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The paper investigates an inverse problem of recovering the internal source from external temperature measurements in photo-thermal effect. The photo-thermal effect actually involves two physical processes: electromagnetic scattering and heat transfer, described by a nonlinear coupled system of Maxwell's equation and the heat transfer equation. The nonlinear coupling term in the system is represented by the square of the modulus of the electromagnetic (missing the phase information of the electromagnetic field), and the absence of this phase information poses a significant challenge to the reconstruction of the internal source. In addition, the interaction and mutual influence of multiple physical fields, including electric field, magnetic field and temperature field, add to the complexity involved in the inversion of the internal source. Based on the potential theory and asymptotic analysis, we prove that the internal source can be uniquely determined up to sign by the external temperature field. This provides a solid theoretical basis for designing the internal source inversion algorithm and further exploring the theoretical aspects of photo-thermal effect.

This paper investigates the simultaneous reconstruction of the running cost function and the internal topological structure within the mean-field games (MFG) system utilizing partial boundary data. The inverse problem is notably challenging due to factors such as nonlinear coupling, the necessity for multi-parameter reconstruction, constraints on probability measures, and the limited availability of measurement information. To address these challenges, we propose an innovative approach grounded in a higher-order linearization method. This method is tailored for inverse problems in MFG systems that involve Dirichlet and Neumann boundary conditions. Initially, we present unique reconstruction results for the cost function and internal topological structure of the MFG system under various homogeneous boundary conditions. Subsequently, we extend these results to accommodate inhomogeneous boundary conditions. These findings greatly enhance our understanding of simultaneous reconstruction in complex MFG systems.

In this note, we study distributed time-varying optimization for a multi-agent system. We first focus on a class of time-varying quadratic cost functions, and develop a new distributed algorithm that integrates an average estimator and an adaptive optimizer, with both bridged by a Dead Zone Algorithm. Based on a composite Lyapunov function and finite escape-time analysis, we prove the closed-loop global asymptotic convergence to the optimal solution under mild assumptions. Particularly, the introduction of the estimator relaxes the requirement for the Hessians of cost functions, and the integrated design eliminates the waiting time required in the relevant literature for estimating global parameter during algorithm implementation. We then extend this result to a more general class of time-varying cost functions. Two examples are used to verify the proposed designs., Comment: 11 pages, 7 figures

In this paper, we develop a general mathematical framework for the electro-osmosis problem to design simultaneous microscale electric and hydrodynamic cloaking in a Hele-Shaw configuration. A novel approach to achieving simultaneously cloaking both the electric and flow fields through a combination of scattering-cancellation technology and an electro-osmosis effect is proposed. In the design, the electric field is manipulated with scattering-cancellation technology while the pressure with electro-osmosis effect. As proof of this concept, the perfect electric and hydrodynamic cloaking conditions are derived for the cloaks with the cross-sectional shape being annulus or confocal ellipses using the layer potential techniques. Furthermore, we also propose an optimization scheme for the design of approximate cloaks within general geometries and prove the well-posedness of the optimization problem. In particular, the conditions that can ensure the simultaneous occurrence of approximate cloaks for general geometries are also established. Our theoretical findings are validated by a variety of numerical results and guide efficiently designing electric-related multiphysics cloaking., Comment: arXiv admin note: substantial text overlap with arXiv:2302.07495

Markov chain Monte Carlo (MCMC) simulations have been widely used to generate samples from the complex posterior distribution in Bayesian inferences. However, these simulations often require multiple computations of the forward model in the likelihood function for each drawn sample. This computational burden renders MCMC sampling impractical when the forward model is computationally expensive, such as in the case of partial differential equation models. In this paper, we propose a novel sampling approach called the geometric optics approximation method (GOAM) for Bayesian inverse problems, which entirely circumvents the need for MCMC simulations. Our method is rooted in the problem of reflector shape design, which focuses on constructing a reflecting surface that redirects rays from a source, with a predetermined density, towards a target domain while achieving a desired density distribution. The key idea is to consider the unnormalized Bayesian posterior as the density on the target domain within the optical system and define a geometric optics approximation measure with respect to posterior by a reflecting surface. Consequently, once such a reflecting surface is obtained, we can utilize it to draw an arbitrary number of independent and uncorrelated samples from the posterior measure for Bayesian inverse problems. In theory, we have shown that the geometric optics approximation measure is well-posed. The efficiency and robustness of our proposed sampler, employing the geometric optics approximation method, are demonstrated through several numerical examples provided in this paper.

