Alexander Malyshev

University of Bergen | UiB · Department of Mathematics

Let a three-dimensional ball intersect a three-dimensional polyhedron given by its triangulated boundary with outward unit normals. We propose a numerical method for approximate computation of the intersection volume by using voxelization of the interior of the polyhedron. The approximation error is verified by comparison with the exact volume of t...

The TV-Stokes denoising model for a vectorial image defines a denoised vector field in the form of the gradient of a scalar function. The dual formulation naturally leads to a Chambolle-type algorithm, where the most time consuming part is application of the orthogonal projector onto the range space of the gradient operator. This application can be...

The total variation regularization of non-convex data terms in continuous variational models can be convexified by the so called functional lifting, which may be considered as a continuous counterpart of Ishikawa’s method for multi-label discrete variational problems. We solve the resulting convex continuous variational problem by the augmented Lag...

A bisection method is used to compute lower and upper bounds on the distance from a quadratic matrix polynomial to the set of quadratic matrix polynomials having an eigenvalue on the imaginary axis. Each bisection step requires to check whether an even quadratic matrix polynomial has a purely imaginary eigenvalue. First, an upper bound is obtained...

Résumé Nous développons une méthode de type bissection pour calculer la distance à l'instabilité de polynômes matriciels quadratiques. Le calcul prend en compte les erreurs d'arrondi.

A method for a model predictive control (MPC) of a system determines entries of an approximate coefficient matrix only at locations identified in a map of locations as significant. The map of locations identifies each location of an entry in the approximate coefficient matrix as either significant or insignificant. The entries are determined using...

A model predictive control (MPC) system for controlling an operation of a machine according to a model of the machine dynamics optimizes a cost function over a time-horizon subject to constraints to produce a sequence of control inputs to control the state of the machine over the time horizon. The machine is control using the first control input in...

One-compartment models are widely used to quantify hemodynamic parameters such as perfusion, blood volume and mean transit time. These parameters are routinely used for clinical diagnosis and monitoring of disease development and are thus of high relevance. However, it is known that common estimation techniques are discretization dependent and valu...

For a Laurent polynomial \(a(\lambda )\), which is Hermitian and positive definite on the unit circle, the Bauer method provides the spectral factorization \(a(\lambda ) = p(\lambda )p{\kern 1pt} {\text{*}}({{\lambda }^{{ - 1}}})\), where \(p(\lambda )\) is a polynomial having all its roots outside the unit circle. Namely, as the size of the banded...

We investigate using Krylov subspace iterative methods in model predictive control (MPC), where the prediction model is given by linear or linearized systems with linear inequality constraints on the state and the input, and the performance index is quadratic. The inequality constraints are treated by the primal-dual interior point method. We indic...

There are several efficient direct solvers for structured systems of linear equations defining search directions in primal-dual interior point methods applied to constrained model predictive control problems. We propose reusing matrix decompositions of direct solvers as preconditioners in Krylov-subspace methods applied to subsequent iterations of...

A Continuation/GMRES Method for Nonlinear Model Predic- tive Control (NMPC), suggested by T. Ohtsuka in 2004, uses the GMRES iterative algorithm to solve a forward di�erence ap- proximation Ax = b of the original NMPC equations on every time step. We propose a new sparse preconditioner for GMRES (or MINRES) of the arithmetic complexity O(N), where...

A Continuation/GMRES Method for Nonlinear Model Predictive Control (NMPC), suggested by T. Ohtsuka in 2004, uses the GMRES iterative algorithm to solve a forward difference approximation Ax = b of the original NMPC equations on every time step. We propose a new sparse preconditioner for GMRES (or MINRES) of the arithmetic complexity O(N), where N i...

We present Newton-Krylov methods for e�cient numerical solution of optimal control problems arising in model predictive control, where the optimal control is discontinuous. As in our earlier work, preconditioned GMRES practically results in an optimal O(N) complexity, where N is a discrete horizon length. E�ects of a warm-start, shifting along the...

