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MicroRNA hsa-miR-29 was connected to a number of malignancies. Its target genes are many, among them Mcl-1 that is expressed in three possible isoforms, one of which is anti-apoptotic and another one pro-apoptotic. Ratio of these two isoforms appears to affect cell response to external stimuli. We have demonstrated that miR-29b enhanced etoposide toxicity in HeLa cell line by modulating this ratio of Mcl-1 isoforms. However, it is not known whether the described miR-29 effect is common to various cancer types or even have the opposite effect. This represents a significant problem for possible future applications. In this report, we demonstrate that miR-29b affects toxicity of 60 μM etoposide in cell lines derived from selected malignancies. The mechanism, however, differs among the cell lines tested. Hep G2 cells demonstrated similar effect of miR-29b on etoposide toxicity as was described in HeLa cells, i.e. modulation of Mcl-1 expression. Target protein down-regulated by miR-29b resulting in enhanced etoposide toxicity in Caco-2 cells was, however, Bcl-2 protein. Moreover, H9c2, Hek-293 and ARPE-19 cell lines selected as a representatives of non-malignant cells, showed no effect of miR-29b on etoposide toxicity. Our data suggest that miR-29b could be a common enhancer of etoposide toxicity in malignant cells due to its modulation of Bcl family proteins. [ABSTRACT FROM AUTHOR]

FETI (finite element tearing and interconnecting) based domain decomposition methods are well-established massively parallel methods for solving huge linear systems arising from discretizing partial differential equations. The first steps of FETI decompose the domain into nonoverlapping subdomains, discretize the subdomains using matching grids, and interconnect the adjacent variables by multipoint constraints. However, the multipoint constraints enforcing identification of the corners’ variables do not have a unique representation and their proper choice and modification can improve the performance of FETI. Here, we briefly review the main options, including orthogonal, fully redundant, or localized constraints, and use the basic linear algebra and spectral graph theory to examine the quantitative effect of their choice on the effective control of the feasibility error and rate of convergence of FETI. [ABSTRACT FROM AUTHOR]

We describe the three-level hybrid domain decomposition TFETI method and show that the condition number of an elastic cluster defined on a fixed cube domain, decomposed into m × m × m subdomains interconnected by the face's rigid body modes and discretized by a regular grid, increases proportionally to m. The estimates are plugged into the analysis of the unpreconditioned H-TFETI (hybrid) method and used to prove its numerical scalability for linear problems. The estimates show that the cost of the coarse problem decreases with m 6 while the number of iterations increases only proportionally to m . Numerical experiments show a large scope of scalability of H-TFETI. The results are also essential for solving huge contact problems. [ABSTRACT FROM AUTHOR]

The unpreconditioned hybrid domain decomposition method was recently shown to be a competitive solver for linear elliptic PDE problems discretized by structured grids. Here, we plug H-TFETI-DP (hybrid total finite element tearing and interconnecting dual primal) method into the solution of huge boundary elliptic variational inequalities. We decompose the domain into subdomains that are discretized and then interconnected partly by Lagrange multipliers and partly by edge averages. After eliminating the primal variables, we get a quadratic programming problem with a well-conditioned Hessian and bound and equality constraints that is effectively solvable by specialized algorithms. We prove that the procedure enjoys optimal, i.e., asymptotically linear complexity. The analysis uses recently established bounds on the spectrum of the Schur complements of the clusters interconnected by edge/face averages. The results extend the scope of scalability of massively parallel algorithms for the solution of variational inequalities and show the outstanding efficiency of the H-TFETI-DP coarse grid split between the primal and dual variables. [ABSTRACT FROM AUTHOR]

Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of 'floating' clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems. [ABSTRACT FROM AUTHOR]

Bounds on the spectrum of Schur complements of subdomain stiffness matrices of the discretized Laplacian with respect to interior variables are important in the convergence analysis of finite element tearing and interconnecting (FETI)‐based domain decomposition methods. Here, we are interested in bounds on the regular condition number of Schur complements of "floating" clusters, that is, of matrices comprising the Schur complements of subdomains with prescribed zero Neumann conditions that are joined on the primal level by edge averages. Using some known results, angles of subspaces, and known bounds on the spectrum of Schur complements associated with square domains, we give bounds on the regular condition number of the Schur complement of some "floating" clusters arising from the discretization and decomposition of 2D Laplacian on domains comprising square subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m square subdomains joined by edge averages increases proportionally to m. The estimates are compared with numerical values and used in the analysis of H‐FETI‐DP methods. Though the research has been motivated by an effort to extend the scope of scalability of FETI‐based solvers to variational inequalities, the experiments indicate that H‐TFETI‐DP with large clusters can be useful for the solution of huge linear elliptic problems discretized by sufficiently regular grids. [ABSTRACT FROM AUTHOR]

