Maya Neytcheva

Uppsala University | UU · Department of Information Technology

We consider the iterative solution of algebraic systems, arising in optimal control problems constrained by a partial differential equation with additional box constraints on the state and the control variables, and sparsity imposed on the control. A nonsymmetric two-by-two block preconditioner is analysed and tested for a wide range of problem, re...

Fully implicit Runge–Kutta methods offer the possibility to use high order accurate time discretization to match space discretization accuracy, an issue of significant importance for many large scale problems of current interest, where we may have fine space resolution with many millions of spatial degrees of freedom and long time intervals. In thi...

The use of high order fully implicit Runge-Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this study we consider s...

We develop an efficient and robust iterative framework suitable for solving the linear system of equations resulting from the spectral element discretisation of the curl-curl equation of the total electric field encountered in geophysical controlled-source electromagnetic applications. We use the real-valued equivalent form of the original complex-...

We present an implementation of a fully stage-parallel preconditioner for Radau IIA type fully implicit Runge--Kutta methods, which approximates the inverse of $A_Q$ from the Butcher tableau by the lower triangular matrix resulting from an LU decomposition and diagonalizes the system with as many blocks as stages. For the transformed system, we emp...

By use of Fourier time series expansions in an angular frequency variable, time-harmonic optimal control problems constrained by a linear differential equation decouples for the different frequencies. Hence, for the analysis of a solution method one can consider the frequency as a parameter. There are three variables to be determined, the state sol...

The radial point interpolation meshfree discretization is a very efficient numerical framework for the analysis of piezoelectricity, in which the fundamental electrostatic equations governing piezoelectric media are solved without mesh generation. Due to the mechanical‐electrical coupling property and the piezoelectric constant, the discrete linear...

We present a method for solving optimal control problems constrained by a partial differential equation, where we simultaneously impose sparsity-promoting L1-regularization on the control as well as box constraints on both the control and the state. We focus on numerical implementation aspects and on preconditioners used when solving the arising li...

In this paper we briefly account for the structure of the matrices, arising in various optimal control problems, constrained by PDEs, and how it can be utilized when constructing preconditioners for the arising linear systems to be solved in the optimization framework.

An efficient preconditioning technique used earlier for two-by-two block matrix systems with square matrix blocks is shown to be applicable also for a state variable box-constrained optimal control problem. The problem is penalized by a standard regularization term for the control variable and for the box-constraint, using a Moreau-Yosida penalizat...

We consider the iterative solution of optimal control problems constrained by the time-harmonic parabolic equations. Due to the time-harmonic property of the control equations, a suitable discretization of the corresponding optimality systems leads to a large complex linear system with special two-by-two block matrix of saddle point form. For this...

The recent development of the high performance computer platforms shows a clear trend towards heterogeneity and hierarchy. In order to utilize the computational power, particular attention must be paid to finding new algorithms or adjust existing ones so that they better match the HPC computer architecture. In this work we consider an alternative to...

The availability of high performance computing resources enables us to perform very large numerical simulations and in this way to tackle challenging real life problems. At the same time, in order to efficiently utilize the computational power at our disposal, the ever growing complexity of the computer architecture poses high demands on the algori...

We consider the solution of block-coupled large-scale linear systems of equations, arising from the finite element approximation of the linear elasticity problem. Due to the large scale of the problems we use properly preconditioned iterative methods, where the preconditioners utilize the underlying block matrix structures, involving inner block so...

The governing dynamics of fluid flow is stated as a system of partial differential equations referred to as the Navier-Stokes system. In industrial and scientific applications, fluid flow control becomes an optimization problem where the governing partial differential equations of the fluid flow are stated as constraints. When discretized, the opti...

Saddle point matrices of a special structure arise in optimal control problems. In this paper we consider distributed optimal control for various types of scalar stationary partial differential equations and compare the efficiency of several numerical solution methods. We test the particular case when the arising linear system can be compressed aft...

In this article we construct an efficient preconditioner for solving the algebraic systems arising from discretized optimal control problems with time-periodic Stokes equations, based on a preconditioning technique for stationary Stokes-constrained optimal control problems, considered in an earlier paper by the authors. A simplified analysis of the...

