Volume 33, Issue 4

This paper investigates the inverse problem of estimating an unknown bilinear control u , applied locally in a reaction-diffusion equation. Our goal is to achieve a temperature profile close to a desired reference at the final time. We formulate the problem as an optimal control framework and analyze the existence, optimality conditions and stability of the solution with respect to the data. An algorithm and some numerical experiments are proposed to show the effectiveness of our approach in steering the system towards a desired state.

Two novel representations of core-EP inverse A ( † ) A^{({\dagger})} are derived for a deficient square matrix A ∈ C n × n A\in C^{n\times n} with ind ⁡ ( A ) = k \operatorname{ind}(A)=k . Based on the two representations, a new method to find the core-EP inverse A ( † ) A^{({\dagger})} is presented through elementary transformation on a partitioned matrix. We summarize the corresponding algorithm for A ( † ) A^{({\dagger})} and also analyze the computational complexities in detail. Our proposed algorithm is faster than that in [J. Ji and Y. Wei, The core-EP, weighted core-EP inverse of matrices and constrained systems of linear equations, Commun. Math. Res. 37 (2021), 1, 86–112] and [K. Manjunatha Prasad and M. D. Raj, Bordering method to compute core-EP inverse, Spec. Matrices 6 (2018), 193–200] by comparing their computational complexities under a certain condition. In the end, the efficiency of our algorithm is demonstrated by two numerical examples.

This paper investigates the convergence rates for Tikhonov regularization of the problem for identifying the coefficient a ⁢ ( x ) ∈ L ∞ ⁢ ( Ω ) a(x)\in L^{\infty}(\Omega) in the frequency-domain acoustic wave equation − div ⁢ ( a ⁢ ∇ u ) − b ⁢ u = f -\mathrm{div}(a\nabla u)-bu=f in general dimensional spaces; here x ∈ Ω ⊂ R d x\in\Omega\subset\mathbb{R}^{d} , d ≥ 1 d\geq 1 , and b ⁢ ( x ) > 0 b(x)>0 . We assume that we know the imprecise measurement data of 𝑢 in the subdomain Ω 1 ⊂ Ω \Omega_{1}\subset\Omega with a measurement error of level δ > 0 \delta>0 , while 𝑢 satisfies the general Robin boundary condition on ∂ Ω 1 \partial\Omega_{1} . We propose to regularize this problem by minimizing a new functional and prove that the functional attains a unique global minimum on the admissible set of a ⁢ ( x ) a(x) . Furthermore, we derive the convergence rate O ⁢ ( δ ) O(\sqrt{\delta}) for the Tikhonov regularized solution with an easily satisfied source condition.

This paper is devoted to establishing the Carleman estimate for the stochastic degenerate wave equation with drift in the weakly degenerate case. Due to the degeneracy, we first study an approximate nondegenerate system. In order to overcome the difficulty arising from the drift term, we introduce an auxiliary function by which the diffusion and the drift term are transformed into a complex whole. After complicated and detailed estimates, we then derive the Carleman estimate for the stochastic degenerate wave equation. As an application for our Carleman estimate, we solve an inverse problem of determining three unknowns simultaneously by some suitable observations for the stochastic degenerate wave equation with drift. More precisely, we obtain a global uniqueness result for our inverse problem.

In this paper, we study a multi-objective inverse initial problem with a Nash strategy constraint for forward stochastic reaction-diffusion equations with dynamic boundary conditions, where both the volume and surface equations are influenced by randomness. The objective is twofold: First, we maintain the state close to prescribed targets in fixed regions using two controls; second, we determine the history of the solution from observations at the final time. To achieve this, we establish new Carleman estimates for forward and backward equations, which are used to prove an interpolation inequality for a coupled forward-backward stochastic system. Consequently, we obtain two results: Backward uniqueness and a conditional stability estimate for the initial conditions.

A significant challenge in achieving accurate results with the OSEM algorithm in emission tomography is the emergence of edge artifacts at high-contrast boundaries. However, the underlying cause of these artifacts is not well understood. This study aims to investigate the mechanism behind their formation. Image reconstruction was modeled using the MLEM algorithm for the one-dimensional case and the OSEM algorithm for the 3D case closely reflecting clinical practice. This was complemented by an analysis of the underlying formulas of these algorithms. The primary mechanism behind the emergence of edge artifacts is not their occurrence at a specific iteration, but the OSEM algorithm’s tendency to preserve these artifacts, along with other short-wavelength disturbances, over numerous iterations. Thus, filtering of the reconstructed images is necessary, at least during intermediate iterations, to eliminate edge artifacts. Furthermore, it has been observed that these artifacts arise from the inherent nature of the OSEM algorithm, rather than from insufficient information in the observed data.

In this paper, the inverse spectral problem of the AKNS operator ℋ ⁢ ( p , q ) {\mathcal{H}(p,q)} defined on the interval [ 0 , 1 ] {[0,1]} with the potential pair ( p , q ) {(p,q)} being a priori known on the subinterval [ 0 , r m ] {[0,\frac{r}{m}]} with 0 < r m < 1 2 {0<\frac{r}{m}<\frac{1}{2}} is considered. An algorithm of reconstruction of the potential from the knowledge of a full spectrum together with a subset of the norming constants is established.

An inverse problem for electrodynamic equations is considered. It is assumed that the electric current depends nonlinearly of the electric tension. This dependence is determined by seven finite functions of space variables. A direct problem for electrodynamic equations with a running plane wave going in direction ν from infinity is stated. Then traces of solutions of this direct problem on some bounded surface in ℝ 3 {\mathbb{R}^{3}} for different ν are used for posing an inverse problem. It is shown that the inverse problem is decomposed in seven separate problems. One of them is the X-ray tomography problem while 6 others are identical one to other integral geometry problems on a family of strait lines with a given weight function. The latter problems are studied and a stability estimate of solutions is found.