Abstract A variationally consistent approximation of contact conditions by means of biorthogonal mortars was introduced by Wohlmuth as a powerful theoretically supported tool for the discretization of contact problems. This approach is especially useful when a potential contact interface is large and curved or when nonmatching grids are applied, but its effective implementation into FETI based algorithms is not straightforward due to the ill conditioning of related inequality constraints. In this paper we review the mortar discretization and theoretical results on scalability of the FETI based algorithm and show that the recently proposed adaptive augmentation can overcome the difficulties caused by the ill-conditioning of constraints. We demonstrate the (weak) numerical scalability by numerical experiments and present the results for a difficult real world problem discretized by mortars that show the power of the new algorithm – the number of iterations required to the solution of this problem discretized by mortars is just one third of that required by the original algorithm for the same problem discretized by node-to-node constraints. Highlights • The paper reviews the bounds on the conditioning of matrices arising from mortar discretization of non-penetration conditions. • The theoretical results support the scalability of FETI methods for the solution of multibody frictionless contact problems are given. • The (weak) scalability is demonstrated by the solution of a variant of the Hertz problem. • Moreover, the solution of a difficult real world problem shows that the new algorithm autperforms its predecessors. [ABSTRACT FROM AUTHOR]

The efficiency of numerical solvers of PDEs depends on the approximation properties of the discretization methods and the conditioning of the resulting linear systems. If applicable, the boundary element methods typically provide better approximation with unknowns limited to the boundary than the Schur complement of the finite element stiffness matrix with respect to the interior variables. Since both matrices correctly approximate the same object, the Steklov-Poincaré operator, it is natural to assume that the matrices corresponding to the same fine boundary discretization are similar. However, this note shows that the distribution of the spectrum of the boundary element stiffness matrix is significantly better conditioned than the finite element Schur complement. The effect of the favorable conditioning of BETI clusters is demonstrated by solving huge problems by H-TBETI-DP and H-TFETI-DP. [ABSTRACT FROM AUTHOR]

The cgSLS (conjugate gradients for symmetric positive semidefinite least-squares) algorithm is presented. The algorithm exploits the cyclic property of invariant Krylov spaces to reduce the least-squares problem with a symmetric positive semidefinite matrix A to the minimization of the related energy function with the Hessian A on the range of A, so that a simple modification of the conjugate gradient (CG) method is applicable. At the same time, the algorithm generates approximations of the projection of the right-hand side to the range of A. The asymptotic rate of convergence of the new algorithm is proved to be the same as that of the CG method for the related consistent problem. An error bound in terms of the square root of the regular condition number of A is also given. The performance of the algorithm is demonstrated by numerical experiments. [ABSTRACT FROM AUTHOR]

We review our recent results in the development of optimal algorithms for the minimization of a strictly convex quadratic function subject to separable convex inequality constraints and/or linear equality constraints. A unique feature of our algorithms is the theoretically supported bound on the rate of convergence in terms of the bounds on the spectrum of the Hessian of the cost function, independent of representation of the constraints. When applied to the class of convex QP or QPQC problems with the spectrum in a given positive interval and a sparse Hessian matrix, the algorithms enjoy optimal complexity, i.e., they can find an approximate solution at the cost that is proportional to the number of unknowns. The algorithms do not assume representation of the linear equality constraints by full rank matrices. The efficiency of our algorithms is demonstrated by the evaluation of the projection of a point to the intersection of the unit cube and unit sphere with hyperplanes. [ABSTRACT FROM AUTHOR]

The MPRGP (modified proportioning with reduced gradient projections) algorithm for minimization of the strictly convex quadratic function subject to bound constraints is adapted to the solution of problems with a semidefinite Hessian A . The adapted algorithm accepts the decrease directions that belong to the null space of A and generates the iterates that are proved to minimize the cost function. The paper examines specific features of the solution of the problems with convex, but not necessarily strictly convex Hessian. The performance of the algorithm is demonstrated by the solution of a semi-coercive contact problem of elasticity and a 3D particle dynamics problem. The results are compared with those obtained by the spectral projected gradient method and the projected-Jacobi method. [ABSTRACT FROM AUTHOR]