We consider large linear systems of algebraic equations arising from the Finite Element approximation of coupled partial differential equations. As case study we focus on the linear elasticity equations, formulated as a saddle point problem to allow for modelling of purely incompressible materials. Using the notion of the so-called \textit{spectral...

Linear systems with two-by-two block matrices are usually preconditioned by block lower- or upper-triangular systems that require an approximation of the related Schur complement. In this work, in the finite element framework, we consider one special such approximation, namely, the element-wise Schur complement. It is sparse and its construction is...

Using the notion of the so-called \textit{spectral symbol} in the Generalized Locally Toeplitz (GLT) setting, we derive the GLT symbol of the sequence of matrices $\{A_n\}$ approximating the elasticity equations. Further, as the GLT class defines an algebra of matrix sequences and Schur complements are obtained via elementary algebraic operation on...

We consider methods for the numerical simulations of variable density incompressible fluids, modelled by the Navier–Stokes equations. Variable density problems arise, for instance, in interfaces between fluids of different densities in multiphase flows such as appearing in porous media problems. We show that by solving the Navier–Stokes equation fo...

In this work we benchmark the performance of a preconditioned iterative method, used in large scale computer simulations of a geophysical application, namely, the elastic Glacial Isostatic Adjustment model. The model is discretized using the finite element method that gives raise to algebraic systems of equations with matrices that are large, spars...

Complex valued linear algebraic systems arise in many important applications. We present analytical and extensive numerical comparisons of some available numerical solution methods. It is advocated, in particular for large scale ill-conditioned problems, to rewrite the complex-valued system in real valued form leading to a two-by-two block system o...

In this work we develop preconditioners for the iterative solution of the large scale algebraic systems, arising in finite element discretizations of microstructures with an arbitrary number of components, described by the diffusive interface model. The suggested numerical techniques are applied to the study of ternary fluid flow processes.

Two-by-two block matrices arise in various applications, such as in domain decomposition methods or when solving boundary value problems discretised by finite elements from the separation of the node set of the mesh into ‘fine’ and ‘coarse’ nodes. Matrices with such a structure, in saddle point form arise also in mixed variable finite element metho...

We consider two-phase flow problems, modelled by the Cahn–Hilliard equation. In this work, the nonlinear fourth-order equation is decomposed into a system of two coupled second-order equations for the concentration and the chemical potential.We analyse solution methods based on an approximate two-by-two block factorization of the Jacobian of the no...

We present two parallel strategies to compute the inverse of a dense matrix, based on the so-called Sherman-Morrison algorithm and demonstrate their efficiency in memory and runtime on multicore CPU and GPU-equipped computers. Our methods are shown to be much more efficient than the direct method to compute the inverse of a nonsingular dense matrix...

This paper deals with preconditioners for the iterative solution of the discrete Oseen's problem with variable viscosity. The motivation of this work originates from numerical simulations of multiphase flow, governed by the coupled Cahn-Hilliard and incompressible Navier-Stokes equations. The impact of variable viscosity on some known preconditioni...

In this work we consider preconditioned iterative solution methods for numerical simulations of multiphase flow problems, modelled by the Cahn-Hilliard equation. We focus on diphasic flows and the construction and efficiency of a preconditioner for the algebraic systems arising from finite element discretizations in space and the θ-method in time....

We consider preconditioned iterative solution methods to solve the algebraic systems of equations arising from finite element discretizations of multiphase flow problems, based on the phase-field model. The aim is to solve coupled physics problems, where both diffusive and convective processes take place simultaneously in time and space. To model t...

The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized Navier-Stokes equations (Oseen’s problem). We utilize the so-called augmented Lagrangian framework, where the original linear system of equations is first transformed to an equivalent one, which la...

In this work we analyse a method to construct numerically efficient and computationally cheap sparse approximations of some of the matrix blocks arising in the block-factorized preconditioners for matrices with a two-by-two block structure. The matrices arise from finite element discretizations of partial differential equations. We consider scalar...

We discuss a methodology to construct sparse approximations of Schur complements of two-by-two block matrices arising in Finite Element discretizations of partial differential equations. Earlier results from [2] are extended to more general symmetric positive definite matrices of two-by-two block form. The applicability of the method for general sy...

