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Multi-level methods are widely used for the solution of large-scale problems, because of their computational advantages and exploitation of the complementarity between the involved sub-problems. After a re-interpretation of multi-level methods from a block-coordinate point of view, we propose a multi-level algorithm for the solution of nonlinear optimization problems and analyze its evaluation complexity. We apply it to the solution of partial differential equations using physics-informed neural networks (PINNs) and consider two different types of neural architectures, a generic feedforward network and a frequency-aware network. We show that our approach is particularly effective if coupled with these specialized architectures and that this coupling results in better solutions and significant computational savings. [ABSTRACT FROM AUTHOR]

This paper presents an investigation into the stability and control aspects of delayed partial differential equation (PDE) systems utilizing the Lyapunov method. PDEs serve as powerful mathematical tools for modeling diverse and intricate systems such as heat transfer processes, chemical reactors, flexible arms, and population dynamics. However, the presence of delays within the feedback loop of such systems can introduce significant challenges, as even minor delays can potentially trigger system instability. To address this issue, the Lyapunov method, renowned for its efficacy in stability analysis, is employed to assess the exponential stability of a specific cohort of delayed PDE systems. By adopting Dirichlet boundary conditions and incorporating delay-dependent techniques such as the Galerkin method and Halanay inequality, the inherent stability properties of these systems are rigorously examined. Notably, the utilization of Dirichlet boundary conditions in this study allows for simplified analysis, and it is worth mentioning that the stability analysis outcomes under Neumann conditions and combined boundary conditions align with those of the Dirichlet boundary conditions discussed herein. Furthermore, this research endeavor delves into the implications of the obtained results in terms of control considerations and convergence rates. The integration of the Galerkin method aids in approximating the behavior of dominant modes within the system, thereby enabling a more comprehensive understanding of stability and control. The exploration of convergence rates provides valuable insights into the speed at which stability is achieved in practice, thus enhancing the practical applicability of the findings. The outcomes of this study contribute significantly to the broader comprehension and effective control of delayed PDE systems. The elucidation of stability behaviors not only provides a comprehensive understanding of the impact of delays but also offers practical insights for the design and implementation of control strategies in various domains. Ultimately, this research strives to enhance the stability and reliability of complex systems represented by PDEs, thereby facilitating their effective utilization across numerous scientific and engineering applications.

Different types of DEs have a wide range of applications in both engineering and science. Typically, when modelling a physical quantity's variation, Recently, it has been found that models based on the theory of fractional-order derivatives and integrals provide an exceptionally good description of a wide range of scientific phenomena. The FDE is a differential equation that contains some derivatives of non-integer powers order. FDEs have become increasingly significant in the theoretical and applied parts of a wide variety of scientific and technical disciplines in recent years. The high-order ODE can be reduced to systems of first order ODEs, which can then be solved. Directly attacking the issue with numerical methods, however, is much more efficient in terms of accuracy, number of function evaluations, and processing time. In this article, the RKM method for solving ordinary differential equations has been introduced. This numerical approach has been generalized to be suitable for solving a class of fractional differential equations (FDEs). However, the developed RKM approach with three- and four-stages for solving sixth-order FODEs is developed. Moreover, this technique was used to solve various test problems, these examples to which the developed method was applied were for various functions with different values of ⍺ and then, the solutions of the developed numerical method were compared with the exact solution, the numerical results proved the efficiency and accuracy of the modified technique. [ABSTRACT FROM AUTHOR]

In this paper, we establish the pricing framework of European options in the presence of the looping contagion risk. First, the looping contagion risk model is transformed into a martingale form and the semi-explicit risk-neutral pricing formula is established for the pricing of European options. Then the pricing partial differential equations (PDEs) are derived and solved by the stable alternating direction implicit (ADI) methods. Moreover, the ADI methods in this paper are proved to be unconditionally stable through Fourier analysis framework. Finally, the Monte Carlo simulations are performed and combined with the numerical solutions of PDEs to compute the desired option prices via the semi-explicit pricing formula. Numerical examples are given to confirm the convergence of the numerical methods and the economic analysis is provided to illustrate the economic effect of the looping contagion risk on option prices. [ABSTRACT FROM AUTHOR]

