PDE Model Research Articles

Output Feedback Energy Control of the Sine-Gordon PDE Model Using Collocated Spatially Sampled Sensing and Actuation

The present investigation focuses on the output energy control of the nonlinear sine-Gordon model in a practical situation where only spatially sampled sensing and actuation are available. The speed-gradient method that has corroborated its utility for the sine-Gordon model, controlled through the manipulable parameters of the external field and boundary actuation, is now generalized to the spatially sampled sensing and actuation, aiming to pump/dissipate the energy of the model to a desired level. Luenberger-type observers are additionally developed in the present nonlinear PDE setting to be involved into the output feedback synthesis over collocated state (position and/or velocity) measurements. Capabilities of the proposed synthesis are illustrated in numerical simulations.

A PDE model for bleb formation and interaction with linker proteins

AbstractThe aim of this paper is to further develop mathematical models for bleb formation in cells, including cell membrane interactions with linker proteins. This leads to nonlinear reaction–diffusion equations on a surface coupled to fluid dynamics in the bulk. We provide a detailed mathematical analysis and investigate some singular limits of the model, connecting it to previous literature. Moreover, we provide numerical simulations in different scenarios, confirming that the model can reproduce experimental results on bleb initiation.

A PDE model for the spatial dynamics of a voles population structured in age

We prove existence and stability of entropy weak solutions for a macroscopic PDE model for the spatial dynamics of a population of voles structured in age. The model consists of a scalar PDE depending on time, t, age, a, and space x=(x1,x2), supplemented with a non-local boundary condition at a=0. The flux is linear with constant coefficient in the age direction but contains a non-local term in the space directions. Also, the equation contains a term of second order in the space variables only. Existence of solutions is established by compensated compactness, see Panov (2009), and we prove stability by a doubling of variables type argument.

Structural inpainting techniques using equations of engineering physics

A comprehensive survey on structure-based inpainting methods based on equations of engineering physics is provided in this work. Diffusion equations, describing the random motion of the micro-particles in physics, have been successfully applied in the image processing areas in the last decades. Nonlinear anisotropic diffusion-based structural inpainting techniques, using variational and non-variational PDE models, are discussed first. Influential variational interpolation approaches, such as those using the Mumford–Shah functional, total variation inpainting and its versions, and Euler’s Elastica inpainting, are addressed here. Second and fourth-order anisotropic diffusion-based interpolation schemes, some of them not following variational principles, are then presented. Our own contributions in this field are also discussed. The applications of the reaction-diffusion equations, like Ginzburg–Landau equation, describing a large variety of physics phenomena, in the inpainting domain, are described next. Structural reconstructions methods using partial differential equations for fluid dynamics, such as the Navier–Stokes equations describing the motion of the viscous fluids, are then considered here. Other state of the art third-order PDE-based structural inpainting algorithms surveyed here are based on the curvature-driven diffusion equations that use the thermal diffusion principle in physics. Fourth-order PDE-based interpolation solutions, such as Cahn–Hilliard inpainting that uses a modified mathematical physics equation describing the phase separation process, are also described.

Analysis and simulation of a PDE model for surface relaxation

We study a fourth-order, singular, nonlinear PDE model for surface relaxation. A weak solution for the model is defined using an inequality formulation. A numerical scheme based on semi-implicit time stepping, mixed finite elements and regularization is proposed to approximate the PDE model. We investigate the convergence of the scheme with respect to the discretization and the regularization parameters. Finally, formal arguments show that the model can be viewed as a gradient flow with respect to an appropriate Riemannian metric.

The discontinuous Galerkin method for discretely observed Asian options

Asian options represent an important subclass of the path‐dependent contracts that are identified by payoff depending on the average of the underlying asset prices over the prespecified period of option lifetime. Commonly, this average is observed at discrete dates, and also, early exercise features can be admitted. As a result, analytical pricing formulae are not always available. Therefore, some form of a numerical approximation is essential for efficient option valuation. In this paper, we study a PDE model for pricing discretely observed arithmetic Asian options with fixed as well as floating strike for both European and American exercise features. The pricing equation for such options is similar to the Black‐Scholes equation with 1 underlying asset, and the corresponding average appears only in the jump conditions across the sampling dates. The objective of the paper is to present the comprehensive methodological concept that forms and improves the valuation process. We employ a robust numerical procedure based on the discontinuous Galerkin approach arising from the piecewise polynomial generally discontinuous approximations. This technique enables a simple treatment of discrete sampling by incorporation of jump conditions at each monitoring date. Moreover, an American early exercise constraint is directly handled as an additional nonlinear source term in the pricing equation. The proposed solving procedure is accompanied by an empirical study with practical results compared to reference values.

