impact location, with a minimum number of accelerometers. The possibility of reconstructing the impact based on a noisy output signal is first demonstrated with simulated measurements of a simple beam structure. Then an experimental investigation of a real beam is performed.

• Journal of Electrical Engineering and Technology
• /
• v.4 no.3
• /
• pp.365-369
• /
• 2009

Identification is considered to be among the main applications of inverse theory and its objective for a given physical system is to use data which is easily observable, to infer some of the geometric parameters which are not directly observable. In this paper, a parameter identification method using inverse problem methodology is proposed. The minimisation of the objective function with respect to the desired vector of design parameters is the most important procedure in solving the inverse problem. The conjugate gradient method is used to determine the unknown parameters, and Tikhonov's regularization method is then used to replace the original ill-posed problem with a well-posed problem. The simulation and experimental results are presented and compared.

• Journal of the Korean Mathematical Society
• /
• v.44 no.6
• /
• pp.1397-1415
• /
• 2007

We consider an inverse heat conduction problem(IHCP) in a quarter plane which appears in some applied subjects. We want to determine the heat flux on the surface of a body from a measured temperature history at a fixed location inside the body. This is a severely ill-posed problem in the sense that arbitrarily "small" differences in the input temperature data may lead to arbitrarily "large" differences in the surface flux. A semi-discrete central difference scheme in time is employed to deal with the ill posed problem. We obtain some error estimates which also give the information about how to choose the step length in time. Some numerical examples illustrate the effects of the proposed method.

• Journal of the Institute of Electronics Engineers of Korea SP
• /
• v.46 no.5
• /
• pp.15-24
• /
• 2009

Compressed sensing addresses the recovery of a sparse vector from its few linear measurements. Recently, the success for the vector case has been extended to the matrix case. Compressed sensing of low-rank matrices solves the ill-posed inverse problem with fie low-rank prior. The problem can be formulated as either the rank minimization or the low-rank approximation. In this paper, we survey recently proposed efficient algorithms to solve these two formulations.

• Communications of the Korean Mathematical Society
• /
• v.16 no.3
• /
• pp.487-508
• /
• 2001

We consider the Cauchy problem for Laplacian. Using the single layer representation, we obtain an equivalent system of boundary integral equations. We show the singular values of the ill-posed Cauchy operator decay exponentially, which means that a small error is exponentially amplified in the solution of the Cauchy problem. We show the decaying rate is dependent on the geometry of he domain, which provides the information on the choice of numerically meaningful modes. We suggest a pseudo-inverse regularization method based on singular value decomposition and present various numerical simulations.

• Communications of the Korean Mathematical Society
• /
• v.33 no.3
• /
• pp.1025-1037
• /
• 2018

We consider an ill-posed problem for the heat equation $u_{xx}=u_t$ in the quarter plane {x > 0, t > 0}. We propose a new method to compute the heat flux $h(t)=u_x(1,t)$ from the boundary temperature g(t) = u(1, t). The operator $g{\mapsto}h=Hg$ is unbounded in $L^2({\mathbb{R}})$, so we approximate h(t) by $h_{\delta}(t)=u_x(1+{\delta},\;t)$, ${\delta}{\rightarrow}0$. When noise is present, the data is $g_{\epsilon}$ leading to a corresponding heat $h_{{\delta},{\epsilon}}$. We obtain an estimate of the error ${\parallel}h-h_{{\delta},{\epsilon}}{\parallel}$, as well as the error when $h_{{\delta},{\epsilon}}$ is approximated by the trapezoidal rule. With an a priori choice rule ${\delta}={\delta}({\epsilon})$ and ${\tau}={\tau}({\epsilon})$, the step size of the trapezoidal rule, the main theorem gives the error of the heat flux as a function of noise level ${\epsilon}$. Numerical examples show that the proposed method is effective and stable.

• Journal of applied mathematics & informatics
• /
• v.35 no.5_6
• /
• pp.439-447
• /
• 2017

This paper investigates the inverse problem of determining an unknown heat radiative coefficient, which is only time-dependent. This is an ill-posed problem, that is, small errors in data may cause huge deviations in determining solution. In this paper, the existence and uniqueness of the problem is established by the second Volterra integral equation theory, and the method of trace-type functional formulation combined with finite difference scheme is studied. One typical numerical example using the proposed method is illustrated and discussed.

• Journal of the Korean Mathematical Society
• /
• v.55 no.4
• /
• pp.849-875
• /
• 2018

The purpose of this paper is to present an operator method of regularization for the problem of finding a solution of a system of nonlinear ill-posed equations with a monotone hemicontinuous mapping and N inverse-strongly monotone mappings in Banach spaces. A regularization parameter choice is given and convergence rate of the regularized solutions is estimated. We also give the convergence and convergence rate for regularized solutions in connection with the finite-dimensional approximation. An iterative regularization method of zero order in a real Hilbert space and two examples of numerical expressions are also given to illustrate the effectiveness of the proposed methods.

• Kyungpook Mathematical Journal
• /
• v.64 no.2
• /
• pp.353-369
• /
• 2024

In this study, we investigate the unknown time-dependent heat source function in inverse problems. We consider three general nonlocal conditions; two classical boundary conditions and one nonlocal over-determination, condition, these genereate six different cases. The finite integration method (FIM), based on numerical integration, has been adapted to solve PDEs, and we use it to discretize the spatial domain; we use backward differences for the time variable. Since the inverse problem is ill-posed with instability, we apply regularization to reduce the instability. We use the first-order Tikhonov's regularization together with the minimization process to solve the inverse source problem. Test examples in all six cases are presented in order to illustrate the accuracy and stability of the numerical solutions.

• Structural Monitoring and Maintenance
• /
• v.3 no.2
• /
• pp.115-127
• /
• 2016

It is a challenging problem of assessing the location and extent of structural damages with vibration measurements. In this paper, an improved Extended Kalman filter (EKF) with Tikhonov regularization is proposed to identify structural damages. The state vector of EKF consists of the initial values of modal coordinates and damage parameters of structural elements, therefore the recursive formulas of EKF are simplified and modal truncation technique can be used to reduce the dimension of the state vector. Then Tikhonov regularization is introduced into EKF to restrain the effect of the measurement noise for improving the solution of ill-posed inverse problems. Numerical simulations of a seven-story shear-beam structure and a simply-supported beam show that the proposed method has good robustness and can identify the single or multiple damages accurately with the unknown initial structural state.