Medical imaging, the process of visual representation of different organs and tissues of the human body, is employed for monitoring the normal as well as abnormal anatomy and physiology of the body. Imaging which can provide healthcare solutions ensuring a regular measurement of various complex diseases plays a critical role in the diagnosis and management of many complex diseases and medical conditions, and the quality of a medical image, which is not a single factor but a composite of contrast, artifacts, distortion, noise, blur, and so forth, depends on several factors such as the characteristics of the equipment, the imaging method in question as well as the imaging variables chosen by the operator. The medical images (ultrasound image, X-rays, CT scans, MRIs, etc.) may lose significant features and become degraded due to the emergence of noise as a result of which the process of improvement pertaining to medical images has become a thought-provoking area of inquiry with challenges related to detecting the speckle noise in the images and finding the applicable solution in a timely manner. The partial differential equations (PDEs), in this sense, can be used extensively in different aspects with regard to image processing ranging from filtering to restoration, segmentation to edge enhancement and detection, denoising in particular, among the other ones. In this research paper, we present a conformable fractional derivative-based anisotropic diffusion model for removing speckle noise in ultrasound images. The proposed model providing to be efficient in reducing noise by preserving the essential image features like edges, corners and other sharp structures for ultrasound images in comparison to the classical anisotropic diffusion model. Furthermore, we aim at proving the viscosity solution of the fractional diffusion model. The finite difference method is used to discretize the fractional diffusion model and classical diffusion models. The peak signal-to-noise ratio (PSNR) is used for the quality of the smooth images. The comparative experimental results corroborate that the proposed, developed and extended mathematical model is capable of denoising and preserving the significant features in ultrasound towards better accuracy, precision and examination within the framework of biomedical imaging and other related medical, clinical, and image-signal related applied as well as computational processes.

Partial differential equations (PDEs) are essential mathematical models for describing a wide range of physical phenomena. Numerically, Physic-Informed Neural Networks (PINNs), a variant of artificial neural networks, present a promising method for solving PDEs. However, due to limitation in accuracy and stability, various adaptive PINN variants have been proposed. We have designed a novel approach that adopted self-adaptive PINN (SA-PINN) with two optimization techniques: the genetic algorithm (GA) and the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. Self-adaptive PINN modifies the weights in the loss function to be fully trainable, enabling the ANN to learn and stabilize the PINN in approximating the difficult regions of the solution. GA initializes the population of ANN trainable parameters to optimize the training process with less number of iterations, while L-BFGS is used to find the best solution accurately. Our proposed approach, named SA-PINN-GA-LBFGS, is tested on solving several benchmark PDE problems including elliptic, parabolic, and hyperbolic types. We compare our results with state-of-the-art methods, demonstrating that SA-PINN-GA-LBFGS provides higher accuracy and greater efficiency.

