2,726 research outputs found
On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. II.Convergence of the Method of Finite Differences
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 -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:1203.486
The Wiener Test for the Removability of the Logarithmic Singularity for the Elliptic PDEs with Measurable Coefficients and Its Consequences
This paper introduces the notion of -regularity (or -irregularity)
of the boundary point (possibly ) of the arbitrary open
subset of the Greenian deleted neigborhood of in
concerning second order uniformly elliptic equations with bounded and
measurable coefficients, according as whether the -harmonic measure of
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 in a class is established in terms of the Wiener test for the
-regularity of . From a topological point of view, the Wiener test
at presents the minimal thinness criteria of sets near in
minimal fine topology. Precisely, the open set is a deleted
neigborhood of in minimal fine topology if and only if is
-irregular. From the probabilistic point of view, the Wiener test presents
asymptotic law for the -Brownian motion near conditioned on the
logarithmic kernel with pole at .Comment: arXiv admin note: text overlap with arXiv:1010.426
Optimal Stefan Problem
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 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
bound and energy estimate for the discrete multiphase Stefan
problem.Comment: 35 page
Frechet Differentiability in Besov Spaces in the Optimal Control of Parabolic Free Boundary Problems
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 -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 page
Identification of Parameters for Large-scale Kinetic Models
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 Table
Evolution of Interfaces for the Nonlinear Parabolic p-Laplacian Type Reaction-Diffusion Equations
We present a full classification of the short-time behaviour of the
interfaces and local solutions to the nonlinear parabolic -Laplacian type
reaction-diffusion equation of non-Newtonian elastic filtration 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 , 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 figur
Optimal Control of Coefficients in Parabolic Free Boundary Problems Modeling Laser Ablation
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
Optimal Control of the Multiphase Stefan Problem
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 -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 bound, and
-energy estimate for the discrete multiphase Stefan problem.Comment: 26 page
Evolution of Interfaces for the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption. II. Fast Diffusion Case
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 Full classification is pursued in terms of the nonlinearity
parameters 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 () was
presented in .Comment: arXiv admin note: text overlap with arXiv:1811.0727
The Second Law For the Transitions Between the Non-equilibrium Steady States
We show that the system entropy change for the transitions between
non-equilibrium steady states arbitrarily far from equilibrium for any
constituting process is given by the relative entropy of the distributions of
these steady states. This expression is then shown to relate to the dissipation
relations of both Vaikuntanathan and Jarzynski [EPL 87, 60005 (2009)] and
Kawai, Parrondo and Van den Broeck [Phys. Rev. Lett. 98, 080602 (2007)] in the
case of energy-conserving driving.Comment: 7 pages, no fig
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