The inverse problem for diffusion equations

In the standard procedure one has to find the concentration from given initial (and boundary) conditions. We investigate here an inverse problem: to find the initial concentration from the known resulting concentration. This problem is of importance particularly for non-linear diffusion equations where the concentration dependenceD D(c) of the diffusion coefficient is analysed in terms of the Boltzmann-like solutionc f(xt^–1/2). This type of solution is, however, closely related to a very special initial condition for the concentration. The disregard of this fact might lead to misleading analysis ofD(c).     

The Kirkendall effect in single-phase multicomponent systems: dependence on drift and entropy distribution

This paper concerns interdiffusion in a diffusion couple and determination of the Kirkendall plane. The “entropy density” model is proposed in which the entropy is used to predict the position of the Kirkendall plane in a multicomponent system. It is shown that the marker position depends on the drift velocity and pressure field only. Application of the model is presented for ternary CoFeNi diffusion couples of three various initial compositions. The concentration profiles and entropy densities are calculated for each diffusion couple. The positions of the Kirkenadl planes are determined and compared with those obtained by velocity-curve and trajectory methods.     

The Adjoint Method for an Inverse Design Problem in the Directional Solidification of Binary Alloys

This paper presents a systematic procedure based on the adjoint method for solving a class of inverse directional alloy solidification design problems in which a desired growth velocityv_fis achieved under stable growth conditions. To the best of our knowledge, this is the first time that a continuum adjoint formulation is proposed for the solution of an inverse problem with simultaneous heat and mass transfer, thermo-solutal convection, and phase change. In this paper, the interfacial stability is considered to imply a sharp solid–liquid freezing interface. This condition is enforced using the constitutional undercooling criterion in the form of an inequality constraint between the thermal and solute concentration gradients,GandG_c, respectively, at the freezing front. The main unknowns of the design problem are the heating and/or cooling boundary conditions on the mold walls. The inverse design problem is formulated as a functional optimization problem. The cost functional is defined by the square of theL₂norm of the deviation of the freezing interface temperature from the temperature corresponding to thermodynamic equilibrium. A continuum adjoint system is derived to calculate the adjoint temperature, concentration, and velocity fields such that the gradient of the cost functional can be expressed analytically. The cost functional minimization process is realized by the conjugate gradient method via the finite element method solutions of the continuum direct, sensitivity, and adjoint problems. The developed formulation is demonstrated with an example of designing the directional solidification of a binary aqueous solution in a rectangular mold such that a stable vertical interface advances from left to right with a desired growth velocity.     

A numerical method for determination of moisture transfer coefficient according to the diffusion moisture profiles

For a set of the measured diffusion moisture profiles, a numerical method for determination of moisture transfer coefficient D(w, t) is suggested. The transfer coefficient is found as a sum of the degree and exponential functions of the moisture concentration w, as opposite to the previous works. The exponent p(t) of the power function depends on time t. The exponential function describes profiles for large times nearby the boundary of the sample, where the moisture evaporation takes place to the atmosphere. A conservative difference scheme for numerical solution of direct problem is suggested. An inverse problem for minimization of an error functional is solved by the Newton method. Thus, a more accurate coincidence of the calculated profiles of the moisture concentration to the measured profiles is gained. The text was submitted by the authors in English.     

On the problem of the metastable region at morphological instability

The study deals with numerical analysis of the morphological stability of a growing round particle with respect to harmonic perturbations of an arbitrary amplitude. Various growth regimes (from diffusion to kinetic-limited) are considered. It is found that the critical size of the particle stability decreases as the perturbation amplitude increases and tends to the value, which was determined analytically elsewhere using the maximum entropy production principle. This result is a crucial argument in support of the hypothesis that the entropy production can be used for analysis of a nonequilibrium phase transitions similarly to thermodynamic potentials in the case of equilibrium phase transitions.     

Adding Prior Information in FWI through Relative Entropy

Full waveform inversion is an advantageous technique for obtaining high-resolution subsurface information. In the petroleum industry, mainly in reservoir characterisation, it is common to use information from wells as previous information to decrease the ambiguity of the obtained results. For this, we propose adding a relative entropy term to the formalism of the full waveform inversion. In this context, entropy will be just a nomenclature for regularisation and will have the role of helping the converge to the global minimum. The application of entropy in inverse problems usually involves formulating the problem, so that it is possible to use statistical concepts. To avoid this step, we propose a deterministic application to the full waveform inversion. We will discuss some aspects of relative entropy and show three different ways of using them to add prior information through entropy in the inverse problem. We use a dynamic weighting scheme to add prior information through entropy. The idea is that the prior information can help to find the path of the global minimum at the beginning of the inversion process. In all cases, the prior information can be incorporated very quickly into the full waveform inversion and lead the inversion to the desired solution. When we include the logarithmic weighting that constitutes entropy to the inverse problem, we will suppress the low-intensity ripples and sharpen the point events. Thus, the addition of entropy relative to full waveform inversion can provide a result with better resolution. In regions where salt is present in the BP 2004 model, we obtained a significant improvement by adding prior information through the relative entropy for synthetic data. We will show that the prior information added through entropy in full-waveform inversion formalism will prove to be a way to avoid local minimums.     

