Prof. Dr. Talal Rahman | Stamford University, Bangladesh (original) (raw)
Papers by Prof. Dr. Talal Rahman
Mathematics and Visualization, 2018
A model combining the first-order and the second-order variational regularizations for the purpos... more A model combining the first-order and the second-order variational regularizations for the purpose of 3D surface reconstruction based on 2D sparse data is proposed. The model includes a hybrid fidelity constraint which allows the initial conditions to be switched flexibly between vectors and elevations. A numerical algorithm based on the augmented Lagrangian method is also proposed. The numerical experiments are presented, showing its excellent performance both in designing cartoon characters, as well as in recovering oriented three dimensional maps from contours or points with elevation information.
Lecture Notes in Computational Science and Engineering, 2018
A complete multidimential TV-Stokes model is proposed based on smoothing a gradient field in the ... more A complete multidimential TV-Stokes model is proposed based on smoothing a gradient field in the first step and reconstruction of the multidimensional image from the gradient field. It is the correct extension of the original two dimensional TV-Stokes to multidimensions. Numerical algorithm using the Chambolle's semi-implicit dual formula is proposed. Numerical results applied to denoising 3D images and movies are presented. They show excellent performance in avoiding the staircase effect, and preserving fine structures.
In this paper, we propose a two-level overlapping additive Schwarz domain decomposition precondit... more In this paper, we propose a two-level overlapping additive Schwarz domain decomposition preconditioner for the symmetric interior penalty discontinuous Galerkin method for the second order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.
In this paper we introduce an additive Schwarz method for a Crouzeix-Raviart Finite Volume Elemen... more In this paper we introduce an additive Schwarz method for a Crouzeix-Raviart Finite Volume Element (CRFVE) discretization of a second order elliptic problem with discontinuous coefficients, where the discontinuities are both inside the subdomains and across and along the subdomain boundaries. We show that, depending on the distribution of the coefficient in the model problem, the parameters describing the GMRES convergence rate of the proposed method depend linearly or quadratically on the mesh parameters H/h. Also, under certain restrictions on the distribution of the coefficient, the convergence of the GMRES method is independent of jumps in the coefficient.
In this paper an overlapping additive Schwarz method with a spectrally enriched coarse space is p... more In this paper an overlapping additive Schwarz method with a spectrally enriched coarse space is proposed. The method is for solving the standard Finite Element discretization of second order elliptic problems in there dimensions with discontinuous coefficients, where the discontinuities are inside subdomains and across subdomain boundaries. In case when the coarse space is large enough the convergence of the PCG method is independent of jumps in the coefficient.
ArXiv, 2021
In this paper, we extend the additive average Schwarz method to solve second order elliptic bound... more In this paper, we extend the additive average Schwarz method to solve second order elliptic boundary value problems with heterogeneous coefficients inside the subdomains and across subdomain interfaces by the mortar technique, where the mortar finite element discretization is on nonmatching meshes. In this two-level method, we enrich the coarse space in two different ways, i.e., by adding eigenfunctions of two variants of the generalized eigenvalue problems. We prove that the condition number for the system of algebraic equations resulting from the extended additive average Schwarz method, corresponding to both coarse spaces, is of the order O(H/h) and independent of jumps of the coefficients, where H and h are the mesh parameters.
ArXiv, 2020
The paper presents a fully coupled TV-Stokes model, and propose an algorithm based on alternating... more The paper presents a fully coupled TV-Stokes model, and propose an algorithm based on alternating minimization of the objective functional whose first iteration is exactly the modified TV-Stokes model proposed earlier. The model is a generalization of the second order Total Generalized Variation model. A convergence analysis is given.
ArXiv, 2020
We propose a set of iterative regularization algorithms for the TV-Stokes model to restore images... more We propose a set of iterative regularization algorithms for the TV-Stokes model to restore images from noisy images with Gaussian noise. These are some extensions of the iterative regularization algorithm proposed for the classical Rudin-Osher-Fatemi (ROF) model for image reconstruction, a single step model involving a scalar field smoothing, to the TV-Stokes model for image reconstruction, a two steps model involving a vector field smoothing in the first and a scalar field smoothing in the second. The iterative regularization algorithms proposed here are Richardson's iteration like. We have experimental results that show improvement over the original method in the quality of the restored image. Convergence analysis and numerical experiments are presented.