Signal reconstruction from a sample using an orthogonal projector onto a guiding subspace is theoretically well justified, but may be difficult to practically implement. We propose more general guiding operators, which increase signal components in the guiding subspace relative to those in a complementary subspace, e.g., iterative low-pass edge-pre...

Newton-Krylov methods for nonlinear Model Predictive Control are pioneered by T. Ohtsuka under the name \C/GMRES". T. Ohtsuka eliminates a system state over the horizon from Karush-Kuhn-Tucker stationarity conditions of a Lagrangian using equations of system dynamics. We propose instead using least squares to �t the state to the dynamics and some c...

We present advances in Newton-Krylov (AKA continuation) methods for NMPC, pioneered by Prof. Ohtsuka. Our main results concern efficient preconditioning, leading to dramatically improved real-time Newton-Krylov MPC optimization. One idea is solving a forward recursion for the state and a backward recursion for the costate approximately, or reusing...

Signal reconstruction from a sample using an orthogonal projector onto a guiding subspace is theoretically well justified, but may be difficult to practically implement. We propose more general guiding operators, which increase signal components in the guiding subspace relative to those in a complementary subspace, e.g., iterative low-pass edge-pre...

We present Newton-Krylov methods for efficient numerical solution of optimal control problems arising in model predictive control, where the optimal control is discontinuous. As in our earlier work, preconditioned GMRES practically results in an optimal $O(N)$ complexity, where $N$ is a discrete horizon length. Effects of a warm-start, shifting alo...

Newton-Krylov methods for nonlinear Model Predictive Control are pioneered by T. Ohtsuka under the name "C/GMRES". Ohtsuka eliminates a system state over the horizon from Karush-Kuhn-Tucker stationarity conditions of a Lagrangian using equations of system dynamics. We propose instead using least squares to fit the state to the dynamics and some con...

Denoising filters, such as bilateral, guided, and total variation filters, applied to images on general graphs may require repeated Application if noise is not small enough. We formulate two acceleration techniques of the resulted iterations: conjugate gradient method and Nesterov's acceleration. We numerically show efficiency of the accelerated no...

We present advances in Newton-Krylov (AKA continuation) methods for NMPC, pioneered by Prof. Ohtsuka. Our main results concern efficient preconditioning, leading to dramatically improved real-time Newton-Krylov MPC optimization. One idea is solving a forward recursion for the state and a backward recursion for the costate approximately, or reusing...

We review some of our recent work on methods for graph-based signal processing: � Iterative acceleration of repeated application of graph-based edge-preserving denoising: bilateral, guided, and total variation. � Edge-enhancing graph-based denoising, using negative graph weights. � Guided signal reconstruction with application to image magni�ficati...

We propose fast O(N) preconditioning, where N is the number of gridpoints on the prediction horizon, for iterative solution of (non)-linear systems appearing in model predictive control methods such as forward-di�erence Newton-Krylov methods. The Continuation/GMRES method for nonlinear model predictive control, suggested by T. Ohtsuka in 2004, is a...

Outline � Graph-based smoothing �lters � Bilateral �lter (BF) � Guided �lter (GF) � Total Variation �lter (TVF) � Preconditioned conjugate gradient (PCG) acceleration � Nesterov's acceleration � Numerical results

• Bilateral (BF) filter • Guided (GF) filter • Preconditioned conjugate gradient iteration • Numerical results Announcement of numerical results: For the test 1D signal of length 4730, the PCG acceleration provides the 9 times speedup for BF and 4 times speedup for GF filters.

We propose fast O(N) preconditioning, where N is the number of gridpoints on the prediction horizon, for iterative solution of (non)-linear systems appearing in model predictive control methods such as forward-difference Newton-Krylov methods. The Continuation/GMRES method for nonlinear model predictive control, suggested by T. Ohtsuka in 2004, is...