Sensitivity of atmospheric-surface parameters retrieval to the spectral stability of channels in thermal IR

Sensitivity of the solutions of forward and inverse problems of the atmospheric-surface parameters retrieval to spectral shifts of channels is considered. The solution of the forward problem is obtained by radiative transfer simulation. The retrieval (solution of the inverse problem) is obtained by linear (optimal interpolation) and non-linear (variational) techniques. It is shown that atmospheric temperature profiles exhibit high sensitivity to the spectral shift, while the humidity profiles are moderately sensitive while the sea surface temperatures retrievals are insensitive. Two approaches are proposed to reduce the effect of channel spectral shift, one is based on channel selection and the other approach is related to proper calibration of the cost function. We performed the numerical simulations using the parameters of AIRS spectrometer to illustrate the sensitivity of forward and inverse problems. The results of the simulation show that the inversion error can be significantly reduced by the proposed techniques.     

Characteristics of exact retrieval of the structure of plane-stratified media using reflected and transmitted pulse signals

We consider the exact iterative algorithm for solution of the direct and inverse scattering problems in 1D inhomogeneous half-space using piecewise-constant approximation of the constitutive parameter profiles (e.g., refractive index). A novel modification of the Gelfand–Levitan equation is derived for solution of the inverse scattering problem. Numerical examples are given and the effects related to a finite frequency band of the sounding pulse are discussed.     

Identifying time-dependent damping and stiffness functions by a simple and yet accurate method

The inverse vibration problem is a mathematical process to determine unknown mechanical parameters from measured vibration data. In this study the data of displacement are chosen in order to identify a time-dependent function of damping or stiffness. However, when both functions are to be identified we require both the data of displacement and velocity. This is the first time that a closed-form estimation method for the inverse vibration problems of estimating time-dependent parameters has been constructed. We are able to transform the inverse vibration problem into an identification problem governed by a parabolic-type partial differential equation (PDE). Then, a one-step group-preserving scheme (GPS) for the semi-discretization of PDE is established, which can be used to derive a closed-form solution for estimating parameters. The new Lie-group estimation method has three further advantages: it does not require any prior information on the functional forms of unknown functions; no initial guesses are required; and no iterations are required. Numerical examples were examined to show that the present approach is highly accurate and efficient even for identifying discontinuous and oscillatory parameters. Against the noise is good when only one function is estimated; however, the present approach is slightly weak against the noise when both functions are identified.     

Estimation of Uncertainty in an Analytical Inverse Heat Conduction Solution

The method of sequential perturbations is applied to find the uncertainty in estimated surface temperature and heat flux from a two-dimensional analytical inverse heat conduction problem related to impinging jet quenching experiments. It is shown that for meaningful uncertainty estimates, the inverse solution itself must be formulated such that it can be interpreted as giving average surface conditions over a small period of time and space. A procedure for estimating the time and space resolution limits of the solution is proposed.     

Beyond Inverse Ising Model: Structure of the Analytical Solution

I consider the problem of deriving couplings of a statistical model from measured correlations, a task which generalizes the well-known inverse Ising problem. After reminding that such problem can be mapped on the one of expressing the entropy of a system as a function of its corresponding observables, I show the conditions under which this can be done without resorting to iterative algorithms. I find that inverse problems are local (the inverse Fisher information is sparse) whenever the corresponding models have a factorized form, and the entropy can be split in a sum of small cluster contributions. I illustrate these ideas through two examples (the Ising model on a tree and the one-dimensional periodic chain with arbitrary order interaction) and support the results with numerical simulations. The extension of these methods to more general scenarios is finally discussed.     

Forward and inverse problems in fundamental and applied magnetohydrodynamics

This minireview summarizes the recent efforts to solve forward and inverse problems as they occur in different branches of fundamental and applied magnetohydrodynamics. For the forward problem, the main focus is on the numerical treatment of induction processes, including self-excitation of magnetic fields in non-spherical domains and/or under the influence of non-homogeneous material parameters. As an important application of the developed numerical schemes, the functioning of the von-Kármán-sodium (VKS) dynamo experiment is shown to depend crucially on the presence of soft-iron impellers. As for the inverse problem, the main focus is on the mathematical background and some initial practical applications of contactless inductive flow tomography (CIFT), in which flow induced magnetic field perturbations are utilized to reconstruct the velocity field. The promises of CIFT for flow field monitoring in the continuous casting of steel are substantiated by results obtained at a test rig with a low-melting liquid metal. While CIFT is presently restricted to flows with low magnetic Reynolds numbers, some selected problems from non-linear inverse dynamo theory, with possible applications to geo- and astrophysics, are also discussed.     

