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We present stability estimates for the inverse source problem of the stochastic Helmholtz equation in two and three dimensions by either near-field or far-field data. The random source is assumed to be a microlocally isotropic generalized Gaussian random function. For the direct problem, by exploring the regularity of the Green function, we demonstrate that the direct problem admits a unique bounded solution with an explicit integral representation, which enhances the existing regularity result. For the case using near-field data, the analysis of the inverse problem employs microlocal analysis to achieve an estimate for the Fourier transform of the micro-correlation strength by the near-field correlation data and a high-frequency tail. The stability follows by showing the analyticity of the data and applying a novel analytic continuation principle. The stability estimate by far-field data is derived by investigating the correlation of the far-field data. The stability estimate consists of the Lipschitz type data discrepancy and the logarithmic stability. The latter decreases as the upper bound of the frequency increases, which exhibits the phenomenon of increasing stability., Comment: 19pages

In this paper, we investigate well-posedness of time-harmonic scattering of elastic waves by unbounded rigid rough surfaces in three dimensions. The elastic scattering is caused by an $L^2$ function with a compact support in the $x_3$-direction, and both deterministic and random surfaces are investigated via the variational approach. The rough surface in a deterministic setting is assumed to be Lipschitz and lie within a finite distance of a flat plane, and the scattering is caused by an inhomogeneous term in the elastic wave equation whose support lies within some finite distance of the boundary. For the deterministic case, a stability estimate of elastic scattering by rough surface is shown at an arbitrary frequency. It is noticed that all constants in {\it a priori} bounds are bounded by explicit functions of the frequency and geometry of rough surfaces. Furthermore, based on this explicit dependence on the frequency together with the measurability and $\mathbb{P}$-essentially separability of the randomness, we obtain a similar bound for the solution of the scattering by random surfaces.

This paper investigates the elastic scattering by unbounded deterministic and random rough surfaces, which both are assumed to be graphs of Lipschitz continuous functions. For the deterministic case, an a priori bound explicitly dependent on frequencies is derived by the variational approach. For the scattering by random rough surfaces with a random source, well-posedness of the corresponding variation problem is proved. Moreover, a similar bound with explicit dependence on frequencies for the random case is also established based upon the deterministic result, Pettis measurability theorem and Bochner's integrability Theorem.

The paper considers direct and inverse elastic scattering from a cavity in homogeneous medium with Dirichlet and Neumann boundary conditions. For direct scattering, existence and uniqueness are derived by variation approach. For inverse scattering, Fr$\acute{\rm e}$chet derivatives of the solution operators are investigated, which give local stability for Dirichlet case.

This paper optimizes path planning for a trunkand-branch topology network in an irregular 2-dimensional manifold embedded in 3-dimensional Euclidean space with application to submarine cable network planning. We go beyond our earlier focus on the costs of cable construction (including labor, equipment and materials) together with additional cost to enhance cable resilience, to incorporate the overall cost of branching units (again including material, construction and laying) and the choice of submarine cable landing stations, where such a station can be anywhere on the coast in a connected region. These are important issues for the economics of cable laying and significantly change the model and the optimization process. We pose the problem as a variant of the Steiner tree problem, but one in which the Steiner nodes can vary in number, while incurring a penalty. We refer to it as the weighted Steiner node problem. It differs from the Euclidean Steiner tree problem, where Steiner points are forced to have degree three; this is no longer the case, in general, when nodes incur a cost. We are able to prove that our algorithm is applicable to Steiner nodes with degree greater than three, enabling optimization of network costs in this context. The optimal solution is achieved in polynomialtime using dynamic programming.

This paper provides an optimized cable path planning solution for a tree-topology network in an irregular 2D manifold in a 3D Euclidean space, with an application to the planning of submarine cable networks. Our solution method is based on total cost minimization, where the individual cable costs are assumed to be linear to the length of the corresponding submarine cables subject to latency constraints between pairs of nodes. These latency constraints limit the cable length and number of hops between any pair of nodes. Our method combines the Fast Marching Method (FMM) and a new Integer Linear Programming (ILP) formulation for Minimum Spanning Tree (MST) where there are constraints between pairs of nodes. We note that this problem of MST with constraints is NP-complete. Nevertheless, we demonstrate that ILP running time is adequate for the great majority of existing cable systems. For cable systems for which ILP is not able to find the optimal solution within an acceptable time, we propose an alternative heuristic algorithm based on Prim's algorithm. In addition, we apply our FMM/ILP-based algorithm to a real-world cable path planning example and demonstrate that it can effectively find an MST with latency constraints between pairs of nodes., Comment: 11 pages, 7 figures

We study the rise and decay times in BP-MoS2 heterojunctions through gate- and wavelength-dependent scanning photocurrent measurements. Our results have shown that the Schottky barrier at the MoS2-metal interface plays an important role in the photoresponse dynamics of the heterojunction. When the MoS2 channel is in the on-state, photo-excited carriers can tunnel through the narrow depletion region at the MoS2-metal interface, leading to a short carrier transit time. A response time constant of 13 {\mu}s has been achieved in both the rising and decaying regions regardless of the incident laser wavelength, which is comparable or higher than those of other BP-MoS2 heterojunctions as well as BP and MoS2 based phototransistors. On the other hand, when the MoS2 channel is in the off-state the resulting sizeable Schottky barrier and depletion width make it difficult for photo-excited carriers to overcome the barrier. This significantly delays the carrier transit time and thus the photoresponse speed, leading to a wavelength-dependent response time since the photo-excited carriers induced by short wavelength photons have a higher probability to overcome the Schottky barrier at the MoS2-metal interface than long wavelength photons. These studies not only shed light on the fundamental understanding of photoresponse dynamics in BP-MoS2 heterojunctions, but also open new avenues for engineering the interfaces between two-dimensional (2D) materials and metal contacts to reduce the response time of 2D materials based optoelectronic devices., Comment: 4 figures and 17 pages