A PDE Model for Smooth Surface Reconstruction from 2D Parallel Slices

In this letter, we propose a PDE model for 3D smooth surfaces reconstruction from 2D parallel slices. One of the terms in the PDE model is derived from a magnitude penalization term, which enforces the gradient magnitude of the volume to the minimum point of a well-defined potential function. This gives us smooth reconstructed surfaces when assuming that the surfaces are given by the zero-isosurfaces of the volumes. The PDE model has a second term whose parameter controls the ability of the reconstructed surfaces to satisfy the slices constraint. At the end of this letter, we show experiments of employing the proposed reconstruction model to reconstruct surfaces. The numerical results demonstrate the utility of the model.

PDE MODELS OF ADDER MECHANISMS IN CELLULAR PROLIFERATION.

Cell division is a process that involves many biochemical steps and complex biophysical mechanisms. To simplify the understanding of what triggers cell division, three basic models that subsume more microscopic cellular processes associated with cell division have been proposed. Cells can divide based on the time elapsed since their birth, their size, and/or the volume added since their birth-the timer, sizer, and adder models, respectively. Here, we propose unified adder-sizer models and investigate some of the properties of different adder processes arising in cellular proliferation. Although the adder-sizer model provides a direct way to model cell population structure, we illustrate how it is mathematically related to the well-known model in which cell division depends on age and size. Existence and uniqueness of weak solutions to our 2+1-dimensional PDE model are proved, leading to the convergence of the discretized numerical solutions and allowing us to numerically compute the dynamics of cell population densities. We then generalize our PDE model to incorporate recent experimental findings of a system exhibiting mother-daughter correlations in cellular growth rates. Numerical experiments illustrating possible average cell volume blowup and the dynamical behavior of cell populations with mother-daughter correlated growth rates are carried out. Finally, motivated by new experimental findings, we extend our adder model cases where the controlling variable is the added size between DNA replication initiation points in the cell cycle.

Phase transitions of biological phenotypes by means of a prototypical PDE model

Abstract The basic investigation is the existence and the (numerical) observability of propagating fronts in the framework of the so-called Epithelial-to-Mesenchymal Transition and its reverse Mesenchymal-to-Epithelial Transition, which are known to play a crucial role in tumor development. To this aim, we propose a simplified one-dimensional hyperbolic-parabolic PDE model composed of two equations, one for the representative of the epithelial phenotype, and the second describing the mesenchymal phenotype. The system involves two positive constants, the relaxation time and a measure of invasiveness, moreover an essential feature is the presence of a nonlinear reaction function, typically assumed to be S-shaped. An identity characterizing the speed of propagation of the fronts is proven, together with numerical evidence of the existence of traveling waves. The latter is obtained by discretizing the system by means of an implicit-explicit finite difference scheme, then the algorithm is validated by checking the capability of the so-called LeVeque–Yee formula to reproduce the value of the speed furnished by the above cited identity. Once such justification has been achieved, we concentrate on numerical experiments relative to Riemann initial data connecting two stable stationary states of the underlying ODE model. In particular, we detect an explicit transition threshold separating regression regimes from invasive ones, which depends on critical values of the invasiveness parameter. Finally, we perform an extensive sensitivity analysis with respect to the system parameters, exhibiting a subtle dependence for those close to the threshold values, and we postulate some conjectures on the propagating fronts.

Using pde model and system dynamics model for describing multi-operation production lines

Two classes of models for describing production flow lines are analyzed. The use of models of these classes for the design of highly efficient control systems of production lines, the technological route of which consists of a large number of technological operations, is analyzed. The division of the technological route into a large number of operations is caused by the development trend of modern production lines. Synchronization of production line equipment performance is provided by an accumulating buffer. A formalized description of the production line was used as a foundation for constructing equations for each models class. The common features of using each models class in the description of production systems, as well as the conditions for their application, are shown. The form of the system dynamics model and PDE model equations is substantiated. The assumption about a deterministic rate of processing parts and the absence of a time delay and feedback between the parameters of technological operations was made when deriving the equations. The use of generalized technological operations in the system dynamics model as a way to reduce the number of model equations is discussed. Two limiting transitions from the PDE model equations to the system dynamics equations are demonstrated. It is shown that the system dynamics equations are a special case of the PDE model equations, the result of the aggregation of production line parameters within the technological operation. The method for constructing level equations for the system dynamics model is substantiated. For production lines with a different number of operations, the solution to the problem of processing parts along a production line is presented. The comparative analysis of the solutions obtained using the system dynamics and PDE model equations is obtained

