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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]

We are concerned with an inverse problem of simultaneously recovering the initial value and source strength in a transmission problem for a parabolic equation from extra measurements. First, we establish a conditional stability of the inverse problem by combining the Carleman estimate with the logarithmic convexity theory. Second, we construct a regularized minimizing functional and transform the inverse problem into an optimization problem. The existence and stability of solutions to the optimization problem are analyzed rigorously. Then, we introduce a surrogate functional and propose an iterative thresholding algorithm for solving the optimization problem. The algorithm is cheap and easy to be implemented numerically. Finally, we present several numerical examples for the two-dimensional (2D) and three-dimensional (3D) cases to show the validity of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper deals with an inverse problem on determining a time dependent potential in a diffusion equation with temporal fractional derivative of variable order from a distributed observation. We shall study the existence, uniqueness and regularity estimates of the solution for the forward problem by utilizing the Freldhom alternative principle for compact operators. Based on a newly established coercivity for fractional derivatives of variable order, we prove a uniqueness result for the inverse potential problem. Numerically, we transform the inverse potential problem into an optimization problem with Tikhonov regularization. An iterative thresholding algorithm is proposed to find the minimizers by a newly constructed adjoint system, whose wellposedness is also verified. Several numerical experiments are presented to show the accuracy and efficiency of the proposed algorithm. [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]

We shall study in this paper the convergence rates of the Tikhonov regularized solutions for the recovery of the radiativities in elliptic and parabolic systems in general dimensional spaces. The conditional stability estimates are first derived. Due to the difficulty of the verification of the existing source conditions or nonlinearity conditions of the inverse radiativity problems in high dimensional spaces, some new variational source conditions are proposed. The conditions are rigorously verified in general dimensional spaces under the conditional stability estimates. We will also derive the reasonable convergence rates under the new source conditions, and the results reveal the explicit relation between the regularity of the radiativities and the convergence rates. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabilized nonlinear minimization system by an appropriately selected Tikhonov regularization. The existence and the stability of the optimization system are demonstrated. The nonlinear optimization problem is approximated by a fully discrete scheme, whose convergence is established under a novel result verified in this study that the H1-norm of the solution to the discrete forward system is uniformly bounded. The iterative thresholding algorithm is proposed to solve the discrete minimization, and several numerical experiments are presented to show the efficiency and the accuracy of the algorithm. [ABSTRACT FROM AUTHOR]

We study the Levenberg-Marquardt (L-M) method for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The L-M method transforms the Tikhonov regularized nonlinear non-convex minimizations into convex minimizations. And the quadratic convergence of the L-M method is rigorously established for the nonlinear elliptic and parabolic inverse problems for the first time, under a simple novel adaptive strategy for selecting regularization parameters during the L-M iteration. Then the surrogate functional approach is adopted to solve the strongly ill-conditioned convex minimizations, resulting in an explicit solution of the minimisation at each L-M iteration for both the elliptic and parabolic cases. Numerical experiments are provided to demonstrate the accuracy, efficiency and quadratic convergence of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the inverse problem on determining the spatial component of the source term in the hyperbolic equation with a time-dependent principal part. Based on a Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to reformulate the inverse problem into a related optimization problem, for which we develop an iterative thresholding algorithm by using the corresponding adjoint system. Numerical examples up to three spatial dimensions are presented to demonstrate the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we shall derive and propose an efficient algorithm for simultaneously reconstructing the Robin coefficient and heat flux in an elliptic system from part of the boundary measurements. The uniqueness of the simultaneous identification is demonstrated. The ill-posed inverse problem is formulated into an output least-squares nonlinear and non-convex minimization with Tikhonov regularization, while the regularizing effects of the regularized system are justified. The Levenberg–Marquardt method is applied to change the non-convex minimization into convex minimization, which will be solved by surrogate functional method so as to get the explicit expression of the minimizer. Numerical experiments are provided to show the accuracy and efficiency of the algorithm. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the convergence rates of the widely used output least squares method with Tikhonov regularization for the identification of the magnetic permeability in a Maxwell system. By introducing some new techniques, we derive a simple and easily interpretable source condition and guarantee the usual convergence rate O() for the regularized solutions, whereis assumed to be the noise level of the data. More importantly, we finally construct the source functions successfully in two-dimensional spaces. [ABSTRACT FROM PUBLISHER]

