| 1. |
- Breiten, Tobias, et al.
(författare)
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Residual-based iterations for the generalized Lyapunov equation
- 2019
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Ingår i: BIT Numerical Mathematics. - : Springer Netherlands. - 0006-3835 .- 1572-9125. ; 59:4, s. 823-852
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Tidskriftsartikel (refereegranskat)abstract
- This paper treats iterative solution methods for the generalized Lyapunov equation. Specifically, a residual-based generalized rational-Krylov-type subspace is proposed. Furthermore, the existing theoretical justification for the alternating linear scheme (ALS) is extended from the stable Lyapunov equation to the stable generalized Lyapunov equation. Further insights are gained by connecting the energy-norm minimization in ALS to the theory of H2-optimality of an associated bilinear control system. Moreover it is shown that the ALS-based iteration can be understood as iteratively constructing rank-1 model reduction subspaces for bilinear control systems associated with the residual. Similar to the ALS-based iteration, the fixed-point iteration can also be seen as a residual-based method minimizing an upper bound of the associated energy norm.
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| 2. |
- Gattami, Ather, et al.
(författare)
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Time localization and capacity of faster-than-nyquist signaling
- 2015
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Ingår i: 2015 IEEE Global Communications Conference, GLOBECOM 2015. - : Institute of Electrical and Electronics Engineers (IEEE). - 9781479959525
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Konferensbidrag (refereegranskat)abstract
- In this paper, we consider communication over the bandwidth limited analog white Gaussian noise channel using non-orthogonal pulses. In particular, we consider non-orthogonal transmission by signaling samples at a rate higher than the Nyquist rate. Using the faster-than- Nyquist (FTN) framework, Mazo showed that one may transmit symbols carried by sinc pulses at a higher rate than that dictated by Nyquist without loosing bit error rate. However, as we will show in this paper, such pulses are not necessarily well localized in time. In fact, assuming that signals in the FTN framework are well localized in time, one can construct a signaling scheme that violates the Shannon capacity bound. We also show directly that FTN signals are in general not well localized in time. We also consider FTN signaling in the case of pulses that are different from the sinc pulses. We show that one may use a precoding scheme of low complexity, in order to remove the intersymbol interference. This leads to the possibility of increasing the number of transmitted samples per time unit and compensate for spectral inefficiency due to signaling at the Nyquist rate of the non sinc pulses. We demonstrate the power of the precoding scheme by simulations.
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| 3. |
- Jarlebring, Elias, et al.
(författare)
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Computational Graphs for Matrix Functions
- 2022
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Ingår i: ACM Transactions on Mathematical Software. - : Association for Computing Machinery (ACM). - 0098-3500 .- 1557-7295. ; 48:4, s. 1-35
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Tidskriftsartikel (refereegranskat)abstract
- Many numerical methods for evaluating matrix functions can be naturally viewed as computational graphs. Rephrasing these methods as directed acyclic graphs (DAGs) is a particularly effective approach to study existing techniques, improve them, and eventually derive new ones. The accuracy of these matrix techniques can be characterized by the accuracy of their scalar counterparts, thus designing algorithms for matrix functions can be regarded as a scalar-valued optimization problem. The derivatives needed during the optimization can be calculated automatically by exploiting the structure of the DAG in a fashion analogous to backpropagation. This article describes GraphMatFun.jl, a Julia package that offers the means to generate and manipulate computational graphs, optimize their coefficients, and generate Julia, MATLAB, and C code to evaluate them efficiently at a matrix argument. The software also provides tools to estimate the accuracy of a graph-based algorithm and thus obtain numerically reliable methods. For the exponential, for example, using a particular form (degree-optimal) of polynomials produces implementations that in many cases are cheaper, in terms of computational cost, than the Pade-based techniques typically used in mathematical software. The optimized graphs and the corresponding generated code are available online.
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| 4. |
- Jarlebring, Elias, et al.
(författare)
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Krylov methods for low-rank commuting generalized Sylvester equations
- 2018
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Ingår i: Numerical Linear Algebra with Applications. - : Wiley. - 1070-5325 .- 1099-1506. ; 25:6
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Tidskriftsartikel (refereegranskat)abstract
- We consider generalizations of the Sylvester matrix equation, consisting of the sum of a Sylvester operator and a linear operator pi with a particular structure. More precisely, the commutators of the matrix coefficients of the operator pi and the Sylvester operator coefficients are assumed to be matrices with low rank. We show (under certain additional conditions) low-rank approximability of this problem, that is, the solution to this matrix equation can be approximated with a low-rank matrix. Projection methods have successfully been used to solve other matrix equations with low-rank approximability. We propose a new projection method for this class of matrix equations. The choice of the subspace is a crucial ingredient for any projection method for matrix equations. Our method is based on an adaption and extension of the extended Krylov subspace method for Sylvester equations. A constructive choice of the starting vector/block is derived from the low-rank commutators. We illustrate the effectiveness of our method by solving large-scale matrix equations arising from applications in control theory and the discretization of PDEs. The advantages of our approach in comparison to other methods are also illustrated.