In this article we derive a priori error estimates for the hphphp-version of the mortar finite elem... more In this article we derive a priori error estimates for the hphphp-version of the mortar finite element method for parabolic initial-boundary value problems. Both semidiscrete and fully discrete methods are analysed in L2L^2L2- and H1H^1H1-norms. The superconvergence results for the solution of the semidiscrete problem are studied in an eqivalent negative norm, with an extra regularity assumption. Numerical experiments are conducted to validate the theoretical findings.
ArXiv, 2020
A complete multidimential TV-Stokes model is proposed based on smoothing a gradient field in the ... more A complete multidimential TV-Stokes model is proposed based on smoothing a gradient field in the first step and reconstruction of the multidimensional image from the gradient field. It is the correct extension of the original two dimensional TV-Stokes to multidimensions. Numerical algorithm using the Chambolle's semi-implicit dual formula is proposed. Numerical results applied to denoising 3D images and movies are presented. They show excellent performance in avoiding the staircase effect, and preserving fine structures.
In this paper we investigate an additive and a hybrid Schwarz method for solving systems of algeb... more In this paper we investigate an additive and a hybrid Schwarz method for solving systems of algebraic equations resulting from the approximation of second order elliptic boundary value problems with (highly) discontinuous coefficients. The discretization is obtained by using the mortar finite element method on nonmatching meshes, a technique which was first introduced by Bernardi-Maday-Patera [BMP94]. Several efficient iterative methods have thereafter been developed for the mortar element, see for example [CW96, Dry96, Dry97, AMW99, CDS99, BDR00, BDW99, GP00, WK01], and the references therein. The work of this paper is a continuation of the work done in [BDR00], where two variants of the additive Schwarz methods were proposed, the average method and the coarse reformulated average method. The reformulated variant is obtained from the average variant by simply replacing its coarse space by the sum of two special coarse spaces, one associated with the subdomains and the other one def...
We propose a new, harmonically enriched multiscale coarse space (HEM) for domain decomposition me... more We propose a new, harmonically enriched multiscale coarse space (HEM) for domain decomposition methods. For a coercive high contrast model problem, we show how to enrich the coarse space so that the method is robust against any variations and discontinuities in the problem parameters both inside subdomains and across and along subdomain boundaries. We prove our results for an enrichment strategy based on solving simple, lower dimensional eigenvalue problems on the interfaces between subdomains, and we call the resulting coarse space the spectral harmonically enriched multiscale coarse space (SHEM). We then also give a variant that performs equally well in practice, and does not require the solve of eigenvalue problems, which we call non-spectral harmonically enriched multiscale coarse space (NSHEM). Our enrichment process naturally reaches the optimal coarse space represented by the full discrete harmonic space, which enables us to turn the method into a direct solver (OHEM). We als...
In this paper, we present two variants of the Additive Schwarz Method (ASM) for a Crouzeix-Raviar... more In this paper, we present two variants of the Additive Schwarz Method (ASM) for a Crouzeix-Raviart finite volume (CRFV) discretization of the second order elliptic problem with discontinuous coefficients, where the discontinuities are only across subdomain boundaries. The resulting system, which is nonsymmetric, is solved using the preconditioned GMRES iteration, where in one variant of the ASM the preconditioner is symmetric while in the other variant it is nonsymmetric. The proposed methods are almost optimal, in the sense that the convergence of the GMRES iteration, in the both cases, depend only poly-logarithmically on the mesh parameters.
ArXiv, 2020
Three dimensional surface reconstruction based on two dimensional sparse information in the form ... more Three dimensional surface reconstruction based on two dimensional sparse information in the form of only a small number of level lines of the surface with moderately complex structures, containing both structured and unstructured geometries, is considered in this paper. A new model has been proposed which is based on the idea of using normal vector matching combined with a first order and a second order total variation regularizers. A fast algorithm based on the augmented Lagrangian is also proposed. Numerical experiments are provided showing the effectiveness of the model and the algorithm in reconstructing surfaces with detailed features and complex structures for both synthetic and real world digital maps.