In this paper, we develop a general mathematical framework for enhanced hydrodynamic near-cloaking of electro-osmotic flow for more complex shapes, which is obtained by simultaneously perturbing the inner and outer boundaries of the perfect cloaking structure. We first derive the asymptotic expansions of perturbed fields and obtain a first-order coupled system. We then establish the representation formula of the solution to the first-order coupled system using the layer potential techniques. Based on the asymptotic analysis, the enhanced hydrodynamic near-cloaking conditions are derived for the control region with general cross-sectional shape. The conditions reveal the inner relationship between the shapes of the object and the control region. Especially, for the shape of a deformed annulus or confocal ellipses cylinder, the cloaking conditions and relationship of shapes are quantified more accurately. Our theoretical findings are validated and supplemented by a variety of numerical results. The results in this paper also provide a mathematical foundation for more complex hydrodynamic cloaking.

Characterizing urban environments with broad coverages and high precision is more important than ever for achieving the UN's Sustainable Development Goals (SDGs) as half of the world's populations are living in cities. Urban building height as a fundamental 3D urban structural feature has far-reaching applications. However, so far, producing readily available datasets of recent urban building heights with fine spatial resolutions and global coverages remains a challenging task. Here, we provide an up-to-date global product of urban building heights based on a fine grid size of 150 m around 2020 by combining the spaceborne lidar instrument of GEDI and multi-sourced data including remotely sensed images (i.e., Landsat-8, Sentinel-2, and Sentinel-1) and topographic data. Our results revealed that the estimated method of building height samples based on the GEDI data was effective with 0.78 of Pearson's r and 3.67 m of RMSE in comparison to the reference data. The mapping product also demonstrated good performance as indicated by its strong correlation with the reference data (i.e., Pearson's r = 0.71, RMSE = 4.60 m). Compared with the currently existing products, our global urban building height map holds the ability to provide a higher spatial resolution (i.e., 150 m) with a great level of inherent details about the spatial heterogeneity and flexibility of updating using the GEDI samples as inputs. This work will boost future urban studies across many fields including climate, environmental, ecological, and social sciences.

This paper studies distributed continuous-time optimization for time-varying quadratic cost functions with uncertain parameters. We first propose a centralized adaptive optimization algorithm using partial information of the cost function. It can be seen that even if there are uncertain parameters in the cost function, exact optimization can still be achieved. To solve this problem in a distributed manner when different local cost functions have identical Hessians, we propose a novel distributed algorithm that cascades the fixed-time average estimator and the distributed optimizer. We remove the requirement for the upper bounds of certain complex functions by integrating state-based gains in the proposed design. We further extend this result to address the distributed optimization where the time-varying cost functions have nonidentical Hessians. We prove the convergence of all the proposed algorithms in the global sense. Numerical examples verify the proposed algorithms., Comment: 11 pages,10 figures

In this paper, we propose and study the utilization of the Dirichlet-to-Neumann (DN) map to uniquely identify the discount functions $r, k$ and cost function $F$ in a stationary mean field game (MFG) system. This study features several technical novelties that make it highly intriguing and challenging. Firstly, it involves a coupling of two nonlinear elliptic partial differential equations. Secondly, the simultaneous recovery of multiple parameters poses a significant implementation challenge. Thirdly, there is the probability measure constraint of the coupled equations to consider. Finally, the limited information available from partial boundary measurements adds another layer of complexity to the problem. Considering these challenges and problems, we present an enhanced higher-order linearization method to tackle the inverse problem related to the MFG system. Our proposed approach involves linearizing around a pair of zero solutions and fulfilling the probability measurement constraint by adjusting the positive input at the boundary. It is worth emphasizing that this technique is not only applicable for uniquely identifying multiple parameters using full-boundary measurements but also highly effective for utilizing partial-boundary measurements.

