Browsing by Subject "inverse problem"

Now showing 1 - 17 of 17

Access status: Open Access ,
A dynamical inverse problem for a parabolic equations
(2006) Maksimov, Vâčeslav
A problem of dynamical reconstruction of unknown distributed or boundary disturbances acting upon nonlinear parabolic equations is discussed. A regularized algorithm which allows us to reconstruct disturbances synchro with the process under consideration is designed. This algorithm is stable with respect to informational noises and computational errors.
Access status: Open Access ,
An inverse backward problem for degenerate two-dimensional parabolic equation
(Wydawnictwa AGH, 2020) Atifi, Khalid; Essoufi, El-Hassan; Khouiti, Bouchra
This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation $\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),$ where $\Omega$ is an open, bounded subset of $\mathbb{R}^2$, $a \in C^1(\bar{\Omega})$ with $a\geqslant 0$ everywhere, and $f\in L^{2}(\Omega \times (0,T))$, with initial and boundary conditions $u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,$ from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach.
Access status: Open Access ,
Classical and weak solutions for semilinear parabolic equations with Preisach hysteresis
(2008) Jais, Mathias
We consider the solvability of the semilinear parabolic differential equation $\frac{\partial u}{\partial t}(x,t)- \Delta u(x,t) + c(x,t)u(x,t) = \mathcal{P}(u) + \gamma (x,t)$ in a cylinder $D=\Omega \times (0,T)$, where $\mathcal{P}$ is a hysteresis operator of Preisach type. We show that the corresponding initial boundary value problems have unique classical solutions. We further show that using this existence and uniqueness result, one can determine the properties of the Preisach operator $\mathcal{P}$ from overdetermined boundary data.
Access status: Open Access ,
Comparing parallel programming environments for the joint inversion of geoelectrical data
(Wydawnictwa AGH, 2009) Pięta, Anna; Bała, Justyna
The article presents the comparison of the implementation of the inverse problem in geoelectrical methods in two different parallel computational environments. Combination of Monte Carlo method and Multistart algorithm was applied in the inversion process. Parallelization was done by fine grain decomposition. Execution time, speed-up and efficiency received for parallel algorithms in both computational environments were presented and analyzed.
Access status: Restricted ,
Estymacja gęstości jako problem odwrotny
(Data obrony: 2016-07-19) Stawiski, Marcin
Wydział Matematyki Stosowanej
Access status: Restricted ,
Estymacja w problemie odwrotnym Hampela
(Data obrony: 2018-09-05) Obrzut, Katarzyna
Wydział Matematyki Stosowanej
Access status: Open Access ,
Inversion of the Riemann-Liouville operator and its dual using wavelets
(2015) Baccar, Cyrine; Hamadi, Nadia Ben; Herch, Hajer; Meherzi, Fatma
We define and study the generalized continuous wavelet transform associated with the Riemann-Liouville operator that we use to express the new inversion formulas of the Riemann-Liouville operator and its dual.
Access status: Restricted ,
Metody bootstrapowe w problemie Wicksella
(Data obrony: 2015-07-02) Sierpiński, Damian
Wydział Matematyki Stosowanej
Access status: Open Access ,
Nonparametric bootstrap confidence bands for unfolding sphere size distributions
(Wydawnictwa AGH, 2021) Wojdyła, Jakub
The stereological inverse problem of unfolding the distribution of spheres radii from measured planar sections radii, known as the Wicksell's corpuscle problem, is considered. The construction of uniform confidence bands based on the smoothed bootstrap in the Wicksell's problem is presented. Theoretical results on the consistency of the proposed bootstrap procedure are given, where the consistency of the bands means that the coverage probability converges to the nominal level. The finite-sample performance of the proposed method is studied via Monte Carlo simulations and compared with the asymptotic (non-bootstrap) solution described in literature.
Access status: Open Access ,
On a dense minimizer of empirical risk in inverse problems
(2016) Podlewski, Jacek; Szkutnik, Zbigniew
Properties of estimators of a functional parameter in an inverse problem setup are studied. We focus on estimators obtained through dense minimization (as opposed to minimization over δ-nets) of suitably defined empirical risk. At the cost of imposition of a sort of local finite-dimensionality assumption, we fill some gaps in the proofs of results published by Klemelä and Mammen [Ann. Statist. 38 (2010), 482-511]. We also give examples of functional classes that satisfy the modified assumptions.
