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We prove the necessary and sufficient condition for the removability of the fundamental singularity, and equivalently for the unique solvability of the singular Dirichlet problem for the heat equation. In the measure-theoretical context the criterion determines whether the $h$-parabolic measure of the singularity point is null or positive. From the probabilistic point of view, the criterion presents an asymptotic law for conditional Brownian motion., Comment: 29 pages

It is well-known that the embedding of the Sobolev space of weakly differentiable functions into H\"{o}lder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions into the H\"{o}lder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. The proof is based on the generalization of the Newton-Leibniz formula to the $n$-dimensional rectangle and inductive application of the Sobolev trace embedding results. The method is applied to prove the embedding of the Sobolev spaces with dominating mixed smoothness into H\"{o}lder spaces., Comment: 16 pages

We consider Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity and potential in the body based on the measurement of the boundary voltages on the $m$ electrodes for a given electrode current. The variational formulation is pursued in the optimal control framework, where electrical conductivity and boundary voltages are control parameters, and the cost functional is the norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. EIT optimal control problem is fully discretized using the method of finite differences. New Sobolev-Hilbert space is introduced, and the convergence of the sequence of finite-dimensional optimal control problems to EIT coefficient optimal control problem is proved both with respect to functional and control in 2- and 3-dimensional domains.

Optimal control of the singular nonlinear parabolic PDE which is a distributional formulation of multidimensional and multiphase Stefan-type free boundary problem is analyzed. Approximating sequence of finite-dimensional optimal control problems is introduced via finite differences. Existence of the optimal control and the convergence of the sequence of discrete optimal control problems both with respect to functional and control is proved. In particular, convergence of the method of finite differences, and existence, uniqueness and stability estimations are established for the singular PDE problem under minimal regularity assumptions on the coefficients.

We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat flux is the control parameter, and the optimality criteria consist of the minimization of the $L_2$-norm declination of the trace of the solution to the PDE problem at the final moment from the given measurement. Sequence of finite-dimensional optimal control problems is introduced through finite differences. We establish existence of the optimal control and prove the convergence of the sequence of discrete optimal control problems to the original problem both with respect to functional and control. Proofs rely on establishing a uniform $L_{\infty}$ bound, and $W_2^{1,1}$-energy estimate for the discrete nonlinear PDE problem with discontinuous coefficient., Comment: 27 pages. arXiv admin note: text overlap with arXiv:1508.00290

This paper presents a full classification of the short-time behavior of the interfaces in the Cauchy problem for the nonlinear second order degenerate parabolic PDE \[ u_t-\Delta u^m +b u^\beta=0, \ x\in \mathbb{R}^N, 01, C,\alpha, \beta >0, b \in \mathbb{R}$. Interface surface $t=\eta(x)$ may shrink, expand or remain stationary depending on the relative strength of the diffusion and reaction terms near the boundary of support, expressed in terms of the parameters $m,\beta, \alpha, sign\ b$ and $C$. In all cases we prove explicit formula for the interface asymptotics, and local solution near the interface., Comment: 17 pages, 1 figure

Inverse problem for the identification of the parameters for large-scale systems of nonlinear ordinary differential equations (ODEs) arising in systems biology is analyzed. In a recent paper in \textit{Mathematical Biosciences, 305(2018), 133-145}, the authors implemented the numerical method suggested by one of the authors in \textit{J. Optim. Theory Appl., 85, 3(1995), 509-526} for identification of parameters in moderate scale models of systems biology. This method combines Pontryagin optimization or Bellman's quasilinearization with sensitivity analysis and Tikhonov regularization. We suggest modification of the method by embedding a method of staggered corrector for sensitivity analysis and by enhancing multi-objective optimization which enables application of the method to large-scale models with practically non-identifiable parameters based on multiple data sets, possibly with partial and noisy measurements. We apply the modified method to a benchmark model of a three-step pathway modeled by 8 nonlinear ODEs with 36 unknown parameters and two control input parameters. The numerical results demonstrate geometric convergence with a minimum of five data sets and with minimum measurements per data set. Software package \textit{qlopt} is developed and posted in GitHub. MATLAB package AMIGO2 is used to demonstrate advantage of \textit{qlopt} over most popular methods/software such as \textit{lsqnonlin}, \textit{fmincon} and \textit{nl2sol}., Comment: 20 pages, 21 Figures, 5 Tables