Medical imaging, the process of visual representation of different organs and tissues of the human body, is employed for monitoring the normal as well as abnormal anatomy and physiology of the body. Imaging which can provide healthcare solutions ensuring a regular measurement of various complex diseases plays a critical role in the diagnosis and management of many complex diseases and medical conditions, and the quality of a medical image, which is not a single factor but a composite of contrast, artifacts, distortion, noise, blur, and so forth, depends on several factors such as the characteristics of the equipment, the imaging method in question as well as the imaging variables chosen by the operator. The medical images (ultrasound image, X-rays, CT scans, MRIs, etc.) may lose significant features and become degraded due to the emergence of noise as a result of which the process of improvement pertaining to medical images has become a thought-provoking area of inquiry with challenges related to detecting the speckle noise in the images and finding the applicable solution in a timely manner. The partial differential equations (PDEs), in this sense, can be used extensively in different aspects with regard to image processing ranging from filtering to restoration, segmentation to edge enhancement and detection, denoising in particular, among the other ones. In this research paper, we present a conformable fractional derivative-based anisotropic diffusion model for removing speckle noise in ultrasound images. The proposed model providing to be efficient in reducing noise by preserving the essential image features like edges, corners and other sharp structures for ultrasound images in comparison to the classical anisotropic diffusion model. Furthermore, we aim at proving the viscosity solution of the fractional diffusion model. The finite difference method is used to discretize the fractional diffusion model and classical diffusion models. The peak signal-to-noise ratio (PSNR) is used for the quality of the smooth images. The comparative experimental results corroborate that the proposed, developed and extended mathematical model is capable of denoising and preserving the significant features in ultrasound towards better accuracy, precision and examination within the framework of biomedical imaging and other related medical, clinical, and image-signal related applied as well as computational processes. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) describe simulation of physical phenomena occurring in different fields of science and engineering. The analysis of solutions of nonlinear wave equations have been gaining a lot of popularity in the last two decades. Such wave equations have many applications in applied mathematics and theoretical physics. The importance of PDEs that explain nonlinear waves defined by Sine-Gordon (SG) equation are crucial. The SG equation is a particular instance of the Klein-Gordon (KG) equation, which is crucial in a number of scientific fields, such as solid state physics, nonlinear optics, and quantum field theory. This equation is also a description of a soliton wave that exists in many physical situations. Different analytical as well as numerical techniques were used to develop the exact and approximate solution of SG equation. In this article, we explore the numerical solution of the one-dimensional nonlinear SG problem using RBF-FD approach. The scheme is a combination of radial basis functions (RBFs) with finite differences (FD) for constructing local spatial approximations to SG equation. For execution of time variable in the given model equation, Runge-Kutta (RK) time steeping approach is utilized. To verify the validity of our method, solutions to some test problems are examined. Accuracy of the proposed scheme is verified through L1, L2, "RMS" and "MAE" error norms. The solutions acquired by suggested RBF-FD approach are also compared to earlier work and the obtained results are batter and in good agreement with the exact solution. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) are essential mathematical models for describing a wide range of physical phenomena. Numerically, Physic-Informed Neural Networks (PINNs), a variant of artificial neural networks, present a promising method for solving PDEs. However, due to limitation in accuracy and stability, various adaptive PINN variants have been proposed. We have designed a novel approach that adopted self-adaptive PINN (SA-PINN) with two optimization techniques: the genetic algorithm (GA) and the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. Self-adaptive PINN modifies the weights in the loss function to be fully trainable, enabling the ANN to learn and stabilize the PINN in approximating the difficult regions of the solution. GA initializes the population of ANN trainable parameters to optimize the training process with less number of iterations, while L-BFGS is used to find the best solution accurately. Our proposed approach, named SA-PINN-GA-LBFGS, is tested on solving several benchmark PDE problems including elliptic, parabolic, and hyperbolic types. We compare our results with state-of-the-art methods, demonstrating that SA-PINN-GA-LBFGS provides higher accuracy and greater efficiency. [ABSTRACT FROM AUTHOR]

