Volume 26, Issue 1

The Sturm–Liouville operator on a star-shaped graph with different types of boundary conditions (Robin and Dirichlet) in different vertices is studied. Asymptotic formulas for the eigenvalues are derived and partial inverse problems are solved: we show that the potential on one edge can be uniquely determined by different parts of the spectrum if the potentials on the other edges are known. We provide a constructive method for the solution of the inverse problems, based on the Riesz basis property of some systems of vector functions.

We focus on the nonhomogeneous backward heat problem of finding the initial temperature θ=θ⁢(x,y)=u⁢(x,y,0){\theta=\theta(x,y)=u(x,y,0)} such that {ut-a⁢(t)⁢(ux⁢x+uy⁢y)=f⁢(x,y,t),(x,y,t)∈Ω×(0,T),u⁢(x,y,t)=0,(x,y)∈∂⁡Ω×(0,T),u⁢(x,y,T)=h⁢(x,y),(x,y)∈Ω¯,\left\{\begin{aligned} \displaystyle u_{t}-a(t)(u_{xx}+u_{yy})&\displaystyle=f% (x,y,t),&\hskip 10.0pt(x,y,t)&\displaystyle\in\Omega\times(0,T),\\ \displaystyle u(x,y,t)&\displaystyle=0,&\hskip 10.0pt(x,y)&\displaystyle\in% \partial\Omega\times(0,T),\\ \displaystyle u(x,y,T)&\displaystyle=h(x,y),&\hskip 10.0pt(x,y)&\displaystyle% \in\overline{\Omega},\end{aligned}\right.\vspace*{-0.5mm} where Ω=(0,π)×(0,π){\Omega=(0,\pi)\times(0,\pi)}. In the problem, the source f=f⁢(x,y,t){f=f(x,y,t)} and the final data h=h⁢(x,y){h=h(x,y)} are determined through random noise data gi⁢j⁢(t){g_{ij}(t)} and di⁢j{d_{ij}} satisfying the regression models gi⁢j⁢(t)=f⁢(Xi,Yj,t)+ϑ⁢ξi⁢j⁢(t),\displaystyle g_{ij}(t)=f(X_{i},Y_{j},t)+\vartheta\xi_{ij}(t),di⁢j=h⁢(Xi,Yj)+σi⁢j⁢εi⁢j,\displaystyle d_{ij}=h(X_{i},Y_{j})+\sigma_{ij}\varepsilon_{ij}, where (Xi,Yj){(X_{i},Y_{j})} are grid points of Ω. The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically.

Adsorption isotherms are the most important parameters in rigorous models of chromatographic processes. In this paper, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem [2]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem. Thus, this method has advantages regarding robustness and computation in reconstruction. In contrast to the traditional CCBM, for the sake of the reduction of computational complexity and computational cost, the recovered adsorption isotherm only corresponds to the real part of the solution of a forward complex initial boundary value problem. Furthermore, we take into account the position of the profiles and apply the momentum criterion to improve the optimization progress. Using Tikhonov regularization, the well-posedness, convergence properties and regularization parameter selection methods are studied. Based on an adjoint technique, we derive the exact Jacobian of the objective function and give an algorithm to reconstruct the adsorption isotherm. Finally, numerical simulations are given to show the feasibility and efficiency of the proposed regularization method.

The development of efficient computational tools for data assimilation and analysis using multi-parameter models is one of the major issues in systems immunology. The mathematical description of the immune processes across different scales calls for the development of multiscale models characterized by a high dimensionality of the state space and a large number of parameters. In this study we consider a standard parameter estimation problem for two models, formulated as ODEs systems: the model of HIV infection and BrdU-labeled cell division model. The data fitting is formulated as global optimization of variants of least squares objective function. A new computational method based on Tensor Train (TT) decomposition is applied to solve the formulated problem. The idea of proposed method is to extract the tensor structure of the optimized functional and use it for optimization. The method demonstrated a better performance in comparison with some other broadly used global optimization techniques.

In this paper, a regularized sampling algorithm for band-limited signals is presented in the two-dimensional case. The convergence of the regularized sampling algorithm is studied and compared with Shannon’s sampling theorem and the Tikhonov regularization method by some examples.

We show that the convergence rate of ℓ1{\ell^{1}}-regularization for linear ill-posed equations is always 𝒪⁢(δ){{\mathcal{O}}(\delta)} if the exact solution is sparse and if the considered operator is injective and weak*-to-weak continuous. Under the same assumptions convergence rates in case of non-sparse solutions are proven. The results base on the fact that certain source-type conditions used in the literature for proving convergence rates are automatically satisfied.

We develop an approach to the analysis of stock market crises based on the generalized nonparametric method. The generalized nonparametric method is based on solvability and regularization of ill-posed inverse problem in Pareto’s demand theory. Our approach allows one to select a few companies that may be considered as the main reason for the crisis. We apply this approach to study the Chinese stock market crash in 2015.

In this paper, we consider a fast multiscale Galerkin method with compression technique for solving Fredholm integral equations of the first kind via the nonstationary iterated Tikhonov regularization. A modified a posteriori regularization parameter choice strategy is established, which leads to optimal convergence rates.

A new combined numerical algorithm for solving inverse problems of epidemiology is described in this paper. The combined algorithm consists of optimization and iterative methods, and determines the parameters specific to a particular population by using the statistical information for a few previous years. The coefficients of the epidemiology model describe particular qualities of the population and the development of the disease. The inverse problem of parameter identification in a mathematical model is reduced to the problem of minimizing an objective function characterizing the square deviation of the statistical data from the experimental data. The combination of statistical and optimization algorithms demonstrates the identification of parameters with an appropriate relative accuracy of 30Ṫhe results can be used by public health organizations to predict the infectious disease epidemic in a given region by comparing the data of simulation with historical data.

In this paper, we consider an inverse coefficient problem for the linearized Korteweg–de Vries (KdV) equation ut+ux⁢x⁢x+(c⁢(x)⁢u)x=0{u_{t}+u_{xxx}+(c(x)u)_{x}=0}, with homogeneous boundary conditions u⁢(0,t)=u⁢(1,t)=ux⁢(1,t)=0{u(0,t)=u(1,t)=u_{x}(1,t)=0}, when the Neumann data g⁢(t):=ux⁢(0,t){g(t):=u_{x}(0,t)}, t∈(0,T){t\in(0,T)}, is given as an available measured output at the boundary x=0{x=0}. The inverse problem is formulated as a minimum problem for the regularized Tikhonov functional 𝒥α⁢(c)=12⁢∥ux⁢(0,⋅;c)-g∥L2⁢(0,T)2+α2⁢∥c′∥L2⁢(0,1)2{\mathcal{J}_{\alpha}(c)=\frac{1}{2}\|u_{x}(0,\cdot\,;c)-g\|^{2}_{L^{2}(0,T)}+% \frac{\alpha}{2}\|c^{\prime}\|^{2}_{L^{2}(0,1)}} with Sobolev norm. Based on a priori estimates for the weak and regular weak solutions of the direct and adjoint problems, it is proved that the input-output operator is compact, which shows the ill-posedness of the inverse problem. Then Fréchet differentiability of the Tikhonov functional and Lipschitz continuity of the Fréchet gradient are proved. It is shown that the last result allows us to use an important advantage of gradient methods when the functional is from the class C1,1⁢(ℳ){C^{1,1}(\mathcal{M})}. In the final part, an existence of a solution of the minimum problem for the regularized Tikhonov functional 𝒥α⁢(c){\mathcal{J}_{\alpha}(c)} is proved.