In this paper, we are concerned with two efficient algorithms for surface construction. Based on the Gauss equations, a discretized nonlinear equation of the formshould be solved in the process of surface construction. We first consider a regularized fixed-point iterative algorithm for solving the discretized equation, in which we determine the actual regularization parameter by the Morozov discrepancy principle. A two-parameter algorithm is employed for solving the Morozov equation, and the convergence of the regularized fixed-point iterative algorithm is demonstrated. Secondly, we also propose the regularizing Levenberg–Marquardt scheme to solve the discretized equation, in which the regularization parameter is chosen to be a small constant. Numerical experiments are provided to demonstrate the robustness and efficiency of the two algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we aim to reconstruct the unknown Robin coefficient and unknown heat flux simultaneously in an elliptic system from Cauchy data on arbitrarily small portion of accessible boundary. A novel logarithmic type stability in terms of the negative Sobolev norms is established. [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 study focused on the coupling heat transfer mechanism and the cooling efficiency of L-shaped two-phase closed thermosyphons (L-shaped TPCTs) in the wide subgrade of permafrost regions. Considering the fact that time–space dynamics change the effects of the air temperature, wind speed, and geotemperature, a coupled air temperature–L-shaped TPCT–subgrade soil heat transfer model was established using the ANSYS 15.0 software platform, and the rationality of the model was verified through measured data. The heat-transfer characteristics of the L-shaped TPCTs and the long-term thermal stability of the subgrade were studied under different inclination angles of the evaporator (α = 15°, 30°, 50°, 70°, and 90°). Then, the cooling effectiveness of a composite subgrade with TPCTs and an XPS insulation board was numerically calculated. The results show that the heat flux of the L-shaped TPCT was the greatest when α = 50°, and the heat flux reached the maximum value of 165.7 W·m−2 in January. The L-shaped TPCT had a relatively good cooling effect on the subgrade as a whole when α = 50° and 70°, but the thawing depth at the center of the subgrade with L-shaped TPCTs reached 9.0 m below the ground surface in the 30th year. The composite subgrade with L-shaped TPCTs/vertical TPCT/XPS insulation board is an effective method to protect the permafrost foundation and improve the long-term thermal stability of the wide subgrade. The maximum heat flux of evaporation section of the L-shaped TPCT is 18.8% higher than that of the vertical TPCT during the working period of the TPCTs of the composite subgrade. [ABSTRACT FROM AUTHOR]

The engineering properties of collapsible loess have significant uncertainty. Accurate prediction of collapsible deformation is crucial for the safety of engineering construction in loess areas. Taking the typical collapsible loess stratum as the research object in Xi'an, based on the random field theory, combined with the Monte Carlo strategy and modulus reduction method, the stochastic finite element analysis of loess self-weight collapsibility is carried out to study the influence of the spatial variability of compression modulus on the self-weight collapsibility of loess. The results show that the loess tends to be stratified and average along the depth direction with the increase of transverse correlation distance. The random field result of self-weight collapsibility considering the spatial variability of compression modulus is significantly greater than the deterministic result of layered average and the calculated value of loess code. Considering the low compression modulus dominance effect of compression modulus with positive skewed distribution of random field, the equivalent characteristic value of the compression modulus calculated by the layered average modeling for the collapsibility evaluation of typical loess strata in Xi'an area is proposed. [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]

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]

One of the major challenges encountered in electrification reconstruction project of Golmud-Lhasa section of Qinghai-Tibet railway of China is ensuring the frost jacking stability of the pile foundation of overhead contact system mast (OCSM). To evaluate the resisting frost jacking performance of OCSM pile in permafrost regions, a theoretical model of OCSM pile in permafrost soils under the effect of frost heave was constructed based on superposition principle and deformation coordination relationship between the pile and soil. The model was solved by using the finite difference method combined with MATLAB. Then, large scale model tests were performed to investigate the thermodynamic behaviors of OCSM piles with different sections (equal section circular pile, straight cone cylindrical pile and curved cone cylindrical pile) in the permafrost soils under freeze-thaw actions, after which the theoretical model were validated. It was found that distribution curves of pile axial force obtained by the calculation method is basically consistent with model test results in the overall trend, implying that the calculation model is reasonable and feasible. During soil freezing, the pile is under tension as a whole, and the axial force is greatest near the depth of frost penetration. The maximum value of tangential frost-heave stress occurs near the ground surface. The total tangential frost heaving force of the curved cone cylindrical pile is the smallest, and the effect of resisting frost jacking is the best. These research results can provide a reference for resisting frost jacking design of OCSM pile in permafrost regions. • A theoretical calculation model of OCSM pile frost jacking in permafrost soils was constructed. • Model tests were performed to evaluate the frost jacking behaviors of OCSM piles. • A curved cone cylindrical pile is proposed which has better effect of resisting frost jacking. • The influence factors of OCSM pile frost jacking characteristics were analyzed. [ABSTRACT FROM AUTHOR]