Computation of injective (or inversion-free) maps is a key task in geometry processing, physical simulation, and shape optimization. Despite being a longstanding problem, it remains challenging due to its highly nonconvex and combinatoric nature. We propose computation of variational quasi-harmonic maps to obtain smooth inversion-free maps. Our work is built on a key observation about inversion-free maps: A planar map is a diffeomorphism if and only if it is quasi-harmonic and satisfies a special Cauchy boundary condition. We hence equate the inversion-free mapping problem to an optimal control problem derived from our theoretical result, in which we search in the space of parameters that define an elliptic PDE. We show that this problem can be solved by minimizing within a family of functionals. Similarly, our discretized functionals admit exactly injective maps as the minimizers, empirically producing inversion-free discrete maps of triangle meshes. We design efficient numerical procedures for our problem that prioritize robust convergence paths. Experiments show that on challenging examples our methods can achieve up to orders of magnitude improvement over state-of-the-art, in terms of speed or quality. Moreover, we demonstrate how to optimize a generic energy in our framework while restricting to quasi-harmonic maps. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) provide a new approach to solving partial differential equations (PDEs), while the properties of coupled physical laws present potential in surrogate modeling. However, the accuracy of PINNs in solving forward problems needs to be enhanced, and solving inverse problems relies on data samples. The smoothed finite element method (S-FEM) can obtain high-fidelity numerical solutions, which are easy to solve for the forward problems of PDEs, but difficult to solve for the inverse problems. To the best of the authors' knowledge, there has been no prior research on coupling S-FEM and PINN. In this paper, a novel approach that couples S-FEM and PINN is proposed. The proposed approach utilizes S-FEM to synthesize high-fidelity datasets required for PINN inversion, while also improving the accuracy of data-independent PINN in solving forward problems. The proposed approach is applied to solve linear elastic and elastoplastic forward and inverse problems. The computational results demonstrate that the coupling of the S-FEM and PINN exhibits high precision and convergence when solving inverse problems, achieving a maximum relative error of 0.2% in linear elasticity and 5.69% in elastoplastic inversion by using approximately 10,000 data points. The coupling approach also enhances the accuracy of solving forward problems, reducing relative errors by approximately 2–10 times. The proposed coupling of the S-FEM and PINN offers a novel surrogate modeling approach that incorporates knowledge and data-driven techniques, enabling it to solve both forward and inverse problems associated with PDEs with high levels of accuracy and convergence. [ABSTRACT FROM AUTHOR]

Methods of reduced order modelling (ROM) of nonlinear PDEs to ODEs based on investigation of flexible arbitrary geometry of the plan nano-plates have been reviewed, modified, and employed. The modified von Kármán equations serve as the paradigmatic object of investigation. They are derived in the framework of the modified coupled stress theory with the inclusion of the von Kármán geometric nonlinearity. The elliptic type PDEs are reduced to a system of ODEs using several methods, and the results are compared in order to quantify them for rectangular nano-plates. The following ROM methods have been used: Fourier method of separation of variables, the Kantorovich-Vlasov method (KVM), the variational iterations method (VIM) known as the extended Kantorovich method (EKM), the Agranovskii–Baglai–Smirnov method (ABSM) in combination with the KVM and VIM. We have found that the solutions obtained by combining VIM with the ABSM can be considered exact solutions of the von Kármán equations. The convergence of the developed iterative procedures is justified by a number of theorems for the static analysis of nanoplates on a rectangular plane. [ABSTRACT FROM AUTHOR]

Optimal control problems are a class of optimisation problems with partial differential equations as constraints. These problems arise in many application areas of science and engineering. The finite element method was used to transform the optimal control problems of an elliptic partial differential equation into a system of linear equations of saddle point form. The main focus of this paper is to characterise and exploit the structure of the coeficient matrix of the saddle point system to build an eficient numerical process. These systems are of large dimension, block, sparse, indefinite and ill conditioned. The numerical solution of saddle point problems is a computational task since well known numerical schemes perform poorly if they are not properly preconditioned. The main task of this paper is to construct a preconditioner the mimic the structure of the system coeficient matrix to accelerate the convergence of the generalised minimal residual method. Explicit expression of the eigenvalue and eigenvectors for the preconditioned matrix are derived. The main outcome is to achieve optimal convergence results in a small number of iterations with respect to the decreasing mesh size h and the changes in fi the regularisation problem parameters. The numerical results demonstrate the effectiveness and performance of the proposed preconditioner compared to the other existing preconditioners and confirm theoretical results. [ABSTRACT FROM AUTHOR]

In this work we develop the standard Hermite interpolation based RBF-generated finite difference (RBF-HFD) method into a new faster and more accurate technique based on partition of unity (PU) method. In the new approach, much fewer number of local linear systems needs to be solved for calculating the stencil weights. This reduces the computational cost of the method, remarkably. In addition, the method is flexible in using different types of PU weight functions, smooth or discontinuous, each results in a different scheme with additional nice properties. We also investigate the scaling property of polyharmonic spline (PHS) kernels to develop a simple and stable algorithm for computing local approximants in PU patches. Experimental results confirm the efficiency and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

