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We propose a globally convergent computational technique for the nonlinear inverse problem of reconstructing the zero-order coefficient in a parabolic equation using partial boundary data. This technique is called the ``reduced dimensional method.'' Initially, we use the polynomial-exponential basis to approximate the inverse problem as a system of 1D nonlinear equations. We then employ a Picard iteration based on the quasi-reversibility method and a Carleman weight function. We will rigorously prove that the sequence derived from this iteration converges to the accurate solution for that 1D system without requesting a good initial guess of the true solution. The key tool for the proof is a Carleman estimate. We will also show some numerical examples. 

Abstract This paper aims to determine the initial conditions for quasi-linear hyperbolic equations that include nonlocal elements. We suggest a method where we approximate the solution of the hyperbolic equation by truncating its Fourier series in the time domain with a polynomial–exponential basis. This truncation effectively removes the time variable, transforming the problem into a system of quasi-linear elliptic equations. We refer to this technique as the ‘time dimensional reduction method.’ To numerically solve this system comprehensively without the need for an accurate initial estimate, we used the newly developed Carleman contraction principle. We show the efficiency of our method through various numerical examples. The time dimensional reduction method stands out not only for its precise solutions but also for its remarkable speed in computation. 

This paper provides a survey of the literature on the application of Multi-agent Systems (MAS) technology for Smartgrids. Smartgrids represent the next generation electric network, as communities are developing self-sufficient and environmentally friendly energy production. As a cyber-physical system, the development of the vision of Smartgrids requires the resolution of major technical problems; this has fed over a decade of research. Due to the stochastic, intermittent nature of renewable energy resources and the heterogeneity of the agents involved in a Smartgrid, demand and supply management, energy trade and control of grid elements constitute great challenges for stable operation. In addition, in order to offer resilience against faults and attacks, Smartgrids should also have restoration, self-recovery and security capabilities. Multi-agent systems (MAS) technology has been a popular approach to deal with these challenges in Smartgrids, due to their ability to support reasoning in a distributed context. This survey reviews the literature concerning the use of MAS models in each of the relevant research areas related to Smartgrids. The survey explores how researchers have utilized agent-based tools and methods to solve the main problems of Smartgrids. The survey also discusses the challenges in the advancement of Smartgrid technology and identifies the open problems for research from the view of multi-agent systems. 

Recent theoretical models and field observations suggest that fluvial bedload flux can be estimated from seismic energy measured within appropriate frequency bands. We present an application of the Tsai et al. (2012, https://doi.org/10.1029/2011gl050255) bedload seismic model to an ephemeral channel located in the semi‐arid southwestern US and incorporate modifications to better estimate bedload flux in this environment. To test the model, we collected streambank seismic signals and directly measured bedload flux during four flash‐floods. Bedload predictions calculated by inversion from the Tsai model underestimated bedload flux observations by one‐to‐two orders of magnitude at low stages. However, model predictions were better for moderate flow depths (>50 cm), where saltation is expected to dominate bedload transport. We explored three differences between the model assumptions and our field conditions: (a) rolling and sliding particles have different impact frequencies than saltating particles; (b) the velocity and angle of impact of rolling particles onto the riverbed differ; and (c) the fine‐grained alluvial character of this and similar riverbeds leads to inelastic impacts, as opposed to the originally conceptualized elastic impacts onto rigid bedrock. We modified the original model to assume inelastic bed impacts and to incorporate rolling and sliding by adjusting the statistical distributions of bedload impact frequency, velocity, and angle. Our modified “multiple‐transport‐mode bedload seismic model” decreased error relative to observations to less than one order of magnitude across all measured flow conditions. Further investigations in other environmental settings are required to demonstrate the robustness and general applicability of the model. 

This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping coefficient. Our approach involves approximating the hyperbolic equation’s solution by its truncated Fourier expansion in the time domain and using the recently developed polynomial-exponential basis. This truncation process facilitates the elimination of the time variable, consequently, yielding a system of quasi-linear elliptic equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman contraction principle. We provide several numerical examples to illustrate the efficacy of our method. The method not only delivers precise solutions but also showcases remarkable computational efficiency.