For the iterative solution of linear systems of equations arising from finite element discretization of elliptic problems there exist well-established techniques to construct numerically efficient and computationally optimal preconditioners. Among those, most often preferred choices are Multigrid methods (geometric or algebraic), Algebraic MultiLev...

We consider element-by-element Schur complement approximations for indefinite and general nonsymmetric matrices of two-by-two block form, as arising in finite element discretized systems of PDEs. The paper provides some analysis of the so-obtained approximation and attempts to quantify the quality of the underlying two-by-two matrix splitting in a...

In this study, we propose several improvements of the Average Information Restricted Maximum Likelihood algorithms for estimating the variance components for genetic mapping of quantitative traits. The improved methods are applicable when two variance components are to be estimated. The improvements are related to the algebraic part of the methods...

This paper deals with an efficient technique for computing high-quality approximations of Schur complement matrices to be used in various preconditioners for the iterative solution of finite element discretizations of elliptic boundary value problems. The Schur complements are based on a two-by-two block decomposition of the matrix, and their appro...

In order to control the accuracy of a preconditioner for an outer iterative process one often involves variable preconditioners. The variability may for instance be due to the use of inner iterations in the construction of the preconditioner. Both the outer and inner iterations may be based on some conjugate gradient type of method, e.g. generalize...

In this paper we consider numerical simulations of the so-called glacial rebound phenomenon and the use of efficient preconditioned iterative solution methods in that context. The problem originates from modeling the response of the solid earth to large scale glacial advance and recession which may have provoked very large earthquakes in Northern S...

The paper is devoted to the numerical solution of both elliptic and parabolic problems by overlapping Schwarz methods. It demonstrates that while the two-level Schwarz method is necessary for the efficient solution of discrete elliptic problems, the one-level Schwarz method can be sufficiently efficient in the parabolic case. The paper includes res...

This paper discusses preconditioners for the iterative solution of nonsymmetric indefinite linear algebraic systems of equations as arising in modeling of the purely elastic part of glacial rebound processes. The iteration scheme is of inner-outer type using a multilevel preconditioner for the inner solver. Numerical experiments are provided showin...

This paper discusses finite element discretization and preconditioning strategies for the iterative solution of nonsymmetric indefinite linear algebraic systems of equations arising in modelling of glacial rebound processes. Some numerical experiments for the purely elastic model setting are provided. Comparisons of the performance of the iterative...

We consider strategies to construct optimal order two- and multilevel hierarchical preconditioners for linear systems as arising from the finite element discretization of self-adjoint second order elliptic problems using non-conforming Crouzeix–Raviart linear elements. In this paper we utilize the hierarchical decompositions, derived in a previous...

Preconditioning techniques based on various multilevel extensions of two-level splittings of finite element (FE) spaces lead to iterative methods which have an optimal rate of convergence and computational complexity with respect to the number of degrees of freedom. This article deals with the construction of algebraic two-level and multilevel prec...

Preconditioners based on various multilevel extensions of two-level finite element methods (FEM) lead to iterative methods which have an optimal order computational complexity with respect to the size of the system. Such methods were first presented in Axelsson and Padiy (SIAM. J. Sci. Stat. Comp. 1990; 20:1807) and Axelsson and Vassilevski (Numer....

Preconditioned iterative solution methods are compared with the direct Gaussian elimination method to solve dense linear systems Ax=b which originate from problems, discretized by boundary element method (BEM) techniques. Numerical experiments are presented and compared with the direct solution method available in a commercial BEM package, which sh...

We consider the finite element discretization of the system of partial dierential equations describing the stress field and the displacements in a (visco)elastic inho- mogeneous layered media in response to a surface load. The underlying physical phenomenon, which is modelled, is glacial advance and recession, and the resulting crustal stress state...

Preconditioning methods for matrices on saddle point form, as typically arising in equality constrained optimization problems, are surveyed. Special consideration is given to two methods: a nearly symmetric block incomplete factorization preconditioning method and a preconditioner on the same saddle point form as the given matrix. Both methods resu...