Denoising filters, such as bilateral, guided, and total variation filters, applied to images on general graphs may require repeated application if noise is not small enough. We formulate two acceleration techniques of the resulted iterations: conjugate gradient method and Nesterov's acceleration. We numerically show efficiency of the accelerated no...

Graph-based spectral denoising is a low-pass filtering using the eigendecomposition of the graph Laplacian matrix of a noisy signal. Polynomial filtering avoids costly computation of the eigendecomposition by projections onto suitable Krylov subspaces. Polynomial filters can be based, e.g., on the bilateral and guided filters. We propose constructi...

Continuation model predictive control (MPC), introduced by T. Ohtsuka in 2004, uses Krylov-Newton approaches to solve MPC optimization and is suitable for nonlinear and minimum time problems. We suggest particle continuation MPC in the case, where the system dynamics or constraints can discretely change on-line. We propose an algorithm for on-line...

Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. Continuation MPC, suggested by T.~Ohtsuka in 2004, uses Krylov-Newton iterations. Continuation MPC is suitable for nonlinear problems and has been recently adopted for minimum...

Graph-based spectral denoising is a low-pass filtering using the eigendecomposition of the graph Laplacian matrix of a noisy signal. Polynomial filtering avoids costly computation of the eigendecomposition by projections onto suitable Krylov subspaces. Polynomial filters can be based, e.g., on the bilateral and guided filters. We propose constructi...

The most efficient signal edge-preserving smoothing filters, e.g., for denoising, are non-linear. Thus, their acceleration is challenging and is often performed in practice by tuning filter parameters, such as by increasing the width of the local smoothing neighborhood, resulting in more aggressive smoothing of a single sweep at the cost of increas...

Contents: � Representative test problem. � Model Predictive Control (MPC). � Reduction to a real-time solution for a nonlinear equation. � Continuation method.1 � Generalized Minimum Residual (GMRES) iterative method. � Jacobi matrices as preconditioners. � Conclusion

Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. A Continuation/GMRES Method for NMPC, suggested by T. Ohtsuka in 2004, uses the GMRES iterative algorithm to solve a forward difference approximation $Ax=b$ of the Continuatio...

The computation of the distance of a quadratic matrix polynomial to the quadratic matrix polynomials that are singular on the unit circle is investigated. The emphasis is placed on backward stable methods that transform the computation of the distance to a palindromic eigenvalue problem for which structure-preserving eigensolvers can be utilized in...

SUMMARY A fast algorithm for solving systems of linear equations with banded Toeplitz matrices is studied. An important step in the algorithm is a novel method for the spectral factorization of the generating function associated with the Toeplitz matrix. The spectral factorization is extracted from the right deflating subspaces corresponding to the...

For a quadratic matrix polynomial, the distance to the set of quadratic matrix polynomials which have singularities on the unit circle is computed using a bisection-based algorithm. The success of the algorithm depends on the eigenvalue method used within the bisection to detect the eigenvalues near the unit circle. To this end, the QZ algorithm al...

We study the convergence behavior of an orthogonal subspace iteration for matrices whose spectrum is partitioned into three groups: the eigenvalues inside, outside, and on the unit circle. The main focus is on symplectic matrices. Numerical experiments are provided to illustrate the theory.

A novel numerical algorithm is proposed for solving systems of linear equations with block-Toeplitz narrow-banded matrices. Its arithmetical cost is about double of that of the block cyclic reduction. The backward roundoff error stability of the method guarantees its reliability for the matrices that are not symmetric positive definite or diagonall...

A fast numerical algorithm for solving systems of linear equations with tridiagonal block Toeplitz matrices is presented. The algorithm is based on a preliminary factorization of the generating quadratic matrix polynomial associated with the Toeplitz matrix, followed by the Sherman–Morrison–Woodbury inversion formula and solution of two bidiagonal...

We introduce a method for approximating the right and left deflating subspaces of a regular matrix pencil corresponding to the eigenvalues inside, on and outside the unit circle. The method extends the iteration used in the context of spectral dichotomy, where the assumption on the absence of eigenvalues on the unit circle is removed. It constructs...