Interacting Particles with Lévy Strategies: Limits of Transport Equations for Swarm Robotic Systems

L\'{e}vy robotic systems combine superdiffusive random movement with emergent collective behaviour from local communication and alignment in order to find rare targets or track objects. In this article we derive macroscopic fractional PDE descriptions from the movement strategies of the individual robots. Starting from a kinetic equation which describes the movement of robots based on alignment, collisions and occasional long distance runs according to a L\'{e}vy distribution, we obtain a system of evolution equations for the fractional diffusion for long times. We show that the system allows efficient parameter studies for a search problem, addressing basic questions like the optimal number of robots needed to cover an area in a certain time. For shorter times, in the hyperbolic limit of the kinetic equation, the PDE model is dominated by alignment, irrespective of the long range movement. This is in agreement with previous results in swarming of self-propelled particles. The article indicates the novel and quantitative modeling opportunities which swarm robotic systems provide for the study of both emergent collective behaviour and anomalous diffusion, on the respective time scales.

A Computation Approach to Chance Constrained Optimization of Boundary-Value Parabolic Partial Differential Equation Systems

Abstract This work studies chance constrained optimization of boundary-value parabolic partial differential equations (CCPDE) with random data, where the PDE model is treated as equality constraint and chance constraints are imposed on inequality constraints involving state variables. Since such a CCPDE problem is generally non-smooth, non-convex and difficult to solve directly, we use our recently proposed smoothing approximation method to solve the problem. As a result, the probability function of the chance constraints is approximated in two different ways by a family of differentiable functions. This leads to two smooth parametric optimization problems IAτ and OAτ, where the feasible sets of IAτ are always subsets (inner approximation) and the feasible sets of OAτ always supersets (outer approximation). The feasible sets of IAτ (resp. OAτ ) converge asymptotically to the feasible set of the CCPDE. Moreover, any limit point of a sequence of optimal solutions of IAτ (resp. OAτ ) is a stationary point of CCPDE. The viability of the approximation approach is numerically demonstrated by optimal thermal cancer treatment as a case study.

Persistence and Propagation of a PDE and Discrete-Time Map Hybrid Animal Movement Model With Habitat Shift Driven by Climate Change

Persistence or extinction of moving animal species is a fundamental question in spatial ecology. This paper focuses on the impact of habitat shift driven by climate change on the persistence and propagation of a population with birth pulse. We first present a class of impulsive reaction-diffusion models with heterogeneous nonlinear reaction in high-dimensional space and study their threshold dynamics. We provide the persistence criterion of the system in bounded domains, and prove the existence, uniqueness, and global attraction of a positive steady state. Then we extend the results from bounded domains to the whole space. Our results indicate how the speed of the shifting habitat edge and impulsive reproduction (or harvesting) rate determine the persistence and extinction of the population. Numerical simulations are presented to illustrate the theoretical results.

External optimal control of fractional parabolic PDEs

In [Antilet al. Inverse Probl.35(2019) 084003.] we introduced a new notion of optimal control and source identification (inverse) problems where we allow the control/source to be outside the domain where the fractional elliptic PDE is fulfilled. The current work extends this previous work to the parabolic case. Several new mathematical tools have been developed to handle the parabolic problem. We tackle the Dirichlet, Neumann and Robin cases. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the control/source either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control/source in the exterior. We introduce the notions of weak and very-weak solutions to the fractional parabolic Dirichlet problem. We present an approach on how to approximate the fractional parabolic Dirichlet solutions by the fractional parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.

Numerical study of Bose–Einstein condensation in the Kaniadakis–Quarati model for bosons

Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting. It is an open question whether this equation has solutions exhibiting condensates in finite time. The main analytical challenge lies in the continuation of exploding solutions beyond their first blow-up time while having a linear diffusion term. We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the Kaniadakis--Quarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model in 3D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional at an exponential rate. Our simulations further indicate that the spatial blow-up profile near the origin follows a universal power law and that transient condensates can occur for sufficiently concentrated initial data.

