Space-dependent Coefficient Research Articles

Simultaneous identification and reconstruction of the space-dependent reaction coefficient and source term

Abstract The inverse problem of simultaneously determining, i.e., identifying and reconstructing, the space-dependent reaction coefficient and source term component from time-integral temperature measurements is investigated. This corresponds to thermal applications in which the heat is generated from a source depending linearly on the temperature, but with unknown space-dependent coefficients. For the resulting nonlinear inverse problem, we first prove the existence of solution based on the Schauder fixed point theorem. Then, under certain additional conditions, the solution is also proved to be unique. For the numerical reconstruction of solution, the problem is reformulated as a least-squares minimisation whose Fréchet gradients with respect to the two unknowns are derived in terms of the solution of an adjoint problem. The conjugate gradient method (CGM) to calculate the numerical solution is developed, and its convergence is proved from the Lipschitz continuity of these gradients. Three numerical examples for one- and two-dimensional inverse problems are illustrated to reveal the accuracy and stability of the solutions applying the CGM regularised by the discrepancy principle when noisy data are inverted.

Critical patch size reduction by heterogeneous diffusion.

Population survival depends on a large set of factors and on how they are distributed in space. Due to landscape heterogeneity, species can occupy particular regions that provide the ideal scenario for development, working as a refuge from harmful environmental conditions. Survival occurs if population growth overcomes the losses caused by adventurous individuals that cross the patch edge. In this work, we consider a single species dynamics in a patch with a space-dependent diffusion coefficient. We show analytically, within the Stratonovich framework, that heterogeneous diffusion reduces the minimal patch size for population survival when contrasted with the homogeneous case with the same average diffusivity. Furthermore, this result is robust regardless of the particular choice of the diffusion coefficient profile. We also discuss how this picture changes beyond the Stratonovich framework. Particularly, the Itô case, which is nonanticipative, can promote the opposite effect, while Hänggi-Klimontovich interpretation reinforces the reduction effect.

Uniqueness for identifying a space-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation from a single boundary point measurement

This paper is focused on a nonlinear inverse problem for identifying a space-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation by the measured data on a single boundary point for one-dimensional case. We give the definition of a weak solution and prove its existence for the corresponding direct problem by using the Fourier method. Based on the Gronwall inequality, analytic continuation and the Laplace transformation, we obtain the uniqueness for the inverse zeroth-order coefficient problem under some simple requirements to the Neumann boundary data.

Convergence rates for the reaction coefficient and the initial temperature identification problems

ABSTRACT In this paper, we investigate the convergence rates of the inverse heat transfer problem of simultaneously determining the space-dependent reaction coefficient and the initial temperature from some additional temperature observations. The strategy is to decouple the original problem into two inverse problems: (i) recover the reaction coefficient; (ii) determine the initial temperature with the estimated reaction coefficient in the previous stage. The Tikhonov regularization method is used to reconstruct the reaction coefficient, and a new source condition is used to derive the convergence rates to the reaction coefficient. Based on the approximated reaction coefficient, the initial temperature as well as its convergence rates can be obtained by using the quasi-reversibility method.

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, II: Microlocal analysis

In this paper we continue the study of non-diagonalisable hyperbolic systems with variable multiplicity started by the authors in [1]. In the case of space dependent coefficients, we prove a representation formula for solutions that allows us to derive results of regularity and propagation of singularities.

Small-time expansion of the Fokker–Planck kernel for space and time dependent diffusion and drift coefficients

We study the general solution of the Fokker–Planck equation in d dimensions with arbitrary space and time dependent diffusion matrix and drift terms. We show how to construct the solution for arbitrary initial distributions as an asymptotic expansion for small time. This generalizes the well-known asymptotic expansion of the heat kernel for the Laplace operator on a general Riemannian manifold. We explicitly work out the general solution to leading and next-to-leading order in this small-time expansion, as well as to next-to-next-to-leading order for vanishing drift. We illustrate our results on a several examples.