We propose a fast algorithm for image denoising, which is based on a dual formulation of a recent denoising model involving the total variation minimization of the tangential vector field under the incompressibility condition stating that the tangential vector field should be divergence free. The model turns noisy images into smooth and visually pl...

Let $ \tilde \lambda $ \tilde \lambda be an approximate eigenvalue of multiplicity m c = n − r of an n × n real symmetric tridiagonal matrix T having nonzero off-diagonal entries. A fast algorithm is proposed (and numerically tested) for deleting m c rows of T−$ \tilde \lambda $ \tilde \lambda I so that the condition number of the r × n matr...

An efficient method is developed for computation of eigenvalues and eigenvectors with high relative precision in the Sturm-Liouville problem with strongly varying coefficients. Accuracy of the method is independent of the traditional condition number. New structured condition numbers for nonmultiple eigenvalues are introduced.

We propose exact and computable formulas for computing condition numbers of the Krylov bases and spaces associated with the Hessenberg-Triangular reduction of a regular linear matrix pencil.

Sensitivity with respect to infinitesimal perturbations of a matrix is analyzed for orthonormal bases in Krylov subspaccs constructed by using the symmetric Lanczos iteration. Exact expressions for the corresponding condition numbers are derived, which makes it possible to efficiently calculate these numbers.

Given a nonnegative weight matrix E defining componentwise structured perturbations |Ẽ|⩽ϵE, we introduce a componentwise pseudospectrum of a square matrix A as contour sets of the function λ∈C↦ρ(|(A−λI)−1|E), where ρ(M) is the spectral radius of a square matrix M, and discuss how to compute it.

We derive exact and computable formulas for the condition numbers characterizing the forward instability in Lanczos bidiagonalization with complete reorthogonalization. One series of condition numbers is responsible for stability of Krylov spaces, the second for stability of orthonormal bases in the Krylov spaces and the third for stability of the...

We develop a unified perturbation theory for the unconstrained linear least squares problem, least squares with linear equality constraints, and least squares with quadratic inequality constraint and Tikhonov regularisation solution. The computable condition numbers are exact with respect to the Frobenius norm. For the 2-norm, the computed bounds m...

Suppose that one knows an accurate approximation to an eigenvalue of a real symmetric tridiagonal matrix. A variant of deflation by the Givens rotations is proposed in order to split off the approximated eigenvalue. Such a deflation can be used instead of inverse iteration to compute the corresponding eigenvector.

In this note we propose an algorithm based on the Lanczos bidiagonalization to approximate the backward perturbation bound for the large sparse linear squares problem. The algorithm requires O\mathcal{O} ((m + n)l) operations where m and n are the size of the matrix under consideration and l <#60;<#60; min(m,n). The import of the proposed algorith...

K.V. Fernando developed an e#cient approach for computation of an eigenvector of a tridiagonal matrix corresponding to an approximate eigenvalue. We supplement Fernando's method with deflation procedures by Givens rotations. These deflations can be used in the Lanczos process and instead of the inverse iteration. 1

A computable expression is given for the backward error of an approximate solution to the problem of least squares over a sphere.

Peters and Wilkinson [4] state that "it is well known that Gauss-Jordan is stable"for a diagonally dominant matrix, but a proof does not seem to have been published[3]. The present note fills this gap. Gauss-Jordan elimination is backward stable formatrices diagonally dominant by rows and not backward stable for matrices diagonallydominant by colum...

We review the Krein-Gel&apos;fand-Lidskii theory of strong stability of Hamiltonian and symplectic matrices in order to find a quantitative measure of the strong stability. As a result, formulas for the 2-norm distance from a strongly stable Hamiltonian or symplectic matrix to the set of unstable matrices are derived. 1 Definition and simple proper...