The paper deals with an effective implementation of some algorithms for the solution of convex QPQC problems with elliptic and other separable constraints with strong curvature. Here we discuss robust quantitative refinement of the Karush–Kuhn–Tucker conditions, extend existing results on the decrease of the cost function along the projected gradient path to separable constraints with elliptic components, and plug them into the existing algorithms for the solution of the QPQC problems with R-linear rate of convergence in the bounds on the spectrum. The results are then extended to the problems with separable inequality and linear equality constraints. The performance of the algorithms is demonstrated on the solution of a problem of two cantilever beams in mutual contact with orthotropic Tresca and Coulomb friction discretized by up to one and half million nodal variables. [ABSTRACT FROM AUTHOR]

Abstract: New convergence results for a variant of the inexact augmented Lagrangian algorithm SMALBE [Z. Dostál, An optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum, Computing 78 (2006) 311–328] for the solution of strictly convex bound and equality constrained quadratic programming problems are presented. The algorithm SMALBE-M presented here uses a fixed regularization parameter and controls the precision of the solution of auxiliary bound constrained problems by a multiple of the norm of violation of the equality constraints and a constant which is updated in order to enforce the increase of Lagrangian function. A nice feature of SMALBE-M is its capability to find an approximate solution of important classes of problems in a number of iterations that is independent of the conditioning of the equality constraints. Here we prove the R-linear rate of convergence of the outer loop of SMALBE-M for any positive regularization parameter after the strong active constraints of the solution are identified. The theoretical results are illustrated by solving two benchmarks, including the contact problem of elasticity discretized by two million of nodal variables. The numerical experiments indicate that the inexact solution of auxiliary problems in the inner loop results in a very small increase of the number of outer iterations as compared with the exact algorithm. The results do not assume independent equality constraints and remain valid when the solution is dual degenerate. [Copyright &y& Elsevier]

Abstract: The FETI based domain decomposition method is adapted to implement the time step of the Newmark scheme for the solution of dynamic contact problems without friction. If the ratio of the decomposition and discretization parameters is kept uniformly bounded, then the cost of the time step is proved to be proportional to the number of nodal variables. The algorithm uses our in a sense optimal MPRGP algorithm for the solution of strictly convex bound constrained quadratic programming problems with optional preconditioning by the conjugate projector to the subspace defined by the trace of the rigid body motions on the artificial subdomain interfaces. The proof of optimality combines the convergence theory of our MPRGP algorithm, the classical bounds on the spectrum of the mass and stiffness matrices, and our theory of the preconditioning by a conjugate projector for nonlinear problems. The results are confirmed by numerical solution of 2D and 3D dynamic contact problems. [Copyright &y& Elsevier]

We propose a modification of our MPGP algorithm for the solution of bound constrained quadratic programming problems so that it can be used for minimization of a strictly convex quadratic function subject to separable convex constraints. Our active set based algorithm explores the faces by conjugate gradients and changes the active sets and active variables by gradient projections, possibly with the superrelaxation steplength. The error estimate in terms of extreme eigenvalues guarantees that if a class of minimization problems has the spectrum of the Hessian matrix in a given positive interval, then the algorithm can find and recognize an approximate solution of any particular problem in a number of iterations that is uniformly bounded. We also show how to use the algorithm for the solution of separable and equality constraints. The power of our algorithm and its optimality are demonstrated on the solution of a problem of two cantilever beams in mutual contact with Tresca friction discretized by almost twelve millions nodal variables. [ABSTRACT FROM AUTHOR]

Abstract: The Cholesky decomposition of a symmetric positive semidefinite matrix A is a useful tool for solving the related consistent system of linear equations or evaluating the action of a generalized inverse, especially when A is relatively large and sparse. To use the Cholesky decomposition effectively, it is necessary to identify reliably the positions of zero rows or columns of the factors and to choose these positions so that the nonsingular submatrix of A of the maximal rank is reasonably conditioned. The point of this note is to show how to exploit information about the kernel of A to accomplish both tasks. The results are illustrated by numerical experiments. [Copyright &y& Elsevier]

Abstract: By combining FETI algorithms of dual-primal type with recent results for bound constrained quadratic programming problems, we develop an optimal algorithm for the numerical solution of coercive variational inequalities. The model problem is discretized using non-penetration conditions of mortar type across the potential contact interface, and a FETI–DP algorithm is formulated. The resulting quadratic programming problem with bound constraints is solved by a scalable algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem. Numerical experiments for non-matching meshes across the contact interface confirm the theoretical scalability of the algorithm. [Copyright &y& Elsevier]