The study of rotating-disk heat-flow problems is relevant to computer storage devices, rotating machineries, heat-storage devices, MHD rotators, lubrication, and food-processing devices. Therefore, this study investigated the effects of a Hall current and motile microorganisms on nanofluid flow generated by the spinning of a disk under multiple slip and thermal radiation conditions. The Buongiorno model of a nonhomogeneous nanofluid under Brownian diffusion and thermophoresis was applied. Using the Taylor series, the effect of Resseland radiation was linearized and included in the energy equation. By implementing the appropriate transformations, the governing partial differential equations (PDEs) were simplified into a two-point ordinary boundary value problem. The classical Runge–Kutta dependent shooting method was used to find the numerical solutions, which were validated using the data available in the literature. The velocity, motile microorganism distribution, temperature, and concentration of nanoparticles were plotted and comprehensively analyzed. Moreover, the density number, Sherwood number, shear stresses, and Nusselt number were calculated. The radial and tangential velocity declined with varying values of magnetic numbers, while the concentration of nanoparticles, motile microorganism distribution, and temperature increased. There was a significant reduction in heat transfer, velocities, and motile microorganism distribution under the multiple slip conditions. The Hall current magnified the velocities and reduced the heat transfer. Thermal radiation improved the Nusselt number, while the thermal slip conditions reduced the Nusselt number. [ABSTRACT FROM AUTHOR]

Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present a feasibility study of applying physics-informed deep learning methods for solving PDEs related to the physical laws of electromagnetics. The methodology uses automatic differentiation, and the loss function is formulated based on the underlying PDE and boundary conditions. The feasibility of the method is shown using three electromagnetic problems of varying complexity and the results show close agreement with the ground truth from a finite-element analysis solver. The application of transfer learning is also explored and results in faster training. Furthermore, a hybrid approach involving physics-based governing equations and labeled data is also introduced to improve the accuracy of the results. [ABSTRACT FROM AUTHOR]

In this article, we present a data-free method to solve differential equation using artificial neural networks (ANN). This method exploits the universal function approximation nature of a neural network to mimic a specified partial differential equation (PDE) and provide its solution. Specifically, use of simple feed-forward artificial neural network (FF-ANN) is shown to demonstrate the solution of 1-D second-order differential equations without use of any prior data. The article also demonstrates that the similarity in two PDEs is akin to similarity in the optimized weight matrices attained after the solution using FF-ANN. This property is then utilized for a transferred learning to enable faster convergence of a new PDE, based on the prior solution of a similar PDE. The concept is shown while considering a general form of second-order PDE, and considering specific cases of scalar in-homogeneous wave equation form and Poisson Equation form. Error convergence below 10−6 is shown and the transferred learning process shows typical time acceleration by a factor of 1.5–3 for the considered equations. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) provide a new approach to solving partial differential equations (PDEs), while the properties of coupled physical laws present potential in surrogate modeling. However, the accuracy of PINNs in solving forward problems needs to be enhanced, and solving inverse problems relies on data samples. The smoothed finite element method (S-FEM) can obtain high-fidelity numerical solutions, which are easy to solve for the forward problems of PDEs, but difficult to solve for the inverse problems. To the best of the authors’ knowledge, there has been no prior research on coupling S-FEM and PINN. In this paper, a novel approach that couples S-FEM and PINN is proposed. The proposed approach utilizes S-FEM to synthesize high-fidelity datasets required for PINN inversion, while also improving the accuracy of data-independent PINN in solving forward problems. The proposed approach is applied to solve linear elastic and elastoplastic forward and inverse problems. The computational results demonstrate that the coupling of the S-FEM and PINN exhibits high precision and convergence when solving inverse problems, achieving a maximum relative error of 0.2% in linear elasticity and 5.69% in elastoplastic inversion by using approximately 10,000 data points. The coupling approach also enhances the accuracy of solving forward problems, reducing relative errors by approximately 2–10 times. The proposed coupling of the S-FEM and PINN offers a novel surrogate modeling approach that incorporates knowledge and data-driven techniques, enabling it to solve both forward and inverse problems associated with PDEs with high levels of accuracy and convergence.