Conditional Karhunen-Loève expansion for uncertainty quantification and active learning in partial differential equation models

We use conditional Karhunen-Loève (KL) expansions to quantify and reduce uncertainty in the coupled linear diffusion equation (a combination of the Darcy law and continuity equations for average pore velocity and hydraulic head) and the advection-dispersion partial differential equation (PDE) with an unknown sparsely sampled space-dependent hydraulic conductivity coefficient k(x)=exp⁡(g(x)) describing fluid flow and transport of a conservative tracer in fully saturated heterogeneous porous media. We achieve reduction in uncertainty by modeling g(x) as a random field with a KL expansion conditioned on g measurements. We consider two approaches for conditioning the KL expansion. In Approach 1, we condition the KL model first and then truncate it. In Approach 2, we first truncate the KL expansion and then condition it. We employ the conditional KL expansion together with the sparse grid collocation method to compute the moments of the resulting stochastic PDEs. The accuracy of the moment solutions is verified against the Monte Carlo solution of the stochastic PDEs. Uncertainty in the flow problem (the variance of hydraulic head) is further reduced by adaptively selecting additional observation locations of g using two active learning methods. Method 1 minimizes the variance of g(x), while Method 2 minimizes the variance of the PDE states. We demonstrate that conditioning leads to dimension reduction of the KL representation of g(x). For the considered equations, we show that Approach 1 provides a more accurate approximation of the conditional log-normal coefficient and solution of the SPDE than Approach 2 for the same number of dimensions in a conditional KL expansion. Furthermore, Approach 2 provides a good estimate for the number of terms of the truncated KL expansion of the conditional field of Approach 1. Finally, we demonstrate that active learning based on Method 2 is more efficient for uncertainty reduction in the SPDE's states (i.e., it leads to a larger reduction of the variance) than active learning using Method 2.

Inverse problem for a Cahn–Hilliard type system modeling tumor growth

In this paper, we address an inverse problem of reconstructing a space-dependent semilinear coefficient in the tumor growth model described by a system of semilinear partial differential equations (PDEs) with Dirichlet boundary condition using boundary-type measurement. We establish a new higher order weighted Carleman estimate for the given system with the help of Dirichlet boundary conditions. By deriving a suitable regularity of solutions for this nonlinear system of PDEs and the new Carleman estimate, we prove Lipschitz-type stability for the tumor growth model.

Periodically poled LiNbO3 crystals from 1D and 2D to 3D

A periodically-poled LiNbO3 (PPLN) crystal features space-dependent second-order nonlinear coefficients, which is one of the most important materials to effectively control nonlinear optical interactions through quasi-phase matching (QPM). By using electric-field poling method, 1D and 2D PPLN crystals have been successfully fabricated for laser frequency conversion, quantum light sources, nonlinear beam shaping and nonlinear optical imaging. Recently, femtosecond laser engineering technique is utilized to prepare 3D domain structures inside LiNbO3 crystal, which provides a promising platform to control nonlinear interacting waves in 3D configuration. After 40 years of developments, PPLN crystals still have exciting prospects in fundamental researches and practical applications for integrated photonic chip, quantum information processing, and so on.

Strauss exponent for semilinear wave equations with scattering space dependent damping

It has been asserted that the semilinear wave equation with scattering damping admits the same Strauss critical exponent as the classical semilinear wave equation. In this paper, we devote to showing that this assertion holds for the space dependent damping wave equation when the decay rate of the space dependent variable coefficients of the damping is greater than 2. In particular, we derive the lifespan upperbound estimates for both sub-critical and critical exponents. Moreover, when the nonlinear term is of derivative type, we also obtain the same Glassey exponent and the same upper bound lifespan as that of the corresponding problem for wave equation without damping. Both results show the “hyperbolicity” of the equation from the blow-up aspect.

Reconstruction of the thermal properties in a wave-type model of bio-heat transfer

Purpose This study aims to at numerically retrieve five constant dimensional thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements. Design/methodology/approach The thermal-wave model of bio-heat transfer is used as an appropriate model because of its realism in situations in which the heat flux is extremely high or low and imposed over a short duration of time. For the numerical discretization, an unconditionally stable finite difference scheme used as a direct solver is developed. The sensitivity coefficients of the dimensionless boundary temperature measurements with respect to five constant dimensionless parameters appearing in a non-dimensionalised version of the governing hyperbolic model are computed. The retrieval of those dimensionless parameters, from both exact and noisy measurements, is successfully achieved by using a minimization procedure based on the MATLAB optimization toolbox routine lsqnonlin. The values of the five-dimensional parameters are recovered by inverting a nonlinear system of algebraic equations connecting those parameters to the dimensionless parameters whose values have already been recovered. Findings Accurate and stable numerical solutions for the unknown thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements are obtained using the proposed numerical procedure. Research limitations/implications The current investigation is limited to the retrieval of constant physical properties, but future work will investigate the reconstruction of the space-dependent blood perfusion coefficient. Practical implications As noise inherently present in practical measurements is inverted, the paper is of practical significance and models a real-world situation. Social implications The findings of the present paper are of considerable significance and interest to practitioners in the biomedical engineering and medical physics sectors. Originality/value In comparison to Alkhwaji et al. (2012), the novelty and contribution of this work are as follows: considering the more general and realistic thermal-wave model of bio-heat transfer, accounting for a relaxation time; allowing for the tissue to have a finite size; and reconstructing five thermally significant dimensional parameters.