Preconditioning by a conjugate projector is combined with the recently proposed modified proportioning with reduced gradient projection (MPRGP) algorithm for the solution of bound-constrained quadratic programming problems. If applied to the partially bound-constrained problems, such as those arising from the application of FETI-based domain decomposition methods to the discretized elliptic boundary variational inequalities, the resulting algorithm is shown to have better bound on the rate of convergence than the original MPRGP algorithm. The performance of the algorithm is illustrated on the solution of a model boundary variational inequality. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The FETI method with a natural coarse grid is combined with recently proposed optimal algorithms for the solution of bound and/or equality constrained quadratic programming problems in order to develop a scalable solver for elliptic boundary variational inequalities such as those describing equilibrium of a system of bodies in mutual contact. A discretized model problem is first reduced by the duality theory of convex optimization to the quadratic programming problem with bound and equality constraints. The latter is then modified by means of orthogonal projectors to the natural coarse grid introduced by Farhat, Mandel, and Roux [Comput. Methods Appl. Mech. Engrg., 115 (1994), pp. 365-385]. Finally, the classical results on linear scalability for linear problems are extended to boundary variational inequalities. The results are validated by numerical experiments. The experiments also confirm that the algorithm enjoys the same parallel scalability as its linear counterpart. [ABSTRACT FROM AUTHOR]

Abstract: We develop an optimal algorithm for the numerical solution of semi-coercive variational inequalities by combining dual-primal FETI algorithms with recent results for bound and equality constrained quadratic programming problems. The discretized version of the model problem, obtained by using the FETI-DP methodology, is reduced by the duality theory of convex optimization to a quadratic programming problem with bound and equality constraints, which is solved by a new algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem. We present convergence bounds that guarantee the scalability of the algorithm. These results are confirmed by numerical experiments. [Copyright &y& Elsevier]

A new variant of the FETI method for numerical solution of elliptic PDE is presented. The basic idea is to simplify inversion of the stiffness matrices of subdomains by using Lagrange multipliers not only for gluing the subdomains along the auxiliary interfaces, but also for implementation of the Dirichlet boundary conditions. Results of numerical experiments are presented which indicate that the new method may be even more efficient then the original FETI. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: We develop an optimal algorithm for the numerical solution of coercive variational inequalities, by combining FETI algorithms of dual-primal type with recent results for bound constrained quadratic programming problems. The discretized version of the model problem, obtained by using the FETI-DP methodology, is reduced by the duality theory of convex optimization to a quadratic programming problem with bound constraints. The resulting problem is solved by a new algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem. We present convergence bounds that guarantee the scalability of the algorithm. These results are confirmed by numerical experiments. [Copyright &y& Elsevier]

Abstract: Theoretical and experimental results concerning FETI based algorithms for contact problems of elasticity are reviewed. A discretized model problem is first reduced by the duality theory of convex optimization to the quadratic programming problem with bound and equality constraints. The latter is then optionally modified by means of orthogonal projectors to the natural coarse space introduced by Farhat and Roux in the framework of their FETI method. The resulting problem is then solved either by special algorithms for bound constrained quadratic programming problems combined with penalty that imposes the equality constraints, or by an augmented Lagrangian type algorithm with the inner loop for the solution of bound constrained quadratic programming problems. Recent theoretical results are reported that guarantee certain optimality and scalability of both algorithms. The results are confirmed by numerical experiments. The performance of the algorithm in solution of more realistic engineering problems by basic algorithm is demonstrated on the solution of 3D problems with large displacements or Coulomb friction. [Copyright &y& Elsevier]

The FETI method with the natural coarse grid is combined with the penalty method to develop an efficient solver for elliptic variational inequalities. A proof is given that a prescribed bound on the norm of feasibility of solution may be achieved with a value of the penalty parameter that does not depend on the discretization parameter and that an approximate solution with the prescribed bound on violation of the Karush–Kuhn–Tucker conditions may be found in a number of steps that does not depend on the discretization parameter. Results of numerical experiments with parallel solution of a model problem discretized by up to more than eight million of nodal variables are in agreement with the theory and demonstrate numerically both optimality of the penalty and scalability of the algorithm presented. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents and analyses an iterative process for the numerical realization of contact problems with Coulomb friction which is based on the method of successive approximations combined with a splitting type approach. Numerical examples illustrate the efficiency of this method. [Copyright &y& Elsevier]