The components of complex analytic functions define solutions for the Laplace's equation, and in a simply connected domain, each solution of this equation is the first component of a complex analytic function. In this paper, we generalize this result; for each PDE of the form Auxx+Buxy+Cuyy=0, and for each affine planar vector field φ, we give an algebra 픸 with unit e = e1, with respect to which the components of all functions of the form L∘φ are all the solutions for this PDE, where L is differentiable in the sense of Lorch with respect to 픸. Solutions are also constructed for the following equations: Auxx+Buxy+Cuyy+Dux+Euy+Fu=0, 3rd‐order PDEs, and 4th‐order PDEs; among these are the bi‐harmonic, the bi‐wave, and the bi‐telegraph equations. [ABSTRACT FROM AUTHOR]

In recent years, compute-in-memory (CIM) has been extensively studied to improve the energy efficiency of computing by reducing data movement. At present, CIM is frequently used in data-intensive computing. Data-intensive computing applications, such as all kinds of neural networks (NNs) in machine learning (ML), are regarded as 'soft' computing tasks. The 'soft' computing tasks are computations that can tolerate low computing precision with little accuracy degradation. However, 'hard' tasks aimed at numerical computations require high-precision computing and are also accompanied by energy efficiency problems. Numerical computations exist in lots of applications, including partial differential equations (PDEs) and large-scale matrix multiplication. Therefore, it is necessary to study CIM for numerical computations. This article reviews the recent developments of CIM for numerical computations. The different kinds of numerical methods solving partial differential equations and the transformation of matrixes are deduced in detail. This paper also discusses the iterative computation of a large-scale matrix, which tremendously affects the efficiency of numerical computations. The working procedure of the ReRAM-based partial differential equation solver is emphatically introduced. Moreover, other PDEs solvers, and other research about CIM for numerical computations, are also summarized. Finally, prospects and the future of CIM for numerical computations with high accuracy are discussed. [ABSTRACT FROM AUTHOR]

We present and mathematically analyze an online adjoint algorithm for the optimization of partial differential equations (PDEs). Traditional adjoint algorithms would typically solve a new adjoint PDE at each optimization iteration, which can be computationally costly. In contrast, an online adjoint algorithm updates the design variables in continuous-time and thus constantly makes progress towards minimizing the objective function. The online adjoint algorithm we consider is similar in spirit to the the pseudo-time-stepping, one-shot method which has been previously proposed. Motivated by the application of such methods to engineering problems, we mathematically study the convergence of the online adjoint algorithm. The online adjoint algorithm relies upon a time-relaxed adjoint PDE which provides an estimate of the direction of steepest descent. The algorithm updates this estimate continuously in time, and it asymptotically converges to the exact direction of steepest descent as t → ∞ . We rigorously prove that the online adjoint algorithm converges to a critical point of the objective function for optimizing the PDE. Under appropriate technical conditions, we also prove a convergence rate for the algorithm. A crucial step in the convergence proof is a multi-scale analysis of the coupled system for the forward PDE, adjoint PDE, and the gradient descent ODE for the design variables. [ABSTRACT FROM AUTHOR]

This paper is concerned with the proportional integral (PI) regulation control of the left Neumann trace of a one-dimensional semilinear wave equation. The control input is selected as the right Neumann trace. The control design goes as follows. First, a preliminary (classical) velocity feedback is applied in order to shift all but a finite number of the eivenvalues of the underlying unbounded operator into the open left half-plane. We then leverage the projection of the system trajectories into an adequate Riesz basis to obtain a truncated model of the system capturing the remaining unstable modes. The controller is computed by applying a classical PI control design scheme to this truncated model. Local stability of the resulting closed-loop infinite-dimensional system is obtained through the study of an adequate Lyapunov function. Finally, an estimate assessing the set point tracking performance of the left Neumann trace is derived. [ABSTRACT FROM AUTHOR]