We prove the short-time asymptotic formula for the interfaces and local solutions near the interfaces for the nonlinear double degenerate reaction-diffusion equation of turbulent filtration with fast diffusion and strong absorption \[ u_t=(|(u^{m})_x|^{p-1}(u^{m})_x)_x-bu^{\beta}, \, 00. \] Full classification is pursued in terms of the nonlinearity parameters $m, p,\beta$ and asymptotics of the initial function near its support. In the case of an infinite speed of propagation of the interface, the asymptotic behavior of the local solution is classified at infinity. A full classification of the short-time behavior of the interface function and the local solution near the interface for the slow diffusion case ($mp>1$) was presented in $\textit{Abdulla et al., Math. Comput. Simul., 153(2018), 59-82}$., Comment: arXiv admin note: text overlap with arXiv:1811.07278

We consider the inverse multiphase Stefan problem with homogeneous Dirichlet boundary condition on a bounded Lipschitz domain, where the density of the heat source is unknown in addition to the temperature and the phase transition boundaries. The variational formulation is pursued in the optimal control framework, where the density of the heat source is a control parameter, and the criteria for optimality is the minimization of the $L_2-$norm declination of the trace of the solution to the Stefan problem from a temperature measurement on the whole domain at the final time. The state vector solves the multiphase Stefan problem in a weak formulation, which is equivalent to Dirichlet problem for the quasilinear parabolic PDE with discontinuous coefficient. The optimal control problem is fully discretized using the method of finite differences. We prove the existence of the optimal control and the convergence of the discrete optimal control problems to the original problem both with respect to cost functional and control. In particular, the convergence of the method of finite differences for the weak solution of the multidimensional multiphase Stefan problem is proved. The proofs are based on achieving a uniform $L_{\infty}$ bound and $W_2^{1,1}$ energy estimate for the discrete multiphase Stefan problem., Comment: 35 pages

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic $p$-Laplacian type reaction-diffusion equation of non-Newtonian elastic filtration \[ u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ 10 \] If the interface is finite, it may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters $p,\beta, sign~b$, and asymptotics of the initial function near its support. In some range of parameters, strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are based on nonlinear scaling laws, and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves. A full description of small-time behaviour of the interfaces and local solutions near the interfaces for slow diffusion case when $p>2$ is presented in a recent paper {\it Abdulla \& Jeli, Europ. J. Appl. Math. 28, 5(2017), 827-853.}, Comment: 19 pages, 1 figure. arXiv admin note: text overlap with arXiv:1605.07279

The Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the electrodes for a given electrode current is analyzed. A PDE constrained optimal control framework in Besov space is pursued, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The state vector is a solution of the second order elliptic PDE in divergence form with bounded measurable coefficients under mixed Neumann/Robin type boundary condition. Existence of the optimal control and Fr\'echet differentiability in the Besov space setting is proved. The formula for the Fr\'echet gradient and optimality condition is derived. Extensive numerical analysis is pursued in the 2D case by implementing the projective gradient method, re-parameterization via principal component analysis (PCA) and Tikhonov regularization., Comment: 29 pages, 11 figures, 1 table

This paper presents computational analysis of the inverse Stefan type free boundary problem, where information on the boundary heat flux is missing and must be found along with the temperature and the free boundary. We pursue optimal control framework introduced in {\it U.G. Abdulla, Inverse Problems and Imaging, 7, 2(2013), 307-340;\ 10, 4(2016), 869--898}, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the quadratic declinations from the available measurements of the temperature distribution at the final moment, phase transition temperature on the free boundary, and the final position of the free boundary. We develop gradient descent algorithm based on Frechet differentiability in Hilbert-Besov spaces complemented with preconditioning or increase of regularity of the Frechet gradient through implementation of the Riesz representation theorem. Three model examples with various levels of complexity are considered. Extensive comparative analysis through implementation of preconditioning and Tikhonov regularization, calibration of preconditioning and regularization parameters, effect of noisy data, comparison of simultaneous identification of control parameters vs. nested optimization is pursued., Comment: 31 pages, 15 figures

We analyze the state constrained inverse Stefan type parabolic free boundary problem as an optimal control problem in the Sobolev-Besov spaces framework. Boundary heat flux, density of heat sources, and free boundary are components of the control vector. Cost functional is the sum of the $L_2$-norm declinations of the temperature measurement at the final moment, the phase transition temperature, the final position of the free boundary, and the penalty term, taking into account the state constraint on the temperature. We prove the existence of optimal control, Frechet differentiability, and optimality condition in the Besov spaces under minimal regularity assumptions on the data. We pursue space-time discretization through finite differences and prove that the sequence of discrete optimal control problems converges to the original problem both with respect to functional and control., Comment: arXiv admin note: text overlap with arXiv:1710.08466

Inverse Stefan problem arising in modeling of laser ablation of biomedical tissues is analyzed, where information on the coefficients, heat flux on the fixed boundary, and density of heat sources are missing and must be found along with the temperature and free boundary. Optimal control framework is employed, where the missing data and the free boundary are components of the control vector, and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary. Discretization by finite differences is pursued, and convergence of the discrete optimal control problems to the original problem is proven.