The paper deals with the numerical solution of the quasi-variational inequality describing the equilibrium of an elastic body in contact with a rigid foundation under Coulomb friction. After a discretization of the problem by mixed finite elements, the duality approach is exploited to reduce the problem to a sequence of quadratic programming problems with box constraints, so that efficient recently proposed algorithms may be applied. A new variant of this method is presented. It combines fixed point with block Gauss–Seidel iterations. The method may be also considered as a new implementation of fixed point iterations for a sequence of problems with given friction. Results of numerical experiments are given showing that the resulting algorithm may be much faster than the original fixed point method and its efficiency is comparable with the solution of frictionless contact problems. [Copyright &y& Elsevier]

Augmented Lagrangian method is a well established tool for the solution of optimization problems with equality constraints. If combined with effective algorithms for the solution of bound constrained quadratic programming problems, it can solve efficiently very large problems with bound and linear equality constraints. The point of this paper is to show that the performance of the algorithm can be essentially improved by enhancing the information on the free set of current iterates into the reorthogonalization of equality constraints. The improvement is demonstrated on the numerical solution of a large problem arising from the application of domain decomposition methods to the solution of discretized elliptic variational inequality describing a variant of Hertz's two-body contact problem. [ABSTRACT FROM AUTHOR]

An alternative approach to the preconditioning of a system of linear equations with a matrix A + ρ CTC that is the sum of a positive definite matrix A and a penalization term is proposed. After showing that there is a gap in the spectrum of A + ρ CTC provided ρ is sufficiently large, a preconditioner for A is applied to A + ρ CTC in such a way that it preserves the gap in the spectrum but still improves the convergence of the conjugate gradient method. A bound on the rate of convergence of the conjugate gradient method with our preconditioning based on the estimates by Axelsson is given that depends neither on ρ nor on the rank of C. Numerical experiments confirm the efficiency of the approach presented. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This contribution deals with the numerical realization of static contact problems with Coulomb friction for three-dimensional elastic bodies. We first introduce auxiliary contact problems with given friction which define a mapping Φ associating with a given slip bound the normal contact stress in the equilibrium state. Solutions to contact problems with Coulomb friction are defined as fixed points of Φ and are computed by using the method of successive approximations. The mathematical model of contact problems with given friction leads to a variational inequality of the second kind. Its discretization is based on the so-called reciprocal variational formulation, i.e., the formulation in terms of the normal and tangential components of stresses on the contact boundary. Unlike the two-dimensional case, constraints imposed on the tangential components of contact stresses are quadratic. The main goal of this contribution is to show how to solve this problem by using existing fast algorithms for simple (box) constraints. Numerical experiments for several variants of our algorithm will be shown and compared. [Copyright &y& Elsevier]

Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities. [ABSTRACT FROM AUTHOR]

We propose an algorithm for the effective solution of quadratic programming (QP) problems arising from model predictive control (MPC). MPC is a modern multivariable control method which gives the solution for a QP problem at each sample instant. Our algorithm combines the active-set strategy with the proportioning test to decide when to leave the actual active set. For the minimization in the face, we use a direct solver implemented by the Cholesky factors updates. The performance of the algorithm is illustrated by numerical experiments, and the results are compared with the state-of-the-art solvers on benchmarks from MPC. [ABSTRACT FROM AUTHOR]

n application of a variant of the parallel domain decomposition method that we call Total FETI or TFETI (Total Finite Element Tearing and Interconnecting) for the solution of contact problems of elasticity to the parallel solution of contact shape optimization problems is described. A unique feature of the TFETI algorithm is its capability to solve large contact problems with optimal, i.e., asymptotically linear complexity. We show that the algorithm is even more efficient for the solution of the contact shape optimization problems as it can exploit effectively a specific structure of the auxiliary problems arising in the semi-analytic sensitivity analysis. Thus the triangular factorizations of the stiffness matrices of the subdomains are carried out in parallel only once for each design step, the evaluation of the components of the gradient of the cost function can be carried out in parallel, and even the evaluation of each component of the gradient itself can be further parallelized using the standard TFETI scheme. Theoretical results which prove asymptotically linear complexity of the solution are reported and documented by numerical experiments. The results of numerical solution of a 3D contact shape optimization problem confirm the high degree of parallelism of the algorithm. [ABSTRACT FROM AUTHOR]