The recent deep learning revolution has created enormous opportunities for accelerating compute capabilities in the context of physics-based simulations. In this article, we propose EikoNet, a deep learning approach to solving the Eikonal equation, which characterizes the first-arrival-time field in heterogeneous 3-D velocity structures. Our grid-free approach allows for rapid determination of the travel time between any two points within a continuous 3-D domain. These travel time solutions are allowed to violate the differential equation—which casts the problem as one of optimization—with the goal of finding network parameters that minimize the degree to which the equation is violated. In doing so, the method exploits the differentiability of neural networks to calculate the spatial gradients analytically, meaning that the network can be trained on its own without ever needing solutions from a finite-difference algorithm. EikoNet is rigorously tested on several velocity models and sampling methods to demonstrate robustness and versatility. Training and inference are highly parallelized, making the approach well-suited for GPUs. EikoNet has low memory overhead and further avoids the need for travel-time lookup tables. The developed approach has important applications to earthquake hypocenter inversion, ray multipathing, and tomographic modeling, as well as to other fields beyond seismology where ray tracing is essential. [ABSTRACT FROM AUTHOR]

This study aims at designing model-based controllers for a hot-melt extrusion process designed for the manufacturing of pills. The control objective is to regulate the output mass flow rate and the active pharmaceutical ingredient (API) concentration. The first part of this study is concerned with the derivation of reduced-order models from a detailed distributed parameter (DP) model involving mass and energy balance partial differential equations. The parameters of these reduced-order models are estimated based on experimental data from a laboratory device presenting a specific screw geometry, and validated against the detailed DP model. The latter is too computationally expensive with regards to process control, but can serve as a process emulator to test control strategies. The second part of this study focuses on several model predictive control strategies, ranging from Smith predictor to NEPSAC controllers. The effectiveness of these control strategies is assessed using numerical simulation and performance indicators such as disturbance rejection and computational cost. [ABSTRACT FROM AUTHOR]

The study of rotating-disk heat-flow problems is relevant to computer storage devices, rotating machineries, heat-storage devices, MHD rotators, lubrication, and food-processing devices. Therefore, this study investigated the effects of a Hall current and motile microorganisms on nanofluid flow generated by the spinning of a disk under multiple slip and thermal radiation conditions. The Buongiorno model of a nonhomogeneous nanofluid under Brownian diffusion and thermophoresis was applied. Using the Taylor series, the effect of Resseland radiation was linearized and included in the energy equation. By implementing the appropriate transformations, the governing partial differential equations (PDEs) were simplified into a two-point ordinary boundary value problem. The classical Runge–Kutta dependent shooting method was used to find the numerical solutions, which were validated using the data available in the literature. The velocity, motile microorganism distribution, temperature, and concentration of nanoparticles were plotted and comprehensively analyzed. Moreover, the density number, Sherwood number, shear stresses, and Nusselt number were calculated. The radial and tangential velocity declined with varying values of magnetic numbers, while the concentration of nanoparticles, motile microorganism distribution, and temperature increased. There was a significant reduction in heat transfer, velocities, and motile microorganism distribution under the multiple slip conditions. The Hall current magnified the velocities and reduced the heat transfer. Thermal radiation improved the Nusselt number, while the thermal slip conditions reduced the Nusselt number.

This article addresses the H∞ stabilization problems for a class of nonlinear distributed parameter systems which is described by the first-order hyperbolic partial differential equations (PDEs). First, the first-order hyperbolic PDE systems are identified as a polynomial fuzzy PDE system and the polynomial fuzzy controller for the polynomial fuzzy PDE system is proposed. By utilizing the proposed homogeneous polynomial Lyapunov functional, Euler’s homogeneous function theorem, and the proposed theorems, a spatial derivative sum-of-squares (SDSOS) exponential stabilization condition is proposed. In addition, a recursive algorithm for the SDSOS exponential stabilization condition is developed to find the feasible solution. Furthermore, in order to reduce the conservatism of the proposed results, a relaxed H∞ stabilization condition for the polynomial fuzzy PDE system is provided. Finally, the nonisothermal plug-flow reactor (PFR) is used to demonstrate the effectiveness and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

