Search

In this article, we discuss a solution to time‐fractional diffusion equation ∂tα(u−u0)+Au=0$$ {\partial}_t^{\alpha}\left(u-{u}_0\right)+ Au=0 $$ with the homogeneous Dirichlet boundary condition, where an elliptic operator −A$$ -A $$ is not necessarily symmetric. We prove that the solution u$$ u $$ is identically zero if its normal derivative with respect to the operator A$$ A $$ vanishes on an arbitrarily chosen subboundary of the spatial domain over a time interval. The proof is based on the Laplace transform and the spectral decomposition for a nonsymmetric elliptic operator. As a direct application, we prove the uniqueness result for an inverse problem on determining the spatial component in the source term by Neumann boundary data on subdoundary. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the identification of the piecewise constant Robin coefficient in an elliptic equation. The iterative regularization method is one of the very effective methods for solving this kind of nonlinear ill-posed inverse problems. But it usually requires to solve numerous amounts of forward solutions during the iterative process, which will cost a lot of computational time in high-dimensional spaces. A reduced basis method is considered to reduce the computational time for solving the forward problems, and its error estimate is also studied. Finally, we propose a reduced basis Landweber algorithm to solve the elliptic inverse Robin problem and present several numerical experiments to demonstrate the accuracy and efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

In order to study the physical and mechanical properties of loess reinforced with F1 ion curing agent, California bearing ratio (CBR) test, unconfined compressive strength test and triaxial unconsolidated undrained test were carried out. Based on the results of scanning electron microscope (SEM), nuclear magnetic resonance (NMR) and X-ray diffraction (XRD), the mechanical properties and micro-pore structure of loess reinforced by F1 ion curing agent were studied. The test results show that the F1 curing agent can reduce the plastic limit, increase the liquid limit, plastic index and maximum dry density of soil by cation exchange, destroy the double layer structure and the hydrophobic effect of sulfonated oil, and improve the water sensitivity and compaction characteristics of soil; F1 can significantly increase the CBR value, unconfined compressive strength and shear strength parameters, and 0.3 L/m³ is the best dosage of loess reinforcement; F1 changes the pore structure and contact form between particles, promotes the aggregation and condensation between particles, reduces the fractal dimension of the pore distribution of microstructures from 1.244 to 1.076, and greatly reduces the total pore area and total pore number of soil. Under the action of rolling, a laminated stack structure with tighter arrangements, larger agglomerates and surface-to-face contact is formed as main contact style, which enhances the compactness and bearing capacity of the soil. [ABSTRACT FROM AUTHOR]

In this paper, we shall present the quadratic convergence of Levenberg-Marquardt (L-M) method for solving general ill-posed nonlinear inverse problems including two parameters to be identified. As the Tikhonov regularized minimizations are non-convex due to the nonlinearity of the nonlinear inverse problems, the L-M method transforms them into convex minimizations. The quadratic convergence of the L-M method is rigorously demonstrated under some basic assumptions. An application of the L-M method for the simultaneous identification of the Robin coefficient and heat flux in an elliptic system is presented, and the surrogate functional technique is introduced to solve the convex but still strongly ill-conditioned minimizations, resulting in an explicit minimizer of the minimization at each L-M iteration. Numerical experiments are presented to illustrate the efficiency and accuracy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this work we are concerned with the analysis on a simultaneous finite element reconstruction of the convection velocity and source strength in a time-dependent convection–diffusion equation. The ill-posed problem is formulated into an output least-squares nonlinear minimization by an appropriately selected Tikhonov regularization. The regularizing effect and mathematical properties of the regularized system are justified and demonstrated. The nonlinear optimization problem is approximated by a fully discrete finite element method, whose convergence is rigorously established. [ABSTRACT FROM AUTHOR]

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux. [ABSTRACT FROM AUTHOR]

We shall derive and propose several efficient domain decomposition methods for solving the nonlinear inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The highly ill-posed inverse problems are transformed into output least-squares nonlinear and non-convex minimizations with classical Tikhonov regularization. The Levenberg–Marquardt method is applied to transform the non-convex minimizations into convex minimizations, which will be solved by several efficient domain decomposition methods. The methods are completely local and the local minimizers have explicit expressions within the subdomains. Several numerical experiments are presented to show the accuracy and efficiency of the methods; in particular, the convergence seems nearly optimal in the sense that the iteration number of the methods is nearly independent of mesh sizes. [ABSTRACT FROM AUTHOR]