In many real physical phenomena, there is heterogeneity, e.g., in some ground flow problems in he... more In many real physical phenomena, there is heterogeneity, e.g., in some ground flow problems in heterogeneousmedia.When somefinite element discretizationmethod is applied to a physical model, one usually obtains a discrete problemwhich is very hard to solve by a preconditioned iterative method like, e.g., Preconditioned Conjugate Gradient (PCG) method. One of the most popular methods of constructing parallel preconditioners are domain decomposition methods, in particular, non-overlapping or overlapping additive Schwarz methods (ASM), cf. e.g., [16]. In Schwarz methods, a crucial role is played by carefully constructed coarse spaces. For multiscale problems with heterogeneous coefficients standard overlapping Schwarz methods with classical coarse spaces fail often to be fast and robust solvers. Therefore we need new coarse spaces which are adaptive to the jumps of the coefficients, i.e. the convergence of the ASM method is independent of the distribution and the magnitude of the coeff...
Lecture Notes in Computer Science, 2009
Lecture Notes in Computer Science
In this paper, we propose a two-step algorithm for denoising digital images with additive noise. ... more In this paper, we propose a two-step algorithm for denoising digital images with additive noise. Observing that the isophote directions of an image correspond to an incompressible velocity field, we impose the constraint of zero divergence on the tangential field. Combined with an energy minimization problem corresponding to the smoothing of tangential vectors, this constraint gives rise to a nonlinear Stokes equation where the nonlinearity is in the viscosity function. Once the isophote directions are found, an image is reconstructed that fits those directions by solving another nonlinear partial differential equation. In both steps, we use finite difference schemes to solve. We present several numerical examples to show the effectiveness of our approach.
Lecture Notes in Computational Science and Engineering, 2014
SIAM Journal on Scientific Computing, 2011
Mathematics and Visualization, 2018
A model combining the first-order and the second-order variational regularizations for the purpos... more A model combining the first-order and the second-order variational regularizations for the purpose of 3D surface reconstruction based on 2D sparse data is proposed. The model includes a hybrid fidelity constraint which allows the initial conditions to be switched flexibly between vectors and elevations. A numerical algorithm based on the augmented Lagrangian method is also proposed. The numerical experiments are presented, showing its excellent performance both in designing cartoon characters, as well as in recovering oriented three dimensional maps from contours or points with elevation information.
Lecture Notes in Computational Science and Engineering, 2018
A complete multidimential TV-Stokes model is proposed based on smoothing a gradient field in the ... more A complete multidimential TV-Stokes model is proposed based on smoothing a gradient field in the first step and reconstruction of the multidimensional image from the gradient field. It is the correct extension of the original two dimensional TV-Stokes to multidimensions. Numerical algorithm using the Chambolle's semi-implicit dual formula is proposed. Numerical results applied to denoising 3D images and movies are presented. They show excellent performance in avoiding the staircase effect, and preserving fine structures.
In this paper, we propose a two-level overlapping additive Schwarz domain decomposition precondit... more In this paper, we propose a two-level overlapping additive Schwarz domain decomposition preconditioner for the symmetric interior penalty discontinuous Galerkin method for the second order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.
In this paper we introduce an additive Schwarz method for a Crouzeix-Raviart Finite Volume Elemen... more In this paper we introduce an additive Schwarz method for a Crouzeix-Raviart Finite Volume Element (CRFVE) discretization of a second order elliptic problem with discontinuous coefficients, where the discontinuities are both inside the subdomains and across and along the subdomain boundaries. We show that, depending on the distribution of the coefficient in the model problem, the parameters describing the GMRES convergence rate of the proposed method depend linearly or quadratically on the mesh parameters H/h. Also, under certain restrictions on the distribution of the coefficient, the convergence of the GMRES method is independent of jumps in the coefficient.