This article deals with the stability analysis of a drilling pipe controlled by a PI controller. The model is a coupled ordinary differential equation/partial differential equation (PDE) and is consequently of infinite dimension. Using recent advances in time-delay systems, we derive a new Lyapunov functional based on a state extension made up of projections of the Riemann coordinates. First, we will provide an exponential stability result expressed using the linear matrix inequality (LMI) framework. This result is dedicated to a linear version of the torsional dynamic. On the other hand, the influence of the nonlinear friction force, which may generate the well-known stick-slip phenomenon, is analyzed through a new stability theorem. Numerical simulations show the effectiveness of the method and that the stick-slip oscillations cannot be weakened using a PI controller. [ABSTRACT FROM AUTHOR]

In recent years, compute-in-memory (CIM) has been extensively studied to improve the energy efficiency of computing by reducing data movement. At present, CIM is frequently used in data-intensive computing. Data-intensive computing applications, such as all kinds of neural networks (NNs) in machine learning (ML), are regarded as ‘soft’ computing tasks. The ‘soft’ computing tasks are computations that can tolerate low computing precision with little accuracy degradation. However, ‘hard’ tasks aimed at numerical computations require high-precision computing and are also accompanied by energy efficiency problems. Numerical computations exist in lots of applications, including partial differential equations (PDEs) and large-scale matrix multiplication. Therefore, it is necessary to study CIM for numerical computations. This article reviews the recent developments of CIM for numerical computations. The different kinds of numerical methods solving partial differential equations and the transformation of matrixes are deduced in detail. This paper also discusses the iterative computation of a large-scale matrix, which tremendously affects the efficiency of numerical computations. The working procedure of the ReRAM-based partial differential equation solver is emphatically introduced. Moreover, other PDEs solvers, and other research about CIM for numerical computations, are also summarized. Finally, prospects and the future of CIM for numerical computations with high accuracy are discussed.

In this paper, a hybrid image denoising algorithm based on directional diffusion is proposed. Specifically, we developed a new noise-removal model by combining the modified isotropic diffusion model and the modified Perona-Malik (PM) model. The novel hybrid model can adapt the diffusion process along the tangential direction of edges in the original image via a new control function based on the patch similarity modulus. In addition, the patch similarity modulus is used as the new structure indicator for the modified Perona-Malik model. The feature of second-order directional derivative of edge's tangential direction allows the proposed model to reduce the aliasing and the noise around edge during edge preserving smoothing. The proposed method is thus able to efficiently preserve the edges, textures, thin lines, weak edges, and fine details, meanwhile preventing the staircase effects. Computer experiments on synthetic image and nature images demonstrate that the proposed model achieves a better performance than the conventional partial differential equations models and some recent advanced models.

Hair loss in the autoimmune disease, alopecia areata (AA), is characterized by the appearance of circularly spreading alopecic lesions in seemingly healthy skin. The distinct spatial patterns of AA lesions form because the immune system attacks hair follicle cells that are in the process of producing hair shaft, catapults the mini-organs that produce hair from a state of growth (anagen) into an apoptosis-driven regression state (catagen), and causes major hair follicle dystrophy along with rapid hair shaft shedding. In this paper, we develop a model of partial differential equations (PDEs) to describe the spatio-temporal dynamics of immune system components that clinical and experimental studies show are primarily involved in the disease development. Global linear stability analysis reveals there is a most unstable mode giving rise to a pattern. The most unstable mode indicates a spatial scale consistent with results of the humanized AA mouse model of Gilhar et al. (Autoimmun Rev 15(7):726–735, 2016) for experimentally induced AA lesions. Numerical simulations of the PDE system confirm our analytic findings and illustrate the formation of a pattern that is characteristic of the spatio-temporal AA dynamics. We apply marginal linear stability analysis to examine and predict the pattern propagation. [ABSTRACT FROM AUTHOR]