Asymptotic and Numerical Analysis of a Stochastic PDE Model of Volume Transmission

Volume transmission is an important neural communication pathway in which neurons in one brain region influence the neurotransmitter concentration in the extracellular space of a distant brain region. In this paper, we apply asymptotic analysis to a stochastic partial differential equation model of volume transmission to calculate the neurotransmitter concentration in the extracellular space. Our model involves the diffusion equation in a three-dimensional domain with interior holes that randomly switch between being either sources or sinks. These holes model nerve varicosities that alternate between releasing and absorbing neurotransmitter according to when they fire action potentials. In the case that the holes are small, we compute analytically the first two nonzero terms in an asymptotic expansion of the average neurotransmitter concentration. The first term shows that the concentration is spatially constant to leading order and that this constant is independent of many details in the problem. Specifically, this constant first term is independent of the number and location of nerve varicosities, neural firing correlations, and the size and geometry of the extracellular space. The second term shows how these factors affect the concentration at second order. Interestingly, the second term is also spatially constant under some mild assumptions. We verify our asymptotic results by high-order numerical simulation using radial-basis-function--generated finite differences.

Front tracking transition system model with controlled moving bottlenecks and probabilistic traffic breakdowns

Cell-based approximations of PDE traffic models are widely used for traffic prediction and control. However, in order to represent the traffic state with good resolution, cell-based models often require a short cell length, which results in a very large number of states. We propose a new transition system traffic model, based on the front tracking method for solving the LWR PDE model. Assuming piecewise-linear flux function and piecewise-constant initial conditions, this model gives an exact solution. Furthermore, it is easier to extend, has fewer states and, although its dynamics are intrinsically hybrid, is faster to simulate than an equivalent cell-based approximation. The model is extended to enable handling moving bottlenecks as well as probabilistic traffic breakdowns and capacity drops at static bottlenecks. A control strategy that utilizes controlled moving bottlenecks for bottleneck decongestion is described and tested in simulation. It is shown that we are able to keep the static bottleneck in free flow by creating controlled moving bottlenecks at specific instances along on the road, and using them to regulate the incoming traffic flow.

Optimal Closures in a Simple Model for Turbulent Flows

In this work we introduce a computational framework for determining optimal closures of the eddy-viscosity type for Large-Eddy Simulations (LES) of a broad class of PDE models, such as the Navier-Stokes equation. This problem is cast in terms of PDE-constrained optimization where an error functional representing the misfit between the target and predicted observations is minimized with respect to the functional form of the eddy viscosity in the closure relation. Since this leads to a PDE optimization problem with a nonstandard structure, the solution is obtained computationally with a flexible and efficient gradient approach relying on a combination of modified adjoint-based analysis and Sobolev gradients. By formulating this problem in the continuous setting we are able to determine the optimal closure relations in a very general form subject only to some minimal assumptions. The proposed framework is thoroughly tested on a model problem involving the LES of the 1D Kuramoto-Sivashinsky equation, where optimal forms of the eddy viscosity are obtained as generalizations of the standard Smagorinsky model. It is demonstrated that while the solution trajectories corresponding to the DNS and LES still diverge exponentially, with such optimal eddy viscosities the rate of divergence is significantly reduced as compared to the Smagorinsky model. By systematically finding {optimal forms of the eddy viscosity within a certain general class of closure} models, thisframework can thus provide insights about the fundamental performance limitations of these models.

A multi-scale epidemic model of Salmonella infection with heterogeneous shedding

Salmonella strains colonize the digestive tract of farm livestock, such as chickens or pigs, without affecting them, and potentially infect food products, representing a threat for human health ranging from food poisoning to typhoid fever. It has been shown that the ability to excrete the pathogen in the environment and contaminate other animals is variable. This heterogeneity in pathogen carriage and shedding results from interactions between the host’s immune response, the pathogen and the commensal intestinal microbiota. In this paper we propose a novel generic multiscale modeling framework of heterogeneous pathogen transmission in an animal population. At the intra-host level, the model describes the interaction between the commensal microbiota, the pathogen and the inflammatory response. Random fluctuations in the ecological dynamics of the individual microbiota and transmission at between-host scale are added to obtain a drift-diffusion PDE model of the pathogen distribution at the population level. The model is further extended to represent transmission between several populations. The asymptotic behavior as well as the impact of control strategies including cleaning and antimicrobial administration are investigated through numerical simulation.

An anisotropic PDE model for image inpainting

A PDE model based on an image specific auto generated multi-well potential function and a 4th order anisotropic diffusion with good performance with respect to loss due to smoothing effects has been proposed for grayscale image inpainting. An unconditionally stable numerical scheme using the notion of convexity splitting in time and Fourier spectral method in space has been derived. The stable scheme is both consistent and convergent. Numerical computations are done for some standard test images and results are compared with the results of other existing models in the literature using image analysis specs such as PSNR, SNR, and SSIM.