Efficient absorption of laser light by nano-porous materials with well-controlled structure

The absorption of high-power nanosecond laser pulses in a porous matter is investigated through theoretical and numerical calculations. The effects of structural properties of the porous target such as size of pores and thickness of solid elements on the laser absorption are investigated. The time and space-dependent absorption coefficient of the laser created partially homogenized plasma is used in a plasma hydrodynamic code to reproduce the laser absorption and plasma formation processes in a porous matter. It is observed that the structural characteristics of the porous material can be optimized to significantly increase the laser energy absorption. For porous targets with pore sizes in the range 30–60 nm a value around 1000 nm could be desirable for the wall thickness to increase the laser absorption efficiency to more than 90%. The results can be advantageous in production of efficient laser absorber targets which are desirable in advanced applications such as inertial confinement fusion and laser-plasma x-ray sources.

Simultaneous reconstruction of space-dependent heat transfer coefficients and initial temperature

Many complex physical phenomena and engineering systems, e.g., in heat exchanges, reflux condensers, combustion chambers, nuclear vessels, etc., due to the high temperatures/high pressures hostile environment involved, possess certain properties which are inaccessible to measure and therefore their influence/determination using inverse analysis is very important and desirable. In this spirit, the purpose of this paper is to mathematically formulate and analyse a new inverse problem in which given measurements of temperature at two different instants, it is required to obtain the space-dependent heat transfer coefficients (HTCs) and the initial temperature. This simultaneous identification is challenging since it is both nonlinear and ill-posed. The uniqueness of solution is established based on the max–min principle for parabolic equations and the contraction mapping principle for the existence and uniqueness of a fixed point. The novel inverse mathematical model that is proposed offers appropriate scientific guidance to the polymer/heat transfer processing industry as to which data to measure/provide in order to be able to reliably determine the desirable HTCs along with the initial temperature, which is in general unknown. Furthermore, for the reconstruction, the surface HTC is determined separately, whilst the variational formulation is introduced for the simultaneous determination of the domain HTC and the initial temperature. The Fréchet gradient of the minimizing objective functional is derived. The numerical reconstruction process is based on the conjugate gradient method (CGM) regularized by the discrepancy principle. Accurate and stable numerical solutions are obtained even in the presence of noise in the input temperature data. Since noisy data are invented, the study models realistic practical situations in which temperature measurements recorded using sensors or thermocouples are inherently contaminated with random noise.

Bilateral Output Feedback Control of Fractional PDEs with Space-Dependent Coefficients

Abstract This paper develops an extension of the bilateral control method for fractional partial differential equations (PDEs) with space-dependent coefficients by output feedback. Using a backstepping transformation, a full state feedback control law is designed. Then the fractional PDE system is folded into two subsystems and Mittag-Leffler convergent state observers of these subsystems are derived. Although the observers are coupled with boundary conditions (BCs), error subsystems are decoupled by assuming some available measurements. Hence, the observer gains are easily obtained. After this, we compose the designed state feedback controller and observers to enable Mittag-Leffler stabilization by output feedback. Finally, a fractional numerical example is provided to support the effectiveness of the proposed synthesis for the case when neither the control kernel nor the estimation kernel has an explicit solution.

Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation

We consider the inverse problem of Holder-stably determining the time- and space-dependent coefficients of the Schrodinger equation on a simple Riemannian manifold with boundary of dimension \begin{document}$ n\geq2 $\end{document} from the knowledge of the Dirichlet-to-Neumann map. Assuming the divergence of the magnetic potential is known, we show that the electric and magnetic potentials can be Holder-stably recovered from these data. Here we also remove the smallness assumption for the solenoidal part of the magnetic potential present in previous results.

Location of sources in reaction-diffusion equations using support vector machines.

The reaction-diffusion equation serves to model systems in the diffusion regime with sources. Specific applications include diffusion processes in chemical reactions, as well as the propagation of species, diseases, and populations in general. In some of these applications the location of an outbreak, for instance, the source point of a disease or the nest of a vector spreading a virus is important. Also important are the environmental parameters of the domain where the process diffuses, namely the space-dependent diffusion coefficient and the proliferation parameter of the process. Determining both, the location of a source and the environmental parameters, define an inverse problem that in turn, involves a partial differential equation. In this paper we classify the values of these parameters using Support Vector Machines (SVM) trained with numerical solutions of the reaction-diffusion problem. Our set up has accuracy of classifying the outbreak location above 90% and 77% of classifying both, the location and the environmental parameters. The approach presented in our analysis can be directly implemented by measuring the population under study at specific locations in the spatial domain as function of time.

