Abstract The efficient numerical solution of the oneāphase linear inverse Stefan and CauchyāStefan problems is a delicate task owing to the problems' susceptibility to the perturbation of the given data. ā¦
Abstract The efficient numerical solution of the oneāphase linear inverse Stefan and CauchyāStefan problems is a delicate task owing to the problems' susceptibility to the perturbation of the given data. In this context, heuristic a posteriori error indicators are constructed for such inverse problems with noisy data in two dimensions (2D). Given a fixed computational effort, the estimator controls the error due to discretization by the method of fundamental solution (MFS). It is accomplished through two meanādriven doubleāfiltering algorithms. Numerical results substantiate the effectiveness of the algorithms.
In this paper we investigate the regularizing behavior of two-phase Stefan problem near initial Lipschitz data.A description of the regularizing phenomena is given in terms of the corresponding space-time scale.I.
In this paper we investigate the regularizing behavior of two-phase Stefan problem near initial Lipschitz data.A description of the regularizing phenomena is given in terms of the corresponding space-time scale.I.
We derive non-asymptotic minimax bounds for the Hausdorff estimation of $d$-dimensional submanifolds $M \subset \mathbb{R}^D$ with (possibly) non-empty boundary $\partial M$. The model reunites and extends the most prevalent $\mathcal{C}^2$-type ā¦
We derive non-asymptotic minimax bounds for the Hausdorff estimation of $d$-dimensional submanifolds $M \subset \mathbb{R}^D$ with (possibly) non-empty boundary $\partial M$. The model reunites and extends the most prevalent $\mathcal{C}^2$-type set estimation models: manifolds without boundary, and full-dimensional domains. We consider both the estimation of the manifold $M$ itself and that of its boundary $\partial M$ if non-empty. Given $n$ samples, the minimax rates are of order $O\bigl((\log n/n)^{2/d}\bigr)$ if $\partial M = \emptyset$ and $O\bigl((\log n/n)^{2/(d+1)}\bigr)$ if $\partial M \neq \emptyset$, up to logarithmic factors. In the process, we develop a Voronoi-based procedure that allows to identify enough points $O\bigl((\log n/n)^{2/(d+1)}\bigr)$-close to $\partial M$ for reconstructing it.
Abstract The displacement of a slightly compressible liquid by another in a porous medium has been modelled. This problem, which involves a moving boundary, has been numerically solved for the ā¦
Abstract The displacement of a slightly compressible liquid by another in a porous medium has been modelled. This problem, which involves a moving boundary, has been numerically solved for the oneādimensional case by using the Galerkin method. State and parameter estimation have been carried out using the extended Kalman filter.
We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of ā¦
We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis upper L squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2(L^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> error of the temperature and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis upper H Superscript negative 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>ā<!-- ā --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2(H^{-1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> error of the enthalpy. Moreover, they allow us to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) and Newton methods. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.
Formulae display:?Mathematical formulae have been encoded as MathML and are displayed in this HTML version using MathJax in order to improve their display. Uncheck the box to turn MathJax off. ā¦
Formulae display:?Mathematical formulae have been encoded as MathML and are displayed in this HTML version using MathJax in order to improve their display. Uncheck the box to turn MathJax off. This feature requires Javascript. Click on a formula to zoom.
In this paper we derive rates of convergence for regularizations of the multidimensional two-phase Stefan problem and use the regularized problems to define backward-difference in time and CĀ° piecewise-linear in ā¦
In this paper we derive rates of convergence for regularizations of the multidimensional two-phase Stefan problem and use the regularized problems to define backward-difference in time and CĀ° piecewise-linear in space Galerkin approximations.We find an L2 rate of convergence of order \^ in the e-regularization and an L rate of convergence of order (h2/e + Ai/ \6F) in the Galerkin estimates which leads to the natural choices Ā£ ~ A4/3, A; ~ A4/3, and a resulting 0(/i2//3) L2 rate of convergence of the numerical scheme to the solution of the differential equation.An essentially 0(h) rate is demonstrated when e = 0 and A/ ~ h2 in our Galerkin scheme under a boundedness hypothesis on the Galerkin approximations.The latter result is consistent with computational experience.
Two a posteriori error estimates are discussed for the p-Laplace problem. Up to errors in their numerical computation, they provide a guaranteed upper bound for the W1,p-seminorm and a weighted ā¦
Two a posteriori error estimates are discussed for the p-Laplace problem. Up to errors in their numerical computation, they provide a guaranteed upper bound for the W1,p-seminorm and a weighted W1,2-seminorm of u ā uh. The first, sharper a posteriori estimate is based on the numerical solution of local interface problems. The standard residual-based error estimate is addressed with emphasis on involved constants, determined as local eigenvalues. Numerical examples that illustrate the performance of these estimators can be found in [3].
This paper proposes a posteriori error estimation of gradient recovery-type for the Ciarlet-Raviart formulation of the first biharmonic problem.By the appropriate modification of Weighted Cleā²ment-type interpolation,the paper gives the proper ā¦
This paper proposes a posteriori error estimation of gradient recovery-type for the Ciarlet-Raviart formulation of the first biharmonic problem.By the appropriate modification of Weighted Cleā²ment-type interpolation,the paper gives the proper scaling of the gradient recovery leading to both lower and upper estimation.
The moving boundary of a melting solid is estimated from temperature and flux measurements performed on the cold side of the solid only. The lack of measurement in the liquid ā¦
The moving boundary of a melting solid is estimated from temperature and flux measurements performed on the cold side of the solid only. The lack of measurement in the liquid phase prevents the moving boundary to be recovered by straightforward use of the direct Stefan solution. The estimation of this moving boundary is an inverse problem that requires particular solving methods. An algorithm is used, based on a regularized least square approach, that is extended to this nonlinear case by a predictive sliding horizon technique.
This paper considers estimation of an unknown state function in the heat equation with state-dependent parameter values. The work is motivated by phase transitions in physical media, e.g., thawing of ā¦
This paper considers estimation of an unknown state function in the heat equation with state-dependent parameter values. The work is motivated by phase transitions in physical media, e.g., thawing of water or foodstuff, welding and casting processes. We point out that known solution to standard Stefan problem solutions can be recovered with this formalism, and then propose a simple phase transition estimator that relies only on boundary measurements. Simulations indicate that the estimates converge for noise-free measurements.
Let x t be a diffusion process observed via a noisy sensor, whose output is yt We consider the problem of evaluating the maximum a posteriori trajectory {xs0ā¤ s ā¤ ā¦
Let x t be a diffusion process observed via a noisy sensor, whose output is yt We consider the problem of evaluating the maximum a posteriori trajectory {xs0ā¤ s ā¤ t Based on results of Stratonovich [1] and Ikeda-Watanabe [2], we show that this estimator is given by the solution of an appropriate variational problem which is a slight modification of the "minimum energy" estimator. We compare our results to the non-linear filtering theory and show that for problems which possess a finite dimensional solution, our approach yields also explicit filters. For linear diffusions observed via linear sensors, these filters are identical to the Kalman-filter