Inverse analysis and regularisation in conditional source-term estimation modelling

Conditional Source-term Estimation (CSE) obtains the conditional species mass fractions by inverting a Fredholm integral equation of the first kind. In the present work, a Bayesian framework is used to compare two different regularisation methods: zeroth-order temporal Tikhonov regulatisation and first-order spatial Tikhonov regularisation. The objectives of the current study are: (i) to elucidate the ill-posedness of the inverse problem; (ii) to understand the origin of the perturbations in the data and quantify their magnitude; (iii) to quantify the uncertainty in the solution using different priors; and (iv) to determine the regularisation method best suited to this problem. A singular value decomposition shows that the current inverse problem is ill-posed. Perturbations to the data may be caused by the use of a discrete mixture fraction grid for calculating the mixture fraction PDF. The magnitude of the perturbations is estimated using a box filter and the uncertainty in the solution is determined based on the width of the credible intervals. The width of the credible intervals is significantly reduced with the inclusion of a smoothing prior and the recovered solution is in better agreement with the exact solution. The credible intervals for temporal and spatial smoothing are shown to be similar. Credible intervals for temporal smoothing depend on the solution from the previous time step and a smooth solution is not guaranteed. For spatial smoothing, the credible intervals are not dependent upon a previous solution and better predict characteristics for higher mixture fraction values. These characteristics make spatial smoothing a promising alternative method for recovering a solution from the CSE inversion process.     

Image quality improvement in ultrasonic nondestructive testing by the maximum entropy method

We consider the possibility of solving the inverse scattering problem in the linear approximation (in the form of a convolution equation) by reducing it to a system of linear algebraic equations and minimizing the residual. Since the problem is an ill-posed one, the Tikhonov regularization proves useful. The possibility of using the entropy of the image estimate as a stabilizing functional is considered, which is the key idea of the maximum entropy method. The single-frequency and multifrequency versions of the method are realized. The advantage of the maximum entropy method over the conventional linear methods of solving the inverse scattering problem is shown. The superresolution and sidelobe suppression abilities of the maximum entropy method are demonstrated. The method is shown to be stable to measurement noise and multiplicative interference in the form of aperture decimation. Examples of the image reconstruction by the maximum entropy method from model and experimental data are presented.     

The influence of the ideal mixing entropy on concentration profiles and diffusion paths in ternary systems

The diffusion profiles and the reaction paths in ternary solid solutions are determined by both thermodynamics and kinetics. The matrix of the diffusion coefficient can be described as the product of the Hessian matrix for the thermodynamic influences and the Onsager matrix for kinetic influences.In this paper the interest is focused on the influence of the ideal part of the Hessian matrix, i.e. the ideal mixing entropy on interdiffusion. The ideal diffusion profiles are calculated by a computer simulation and they are compared with experimental results from the literature. These comparisons reveal that in most cases the qualitative shape of the diffusion profiles and of the reaction paths can be considered as caused by the ideal mixing entropy. Surprisingly, the shape of the diffusion profiles turns out to depend on the component that was chosen as the so-called solvent of the ternary mixture. This means that the ideal reaction paths do not show the triangular symmetry expected for an ideal ternary system. Especially, reaction paths between starting positions showing the same concentration of one of the three components do not run along straight lines.     

Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term.     

The Diffusion Kernel Filter

A particle filter method is presented for the discrete-time filtering problem with nonlinear Itô stochastic ordinary differential equations (SODE) with additive noise supposed to be analytically integrable as a function of the underlying vector-Wiener process and time. The Diffusion Kernel Filter is arrived at by a parametrization of small noise-driven state fluctuations within branches of prediction and a local use of this parametrization in the Bootstrap Filter. The method applies for small noise and short prediction steps. With explicit numerical integrators, the operations count in the Diffusion Kernel Filter is shown to be smaller than in the Bootstrap Filter whenever the initial state for the prediction step has sufficiently few moments. The established parametrization is a dual-formula for the analysis of sensitivity to gaussian-initial perturbations and the analysis of sensitivity to noise-perturbations, in deterministic models, showing in particular how the stability of a deterministic dynamics is modeled by noise on short times and how the diffusion matrix of an SODE should be modeled (i.e. defined) for a gaussian-initial deterministic problem to be cast into an SODE problem. From it, a novel definition of prediction may be proposed that coincides with the deterministic path within the branch of prediction whose information entropy at the end of the prediction step is closest to the average information entropy over all branches. Tests are made with the Lorenz-63 equations, showing good results both for the filter and the definition of prediction.     

Numerical determination of the density profile of an inhomogeneous membrane: solution of the inverse vibration problem

Given one or more vibrational modes of a membrane, the free vibration equation can be applied to infer the mass surface density. This paper considers determining the surface density of an inhomogeneous membrane from digitized holographic projections (interferograms) of the modeshapes. Spatially discrete numerical models of the membrane surface are presented, which can be used to solve both forward and inverse vibration problems. The accuracy of the discrete models is examined for exactly solvable free vibration problems involving inhomogeneous membranes. For the solution of the inverse problem, error estimates are given for the mass surface density deduced from modeshape interferograms. The practicability of the method is investigated using simulated experimental data for membranes with composite and continuously inhomogeneous density profiles. Strategies are discussed for reducing errors in the reconstructed densities.