Backstepping Control of Coupled Linear Parabolic PDEs With Space and Time Dependent Coefficients

In this article, the backstepping design of stabilizing state feedback controllers for coupled linear parabolic PDEs with spatially varying distinct diffusion coefficients as well as space and time dependent reaction is presented. The selected target system is a cascade of exponentially stable, time-invariant systems, with time dependent couplings, that is uniformly exponentially stable with a prescribed rate of convergence. To determine the state feedback controller, the kernel equations are derived, which results in a set of coupled PDEs for a time dependent and spatially varying kernel. For this, the method of successive approximations is extended from the time-invariant case to the present problem. The applicability of the method is demonstrated by the stabilization of two coupled unstable parabolic PDEs with space and time dependent coefficients.

Simultaneous reconstruction of the spatially-distributed reaction coefficient, initial temperature and heat source from temperature measurements at different times

In many practical situations concerned with high temperatures/pressures/loads and/or hostile environments, certain properties of the physical medium, geometry, boundary and/or initial conditions are not known and their direct measurement can be very inaccurate or even inaccessible. In such situations, one can adopt an inverse approach and try to infer the unknowns from some extra accessible measurements of other quantities that may be available. However, the simultaneous identification of several non-constant physical properties along with initial and/or boundary conditions is very challenging, especially when it cannot be decoupled, as it combines both nonlinear as well as ill-posedness features. One such new inverse problem concerning the identification of the space-dependent reaction coefficient, the initial temperature and the source term from measured temperatures at two instants t1,t2 and at the final time T, where 0<t1<t2<T, is investigated in this paper. Insight into the uniqueness of solution is gained by considering various particular cases. Moreover, as in practice the input temperature data are usually noise polluted due to the errors that are inherently present, their influence on the solution of inversion has to be assessed. As such, the least-squares objective functional modelling the gap between the measured and computed data is minimized to obtain the quasi-solution to the inverse problem, and the Fréchet gradients are obtained. The conjugate gradient method (CGM) with the Fletcher–Reeves formula is applied to estimate the three unknown coefficients numerically. Numerical examples are illustrated to show that accurate and stable numerical solutions are obtained using the CGM regularized by the discrepancy principle.

Stability of a general HIV-1 reaction–diffusion model with multiple delays and immune response

The aim of this paper is to study a general HIV-1 reaction–diffusion model with space-dependent diffusion coefficients and antibody immune response. The model contains four discrete time delays and three categories of infected cells: Latently infected cells, short-lived productively infected cells and long-lived productively infected cells. The effect of highly active antiretroviral therapies (HAART) used to treat HIV-1 is incorporated into the model. The existence, non-negativity and boundedness of solutions are studied in order to verify the well-posedness of the model. All possible equilibria of the model are evaluated and the threshold parameters needed for their existence and stability are investigated. The global asymptotic stability of the corresponding equilibria are proved by building suitable Lyapunov functionals. The numerical simulations are conducted in order to confirm the theoretical results and show the behavior of solutions in space and time.

Numerical investigation of spatio-temporal dynamics of nonlinear laser-induced plasma absorption coefficients in aqueous media

A numerical study of spatial and temporal dynamics of nonlinear absorption coefficients of laser-induced plasma in distilled water was conducted using a nonlinear time- and space-dependent model. By taking into account the transient nature of the plasma as well as changes of the intensity-dependent refractive index of liquid water, it was evaluated the time–space-dependent absorption coefficient of the plasma for laser pulses of 30-ps, 1064-nm at focusing angles between 4° and 28° and energies in the wide range of 0.1–800 μJ. In this study, the simultaneous effects of the main mechanisms of ionization including multiphoton and cascade ionization, self-focusing, and plasma-defocusing were considered on the spatio-temporal dynamics of the plasma absorption coefficients using a paraxial-ray-tracing procedure. The results of the simulation and the reported data from other studies showed a qualitatively similar trend in the average plasma absorption coefficients. It was observed that the normalized energy for which the largest absorption coefficient of the laser-induced plasma was obtained is roughly independent of the radiation angle. It was also found that the ratio of the distance to the focal point of a region of plasma, where the largest absorption coefficient occurs, to the Rayleigh length is almost identical at all the angles of radiation. The outcomes of this study can be further used in time-based models describing the laser interaction with aqueous media such as biological tissues to account for the plasma formation effects on the propagating laser pulse.