Preconditioning methods for matrices on saddle point form, as typically arising in constrained optimization problems, are surveyed. Special consideration is given to two methods: a nearly symmetric block incomplete factorization preconditioning method and an indefinite matrix preconditioner. Both methods result in eigenvalues with positive real par...

We survey preconditioning methods for matrices on saddle point form, as typically arising in constrained optimization problems. Special consideration is given to indefinite matrix preconditioners and a preconditioner which results in a symmetric positive definite matrix, which latter may enable the use of the standard conjugate gradient (CG) method...

We describe a numerical procedure for solving the stationary two-dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a s...

This presentation deals with the construction of robust preconditioned iterative solution methods for discrete linear elasticity problem. A preconditioned iterative solution strategy, which makes use of the explicit form of the Schur complement system with respect to the coarse level degrees of freedom, is defined and compared with various direct...

This study is devoted to the numerical solution of 3D elasticity problems in multilayer media. The problem is described by a coupled system of second-order nonlinear elliptic partial differential equations with strongly varying coefficients. The boundary value problem is discretized by trilinear finite elements.The goal of the paper is to analyze t...

This paper describes an elimination strategy for solving finite element elliptic equations based on an interface domain decomposition ordering of the degrees of freedom that allows for high parallelism and a nearly optimal order of computational complexity of the resulting solutions algorithm.

We describe a numerical procedure for solving the stationary two-dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect-correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a s...

This presentation describes the contribution of the group at KUN to Work-packages 2 and 3 in the following directions: (1) study of the properties and the behavior of variable versus fixed preconditioners, based on the so-called separate displacement component (SDC) formulation of the linear elasticity problem; (2) solution of problems with nearly...

Four fundamental aspects of efficient massively parallel computation are discussed: (i) the need for massively parallel computations and,...

The bordering method intended for the construction of matrices-preconditioners for solving systems of linear algebraic equations with the almost degenerated matrix using the method of conjugate gradients is considered. Efficiency of the preliminary reduction of an assumed system to the degenerated one and solving the last using the method of conjug...

In this presentation we discuss the Algebraic Multilevel Iteration method and some aspects of its implementation for massively parallel distributed memory computer systems (CM-2, CM-200). Numerical results are presented and compared with those obtained from a serial implementation of the method (IBM RS/6000, SUN SPARC work stations).

Very large scale mathematical modelling such as accurate modelling using Navier's equations of elasticity needs both massively parallel computing and scalable algorithms. It is shown in this paper that efficient methods must be scalable with respect to the speedup measured as the ratio of the computing time of the best sequential algorithm on one p...

Thesis (Ph. D.)--Katholieke Universiteit Nijmegen, 1995. Includes bibliographical references (p. 221-237).

The numerical solution of elliptic selfadjoint second‐order boundary value problems leads to a class of linear systems of equations with symmetric, positive definite, large and sparse matrices which can be solved iteratively using a preconditioned version of some algorithm. Such differential equations originate from various applications such as hea...

A survey of the recently developed family of algebraic multilevel iteration methods is presented. The theory of the methods, together with the classes of problems considered, is exposed following the chronology of its derivation. This includes the computational complexity of the methods. A summary of some of the applications and illustrating numeri...

The behavior of iterative solution methods depends much on the preconditioner used. Some preconditioning methods, such as the diagonally preconditioned conjugate gradient method, parallelize easily but can converge very slowly. It is shown that they can not complete with optimal order methods, even if the latter are run on a single processor. Full...

In this work we consider block-factorized preconditioners for the iterative solution of systems of linear algebraic equations arising from finite element discretizations of scalar and vector partial differential equations of elliptic type. For the construction of the preconditioners we utilize a general two-level standard finite element framework a...

Preconditioned iterative solution methods are compared with the direct Gaussian elimi-nation method to solve dense linear systems Ax = b which originate from crack propagation problems, modeled and discretized by boundary element (BEM) techniques. Numerical experiments are presented and compared with the direct solution method available in a commer...

We contribute two parallel strategies to compute the exact and approximate inverse of a dense matrix, based on the so-called inverse Sherman-Morrison algorithm and demonstrate their efficiencies on multicore CPU and GPU-equipped computers. Our meth-ods are shown to be much better than a common matrix inverse method, yielding up to 12 times faster p...