In this paper an overlapping additive Schwarz method with a spectrally enriched coarse space is p... more In this paper an overlapping additive Schwarz method with a spectrally enriched coarse space is proposed. The method is for solving the standard Finite Element discretization of second order elliptic problems in there dimensions with discontinuous coefficients, where the discontinuities are inside subdomains and across subdomain boundaries. In case when the coarse space is large enough the convergence of the PCG method is independent of jumps in the coefficient.
ArXiv, 2021
In this paper, we extend the additive average Schwarz method to solve second order elliptic bound... more In this paper, we extend the additive average Schwarz method to solve second order elliptic boundary value problems with heterogeneous coefficients inside the subdomains and across subdomain interfaces by the mortar technique, where the mortar finite element discretization is on nonmatching meshes. In this two-level method, we enrich the coarse space in two different ways, i.e., by adding eigenfunctions of two variants of the generalized eigenvalue problems. We prove that the condition number for the system of algebraic equations resulting from the extended additive average Schwarz method, corresponding to both coarse spaces, is of the order O(H/h) and independent of jumps of the coefficients, where H and h are the mesh parameters.
ArXiv, 2020
The paper presents a fully coupled TV-Stokes model, and propose an algorithm based on alternating... more The paper presents a fully coupled TV-Stokes model, and propose an algorithm based on alternating minimization of the objective functional whose first iteration is exactly the modified TV-Stokes model proposed earlier. The model is a generalization of the second order Total Generalized Variation model. A convergence analysis is given.
ArXiv, 2020
We propose a set of iterative regularization algorithms for the TV-Stokes model to restore images... more We propose a set of iterative regularization algorithms for the TV-Stokes model to restore images from noisy images with Gaussian noise. These are some extensions of the iterative regularization algorithm proposed for the classical Rudin-Osher-Fatemi (ROF) model for image reconstruction, a single step model involving a scalar field smoothing, to the TV-Stokes model for image reconstruction, a two steps model involving a vector field smoothing in the first and a scalar field smoothing in the second. The iterative regularization algorithms proposed here are Richardson's iteration like. We have experimental results that show improvement over the original method in the quality of the restored image. Convergence analysis and numerical experiments are presented.
In this article we derive a priori error estimates for the hphphp-version of the mortar finite elem... more In this article we derive a priori error estimates for the hphphp-version of the mortar finite element method for parabolic initial-boundary value problems. Both semidiscrete and fully discrete methods are analysed in L2L^2L2- and H1H^1H1-norms. The superconvergence results for the solution of the semidiscrete problem are studied in an eqivalent negative norm, with an extra regularity assumption. Numerical experiments are conducted to validate the theoretical findings.
ArXiv, 2020
A complete multidimential TV-Stokes model is proposed based on smoothing a gradient field in the ... more A complete multidimential TV-Stokes model is proposed based on smoothing a gradient field in the first step and reconstruction of the multidimensional image from the gradient field. It is the correct extension of the original two dimensional TV-Stokes to multidimensions. Numerical algorithm using the Chambolle's semi-implicit dual formula is proposed. Numerical results applied to denoising 3D images and movies are presented. They show excellent performance in avoiding the staircase effect, and preserving fine structures.
In this paper we investigate an additive and a hybrid Schwarz method for solving systems of algeb... more In this paper we investigate an additive and a hybrid Schwarz method for solving systems of algebraic equations resulting from the approximation of second order elliptic boundary value problems with (highly) discontinuous coefficients. The discretization is obtained by using the mortar finite element method on nonmatching meshes, a technique which was first introduced by Bernardi-Maday-Patera [BMP94]. Several efficient iterative methods have thereafter been developed for the mortar element, see for example [CW96, Dry96, Dry97, AMW99, CDS99, BDR00, BDW99, GP00, WK01], and the references therein. The work of this paper is a continuation of the work done in [BDR00], where two variants of the additive Schwarz methods were proposed, the average method and the coarse reformulated average method. The reformulated variant is obtained from the average variant by simply replacing its coarse space by the sum of two special coarse spaces, one associated with the subdomains and the other one def...
