619 research outputs found
Principal pivot transforms of quasidefinite matrices and semidefinite lagrangian subspaces
No full textLagrangian subspaces are linear subspaces that appear naturally in control theory applications, and especially in the context of algebraic Riccati equations. In this paper, a class of semidefinite Lagrangian subspaces is introduced and it is shown that these subspaces can be represented by a subset I ⊆ 1, 2,..., n and a Hermitian matrix X ∈ Cn×n with the property that the submatrix XII is negative semidefinite and the submatrix XIcIc is positive semidefinite. A matrix X with these definiteness properties is called I-semidefinite and it is a generalization of a quasidefinite matrix. Under mild hypotheses which hold true in most applications, the Lagrangian subspace associated to the stabilizing solution of an algebraic Riccati equation is semidefinite, and in addition it is shown that there is a bijection between Hamiltonian and symplectic pencils and semidefinite Lagrangian subspaces; hence, this structure is ubiquitous in control theory. The (symmetric) principal pivot transform (PPT) is a map used by Mehrmann and Poloni [V. Mehrmann and F. Poloni. Doubling algorithms with permuted Lagrangian graph bases. SIAM J. Matrix Anal. Appl., 33:780–805, 2012. to convert between two different pairs (I,X) and (J,X ′) representing the same Lagrangian subspace. For a semidefinite Lagrangian subspace, it is proven that the symmetric PPT of an I-semidefinite matrix X is a J -semidefinite matrix X ′, and an implementation of the transformation X → X ′ that both makes use of the definiteness properties of X and guarantees the definiteness of the submatrices of X ′ in finite arithmetic is derived. The resulting formulas are used to obtain a semidefiniteness-preserving version of an optimization algorithm introduced by Mehrmann and Poloni to compute a pair (Iopt,Xopt) with Mopt = maxi,j |(Xopt)ij | as small as possible. Using semidefiniteness allows one to obtain a stronger inequality on M with respect to the general case
Iterative and doubling algorithms for Riccati‐type matrix equations: A comparative introduction
No full textWe review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of doubling: they construct the iterate (Formula presented.) of another naturally-arising fixed-point iteration (Xh) via a sort of repeated squaring. The equations we consider are Stein equations X − A∗ X A = Q, Lyapunov equations A∗ X + X A + Q = 0, discrete-time algebraic Riccati equations X = Q + A∗ X(I + G X)−1A, continuous-time algebraic Riccati equations Q + A∗ X + X A − X G X = 0, palindromic quadratic matrix equations A + Q Y + A∗Y2 = 0, and nonlinear matrix equations X + A∗ X−1A = Q. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory
A new algorithm for time-dependent first-return probabilities of a fluid queue
No full text(non presente
Comparison theorems for splittings of M-matrices in (block) Hessenberg form
No full textSome variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix ρ(PGS) ≥ ρ(PS) ≥ ρ(PAGS) , where PGS, PS, PAGS are the iteration matrices of the Gauss–Seidel, staircase, and anti-Gauss–Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation
Nearest Ω -stable matrix via Riemannian optimization
No full textWe study the problem of finding the nearest Ω-stable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set Ω. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity
Algorithms for quadratic matrix and vector equations
No full textThis book is devoted to studying algorithms for the solution of a class of quadratic matrix and vector equations. These equations appear, in different forms, in several practical applications, especially in applied probability and control theory. The equations are first presented using a novel unifying approach; then, specific numerical methods are presented for the cases most relevant for applications, and new algorithms and theoretical results developed by the author are presented. The book focuses on “matrix multiplication-rich” iterations such as cyclic reduction and the structured doubling algorithm (SDA) and contains a variety of new research results which, as of today, are only available in articles or preprints
The case of gente Rutheni, natione Poloni in Galicia
No full textIn this article, the author seeks to answer the question, since when the term gente Rutheni, natione
Poloni was used in public space in Galicia. This is the starting point to search for the answer
to the question of when the Ruthenians of Polish nationality in Galicia produced the idea defi ning
their identity and worldview. The author tries to reveal when the Ruthenians of Polish nationality
made an unsuccessful attempt to carry out their political demands, and how they were perceived
in the Polish-Ruthenian society of Galicia. Eventually gente Rutheni, natione Poloni disappeared
in the historical process because they had not created an explicit ideology and had not developed
a compact elite representing the group of Ruthenians of Polish nationality in the public space.
Outstanding individuals from this group functioned within Galician society, but they were more of
a tool (the subject) of the Polish policy, rather than an entity
Genetic Algorithm with redundancies for Vehicle Scheduling Problem
No full textA real vehicle scheduling problem conceming the urban public A
transportation system of the city of Mestre (Venice) has been approached by a Genetic Algorithm enhanced using redundancies. Redundant alleles fix the string at cross-over positions in order to improve solution feasibility. The scheduling problem has been studied both as a single and as a multiple objective optimisation problem. A significant reduction of resources as compared to the currently used solution has been achieved
Gas Assisted Injection Molding optimization with M.O.G.A.
No full textThis paper describes the philosophy and architecture of a multi-objective design tool called modeFRONTIER. The paper explains the important concepts that lie behind the tool and demonstrates its use by tackling two multiple criteria design problems. Several technological components needs to be harmonised in order to do be able to face real multi-disciplinary and multi-criteria optimisation: IT infrastructure, efficient optimisation tools, decision making and statistical tools.
The example problem deals with gas-assisted plastic injection moulding. The optimisation is done using CADMould for the simulation of the process. While the geometry of the component is fixed, the plastic and gas injection point and process parameters like pressure, temperature, timing of the process fases are automatically found by the optimiser.
The two examples are related to different type of problems that can be faced by means of distributed computing: parallel execution of design tasks to improve the efficiency of the optimisation algorithms and handling of different simulation codes and platforms in the same design task
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