In this paper, we propose and study several inverse problems of identifying/determining unknown coefficients for a class of coupled PDE systems by measuring the average flux data on part of the underlying boundary. In these coupled systems, we mainly consider the non-negative solutions of the coupled equations, which are consistent with realistic settings in biology and ecology. There are several salient features of our inverse problem study: the drastic reduction of the measurement/observation data due to averaging effects, the nonlinear coupling of multiple equations, and the non-negative constraints on the solutions, which pose significant challenges to the inverse problems. We develop a new and effective scheme to tackle the inverse problems and achieve unique identifiability results by properly controlling the injection of different source terms to obtain multiple sets of mean flux data. The approach relies on certain monotonicity properties which are related to the intrinsic structures of the coupled PDE system. We also connect our study to biological applications of practical interest.

In this paper, a deep reinforcement learning based method is proposed to obtain optimal policies for optimal infinite-horizon control of probabilistic Boolean control networks (PBCNs). Compared with the existing literatures, the proposed method is model-free, namely, the system model and the initial states needn't to be known. Meanwhile, it is suitable for large-scale PBCNs. First, we establish the connection between deep reinforcement learning and optimal infinite-horizon control, and structure the problem into the framework of the Markov decision process. Then, PBCNs are defined as large-scale or small-scale, depending on whether the memory of the action-values exceeds the RAM of the computer. Based on the newly introduced definition, Q-learning (QL) and double deep Q-network (DDQN) are applied to the optimal infinite-horizon control of small-scale and large-scale PBCNs, respectively. Meanwhile, the optimal state feedback controllers are designed. Finally, two examples are presented, which are a small-scale PBCN with 3 nodes, and a large-scale one with 28 nodes. To verify the convergence of QL and DDQN, the optimal control policy and the optimal action-values, which are obtained from both the algorithms, are compared with the ones based on a model-based method named policy iteration. Meanwhile, the performance of QL is compared with DDQN in the small-scale PBCN.

In this paper, we develop a general mathematical framework for perfect and approximate hydrodynamic cloaking and shielding of electro-osmotic flow, which is governed by a coupled PDE system via the field-effect electro-osmosis. We first establish the representation formula of the solution of the coupled system using the layer potential techniques. Based on Fourier series, the perfect hydrodynamic cloaking and shielding conditions are derived for the control region with the cross-sectional shape being annulus or confocal ellipses. Then we further propose an optimization scheme for the design of approximate cloaks and shields within general geometries. The well-posedness of the optimization problem is proved. In particular, the condition that can ensure the occurrence of approximate cloaks and shields for general geometries are also established. Our theoretical findings are validated and supplemented by a variety of numerical results. The results in this paper also provide a mathematical foundation for more complex hydrodynamic cloaking and shielding.

The current quantum reinforcement learning control models often assume that the quantum states are known a priori for control optimization. However, full observation of quantum state is experimentally infeasible due to the exponential scaling of the number of required quantum measurements on the number of qubits. In this paper, we investigate a robust reinforcement learning method using partial observations to overcome this difficulty. This control scheme is compatible with near-term quantum devices, where the noise is prevalent and predetermining the dynamics of quantum state is practically impossible. We show that this simplified control scheme can achieve similar or even better performance when compared to the conventional methods relying on full observation. We demonstrate the effectiveness of this scheme on examples of quantum state control and quantum approximate optimization algorithm. It has been shown that high-fidelity state control can be achieved even if the noise amplitude is at the same level as the control amplitude. Besides, an acceptable level of optimization accuracy can be achieved for QAOA with noisy control Hamiltonian. This robust control optimization model can be trained to compensate the uncertainties in practical quantum computing.