We provide several optimal backward perturbation bounds for the linear least squares problem with a matrix of deficient rank whose solution is defined by means of the truncated singular value decomposition. 1 Introduction The paper [10] has initiated large interest in backward perturbation bounds for the linear least squares problem [5], [7], [8],...

We prove that the 2-norm distance from an matrix A to the matrices that have a multiple eigenvalue is equal to where the singular values are ordered nonincreasingly. Therefore, the 2-norm distance from A to the set of matrices with multiple eigenvalues is

We prove that the 2-distance from an n Theta n matrix A to the matrices that have a multiple eigenvalue is equal to max fl0 oe 2nGamma1 / A Gamma I flI 0 A Gamma I ! ; where the singular values oe k are ordered nonincreasingly. 1 Introduction Given a matrix A 2 C nThetan , n 2, let us define the following parameter: rsep(A) = min n kDeltak 2 j Delt...

The distance rstab(A) of a stable matrix A to the set of unstable matrices and the norm of the exponential of matrices constitute two important topics in stability theory. We treat in this note the case of large matrices. The method proposed partitions the matrix into two blocks: a small block in which the stability is studied and a large block who...

Introduction Given a matrix A 2 C , n 2, let us define the following parameter: ; A + Delta has a multiple eigenvalue : (1) Estimation of this parameter and its relation to the sensitivity of eigenvalues are much discussed in papers [6, 10, 11] and references therein. The main result of the present paper is the derivation of the formula ; (2) where...

This chapter discusses the main results and methods in the theory of matrix equations, with emphasis on algebraic aspects of the theory of matrix equations and matrix polynomials. The problem of factorization of matrix polynomials is essentially a problem concerning special systems of matrix equations. The invariant subspaces of certain associated...

We first describe an algorithm that reduces a matrix A to a block diagonal form using only well conditioned transformations. The spectral properties of A are then carried out from the resulting block diagonal matrix. We show in particular that the spectral portrait of A can be obtained cheaply from that of the block diagonal matrix.

: We derive new estimates of the spectral dichotomy for matrices and matrix pencils which are based upon estimates of the restrictions of Green functions associated with the spectrum dichotomy problem onto the stable and unstable invariant subspaces and estimates of angles between these subspaces. Key-words: spectral dichotomy, condition number (R'...

: We describe parallel programs for computation of spectral portraits of matrices on Paragon and Connection Machine 5. The method used consists of bidiagonal reduction of a complex square matrix by unitary Householder transformations and computation of the minimal singular value of the resulting real bidiagonal matrix by the bisection procedure emp...

: This paper presents some practical and guaranteed ways of studying the discrete-time/ continuous-time stability quality of large sparse matrices. The methods use projection techniques for computing an invariant subspace associated with a few outermost eigenvalues (those with largest real parts for the continuous-time case and with largest magnitu...

: We discuss two spectral dichotomy techniques: one for computing an invariant subspace of a nonsymmetric matrix associated with the eigenvalues inside and outside a given parabola. Another for computing a right deflating subspace of a regular matrix pencil associated with the eigenvalues inside and outside a given ellipse. The techniques use matri...

An analysis of the eigenvalue structure of stably symmetrizable and skew- symmetrizable matrices with the help of averaging is suggested.

An algorithm is described for computing the deflating subspaces of a regular linear matrix pencil λB - A. More precisely, the algorithm is intended to compute the projection matrices P and I - P onto the deflating subspaces of matrix pencils corresponding to the eigenvalues inside and outside the unit circle. This algorithm can be considered as an...

A method for a high-precision spectral decomposition of symmetric matrices is studied. The method is based upon accurate calculation of the solution for Riccati equations arising in the refinement process. Roundoff errors are analyzed in detail. Convergence rates are estimated.

The least squares solutions for linear systems with matrices of deficient rank are improved by means of the augmented system approach. The singular subspaces corresponding to the smallest singular values are refined by solving special Riccati equations. Condition number analysis is carried out for all the included systems of linear equations.