We propose a new, harmonically enriched multiscale coarse space (HEM) for domain decomposition me... more We propose a new, harmonically enriched multiscale coarse space (HEM) for domain decomposition methods. For a coercive high contrast model problem, we show how to enrich the coarse space so that the method is robust against any variations and discontinuities in the problem parameters both inside subdomains and across and along subdomain boundaries. We prove our results for an enrichment strategy based on solving simple, lower dimensional eigenvalue problems on the interfaces between subdomains, and we call the resulting coarse space the spectral harmonically enriched multiscale coarse space (SHEM). We then also give a variant that performs equally well in practice, and does not require the solve of eigenvalue problems, which we call non-spectral harmonically enriched multiscale coarse space (NSHEM). Our enrichment process naturally reaches the optimal coarse space represented by the full discrete harmonic space, which enables us to turn the method into a direct solver (OHEM). We als...
In this paper, we present two variants of the Additive Schwarz Method (ASM) for a Crouzeix-Raviar... more In this paper, we present two variants of the Additive Schwarz Method (ASM) for a Crouzeix-Raviart finite volume (CRFV) discretization of the second order elliptic problem with discontinuous coefficients, where the discontinuities are only across subdomain boundaries. The resulting system, which is nonsymmetric, is solved using the preconditioned GMRES iteration, where in one variant of the ASM the preconditioner is symmetric while in the other variant it is nonsymmetric. The proposed methods are almost optimal, in the sense that the convergence of the GMRES iteration, in the both cases, depend only poly-logarithmically on the mesh parameters.
ArXiv, 2020
Three dimensional surface reconstruction based on two dimensional sparse information in the form ... more Three dimensional surface reconstruction based on two dimensional sparse information in the form of only a small number of level lines of the surface with moderately complex structures, containing both structured and unstructured geometries, is considered in this paper. A new model has been proposed which is based on the idea of using normal vector matching combined with a first order and a second order total variation regularizers. A fast algorithm based on the augmented Lagrangian is also proposed. Numerical experiments are provided showing the effectiveness of the model and the algorithm in reconstructing surfaces with detailed features and complex structures for both synthetic and real world digital maps.
In many real physical phenomena, there is heterogeneity, e.g., in some ground flow problems in he... more In many real physical phenomena, there is heterogeneity, e.g., in some ground flow problems in heterogeneousmedia.When somefinite element discretizationmethod is applied to a physical model, one usually obtains a discrete problemwhich is very hard to solve by a preconditioned iterative method like, e.g., Preconditioned Conjugate Gradient (PCG) method. One of the most popular methods of constructing parallel preconditioners are domain decomposition methods, in particular, non-overlapping or overlapping additive Schwarz methods (ASM), cf. e.g., [16]. In Schwarz methods, a crucial role is played by carefully constructed coarse spaces. For multiscale problems with heterogeneous coefficients standard overlapping Schwarz methods with classical coarse spaces fail often to be fast and robust solvers. Therefore we need new coarse spaces which are adaptive to the jumps of the coefficients, i.e. the convergence of the ASM method is independent of the distribution and the magnitude of the coeff...
Lecture Notes in Computer Science, 2009
Lecture Notes in Computer Science
In this paper, we propose a two-step algorithm for denoising digital images with additive noise. ... more In this paper, we propose a two-step algorithm for denoising digital images with additive noise. Observing that the isophote directions of an image correspond to an incompressible velocity field, we impose the constraint of zero divergence on the tangential field. Combined with an energy minimization problem corresponding to the smoothing of tangential vectors, this constraint gives rise to a nonlinear Stokes equation where the nonlinearity is in the viscosity function. Once the isophote directions are found, an image is reconstructed that fits those directions by solving another nonlinear partial differential equation. In both steps, we use finite difference schemes to solve. We present several numerical examples to show the effectiveness of our approach.
Lecture Notes in Computational Science and Engineering, 2014
SIAM Journal on Scientific Computing, 2011