Access status: Open Access ,
On the computational cost and complexity of stochastic inverse solvers
(Wydawnictwa AGH, 2016) Faliszewski, Piotr; Smołka, Maciej; Schaefer, Robert; Paszyński, Maciej
The goal of this paper is to provide a starting point for investigations into a mainly underdeveloped area of research regarding the computational cost analysis of complex stochastic strategies for solving parametric inverse problems. This area has two main components: solving global optimization problems and solving forward problems (to evaluate the misfit function that we try to minimize). For the first component, we pay particular attention to genetic algorithms with heuristics and to multi-deme algorithms that can be modeled as ergodic Markov chains. We recall a simple method for evaluating the first hitting time for the single-deme algorithm and we extend it to the case of HGS, a multi-deme hierarchic strategy. We focus on the case in which at least the demes in the leaves are well tuned. Finally, we also express the problems of finding local and global optima in terms of a classic complexity theory. We formulate the natural result that finding a local optimum of a function is an NP-complete task, and we argue that finding a global optimum is a much harder, DP-complete, task. Furthermore, we argue that finding all global optima is, possibly, even harder (#P-hard) task. Regarding the second component of solving parametric inverse problems (i.e., regarding the forward problem solvers), we discuss the computational cost of hp-adaptive Finite Element solvers and their rates of convergence with respect to the increasing number of degrees of freedom. The presented results provide a useful taxonomy of problems and methods of studying the computational cost and complexity of various strategies for solving inverse parametric problems. Yet, we stress that our goal was not to deliver detailed evaluations for particular algorithms applied to particular inverse problems, but rather to try to identify possible ways of obtaining such results.
Access status: Open Access ,
On the numerical solution of one inverse problem for a linearized two-dimensional system of Navier-Stokes equations
(Wydawnictwa AGH, 2022) Dženaliev, Muvašarhan Tanabievič; Ramazanov, Murat; Ergaliev, Madi Gabidenovič
The paper studies the numerical solution of the inverse problem for a linearized two-dimensional system of Navier-Stokes equations in a circular cylinder with a final overdetermination condition. For a biharmonic operator in a circle, a generalized spectral problem has been posed. For the latter, a system of eigenfunctions and eigenvalues is constructed, which is used in the work for the numerical solution of the inverse problem in a circular cylinder with specific numerical data. Graphs illustrating the results of calculations are presented.
Access status: Restricted ,
Pasy ufności w modelu regresji odwrotnej
(Data obrony: 2017-09-29) Ruszkowicz, Faustyna
Wydział Matematyki Stosowanej
Access status: Restricted ,
Problem odwrotny do metody Monte Carlo
(Data obrony: 2016-05-05) Kozioł, Piotr
Wydział Matematyki Stosowanej
Access status: Open Access ,
Unfolding spheres size distribution from linear sections with B-splines and EMDS algorithm
(2007) Szkutnik, Zbigniew
The stereological problem of unfolding spheres size distribution from linear sections is formulated as a problem of inverse estimation of a Poisson process intensity function. A singular value expansion of the corresponding integral operator is given. The theory of recently proposed $B$-spline sieved quasi-maximum likelihood estimators is modified to make it applicable to the current problem. Strong $L^{2}$-consistency is proved and convergence rates are given. The estimators are implemented with the recently proposed EMDS algorithm. Promising performance of this new methodology in finite samples is illustrated with a numerical example. Data grouping effects are also discussed.
Access status: Restricted ,
Wersja warstwowa problemu stereologicznego Wicksella
(Data obrony: 2016-11-14) Skała, Anna
Wydział Matematyki Stosowanej
Access status: Restricted ,
Zadanie odwrotne w geofizyce otworowej – rozwiązania dla pojedynczych profilowań geofizyki otworowej i dla kompleksowej interpretacji
(Data obrony: 2010-10-04) Wilk, Monika
Wydział Geologii, Geofizyki i Ochrony Środowiska
The topic of the M.Sc. Thesis was inverse problem in well logging. Several methods of solving it were presented. There were shown some examples of interpretation made for single log as well as for the set of logs. Conventional methods were based on solving single equation and using the program ULTRA for comprehensive interpretation. Paralelly, the author shows efforts to make the interpretation using alternative methods. In case of single log spectra analysis and deconvolution of registered signals were considered. For comprehensive interpretation an expert system was used. All the results were compared with the interpretation from ULTRA program.