We prove the short-time asymptotic formula for the interfaces and local solutions near the interfaces for the nonlinear double degenerate reaction-diffusion equation of turbulent filtration with strong absorption \[ u_t=\Big(|(u^{m})_x|^{p-1}(u^{m})_x\Big)_x-bu^{\beta}, \, mp>1, \, \beta >0. \] Full classification is pursued in terms of the nonlinearity parameters $m, p,\beta$ and asymptotics of the initial function near its support. Numerical analysis using a weighted essentially nonoscillatory (WENO) scheme with interface capturing is implemented, and comparison of numerical and analytical results is presented., Comment: 29 pages, 12 fugures

This paper presents full classification of second minimal odd periodic orbits of a continuous endomorphisms on the real line. A $(2k+1)$-periodic orbit ($k\geq 3$) is called second minimal for the map $f$, if $2k-1$ is a minimal period of $f$ in the Sharkovskii ordering. We prove that there are $4k-3$ types of second minimal $(2k+1)$-orbits, each characterized with unique cyclic permutation and directed graph of transitions with accuracy up to inverses., Comment: 32 pages, 39 figures, 2 tables

This paper introduces the notion of second minimal $n$-periodic orbit of the continuous map on the interval according as if $n$ is a successor of the minimal period of the map in Sharkovski ordering. We pursue classification of second minimal $7$-orbits in terms of cyclic permutations and digraphs. It is proved that there are 9 types of second minimal orbits with accuracy up to inverses. The result is applied to the problem on the distribution of periodic windows within the chaotic regime of the bifurcation diagram of the one-parameter family of unimodal maps. It is revealed that by fixing the maximum number of appearances of the periodic windows there is a universal pattern of distribution. In particular, the first appearance of all the orbits is always a minimal orbit, while the second appearance is a second minimal orbit. It is observed that the second appearance of 7-orbit is a second minimal 7-orbit with Type 1 digraph. The reason for the relevance of the Type 1 second minimal orbit is the fact that the topological structure of the unimodal map with single maximum is equivalent to the structure of the Type 1 piecewise monotonic endomorphism associated with the second minimal 7-orbit. Yet another important report of this paper is the revelation of the universal pattern dynamics with respect to increased number of appearances., Comment: 44 pages, 25 figures, 8 tables

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic $p$-Laplacian type reaction-diffusion equation of non-Newtonian elastic filtration \[ u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ p>2, \beta >0 \] The interface may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters $p,\beta, sign~b$, and asymptotics of the initial function near its support. In all cases, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. The methods of the proof are based on nonlinear scaling laws, and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves., Comment: 22 pages, 1 figure

We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and density of the sources are missing and must be found along with the temperature and the free boundary. We pursue optimal control framework where boundary heat flux, density of sources, and free boundary are components of the control vector. The optimality criteria consists of the minimization of the $L_2$-norm declinations of the temperature measurements at the final moment, phase transition temperature, and final position of the free boundary. We prove the Frechet differentiability in Besov spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov spaces for the numerical solution of the inverse Stefan problem., Comment: 21 pages

This paper introduces the notion of $log$-regularity (or $log$-irregularity) of the boundary point $\zeta$ (possibly $\zeta=\infty$) of the arbitrary open subset $\Omega$ of the Greenian deleted neigborhood of $\zeta$ in $R^2$ concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the $log$-harmonic measure of $\zeta$ is null (or positive). A necessary and sufficient condition for the removability of the logarithmic singularity, that is to say for the existence of a unique solution to the Dirichlet problem in $\Omega$ in a class $O(\log |\cdot - \zeta|)$ is established in terms of the Wiener test for the $log$-regularity of $\zeta$. From a topological point of view, the Wiener test at $\zeta$ presents the minimal thinness criteria of sets near $\zeta$ in minimal fine topology. Precisely, the open set $\Omega$ is a deleted neigborhood of $\zeta$ in minimal fine topology if and only if $\zeta$ is $log$-irregular. From the probabilistic point of view, the Wiener test presents asymptotic law for the $log$-Brownian motion near $\zeta$ conditioned on the logarithmic kernel with pole at $\zeta$., Comment: arXiv admin note: text overlap with arXiv:1010.4263

We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $L_2$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $L_{\infty}$ bound, and $W_2^{1,1}$-energy estimate for the discrete multiphase Stefan problem., Comment: 26 pages

We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. In {\it Inverse Problems and Imaging, 7, 2(2013), 307-340} we proved well-posedness in Sobolev spaces framework and convergence of time-discretized optimal control problems. In this paper we perform full discretization and prove convergence of the discrete optimal control problems to the original problem both with respect to cost functional and control., Comment: 33 pages. arXiv admin note: substantial text overlap with arXiv:1203.4866