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| 5. |
- Larsson, Martin, et al.
(författare)
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Towards an Indoor Testbed for Mobile Networked Control Systems
- 2011
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Konferensbidrag (refereegranskat)abstract
- In this paper, we consider the design of an indoor testbed composed of multiple aerial and ground unmanned vehicles for experimentation in Mobile Networked Control Systems. Taking several motivational aspects from both research and education into account, we propose an architecture to cope with the scale and mobility aspects of the overall system. Currently, the testbed is composed of several low-cost ARdrones quadrotors, small-scale heavy duty vehicles, wireless sensor nodes and a vision-based localization system. As an example, the automatic control of an ARdrone is shown.
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| 6. |
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| 7. |
- Ringh, Emil, et al.
(författare)
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Nonlinearizing two-parameter eigenvalue problems
- 2021
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Ingår i: SIAM Journal on Matrix Analysis and Applications. - : Society for Industrial & Applied Mathematics (SIAM). - 0895-4798 .- 1095-7162. ; 42:2, s. 775-799
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Tidskriftsartikel (refereegranskat)abstract
- We investigate a technique to transform a linear two-parameter eigenvalue problem, into a nonlinear eigenvalue problem (NEP). The transformation stems from an elimination of one of the equations in the two-parameter eigenvalue problem, by considering it as a (standard) generalized eigenvalue problem. We characterize the equivalence between the original and the nonlinearized problem theoretically and show how to use the transformation computationally. Special cases of the transformation can be interpreted as a reversed companion linearization for polynomial eigenvalue problems, as well as a reversed (less known) linearization technique for certain algebraic eigenvalue problems with square-root terms. Moreover, by exploiting the structure of the NEP we present algorithm specializations for NEP-methods, although the technique also allows general solution methods for NEPs to be directly applied. The nonlinearization is illustrated in examples and simulations, with focus on problems where the eliminated equation is of much smaller size than the other two-parameter eigenvalue equation. This situation arises naturally in domain decomposition techniques. A general error analysis is also carried out under the assumption that a backward stable eigensolver is used to solve the eliminated problem, leading to the conclusion that the error is benign in this situation.
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| 8. |
- Ringh, Emil, 1989-
(författare)
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Numerical methods for Sylvester-type matrix equations and nonlinear eigenvalue problems
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Linear matrix equations and nonlinear eigenvalue problems (NEP) appear in a wide variety of applications in science and engineering. Important special cases of the former are the Lyapunov equation, the Sylvester equation, and their respective generalizations. These appear, e.g., as Gramians to linear and bilinear systems, in computations involving block-triangularization of matrices, and in connection with discretizations of some partial differential equations. The NEP appear, e.g., in stability analysis of time-delay systems, and as results of transformations of linear eigenvalue problems.This thesis mainly consists of 4 papers that treats the above mentioned computational problems, and presents both theory and methods. In paper A we consider a NEP stemming from the discretization of a partial differential equation describing wave propagation in a waveguide. Some NEP-methods require in each iteration to solve a linear system with a fixed matrix, but different right-hand sides, and with a fine discretization, this linear solve becomes the bottleneck. To overcome this we present a Sylvester-based preconditioner, exploiting the Sherman–Morrison–Woodbury formula.Paper B treats the generalized Sylvester equation and present two main results: First, a characterization that under certain assumptions motivates the existence of low-rank solutions. Second, a Krylov method applicable when the matrix coefficients are low-rank commuting, i.e., when the commutator is of low rank.In Paper C we study the generalized Lyapunov equation. Specifically, we extend the motivation for applying the alternating linear scheme (ALS) method, from the stable Lyapunov equation to the stable generalized Lyapunov equation. Moreover, we show connections to H2-optimal model reduction of associated bilinear systems, and show that ALS can be understood to construct a rank-1 model reduction subspace to such a bilinear system related to the residual. We also propose a residual-based generalized rational-Krylov-type subspace as a solver for the generalized Lyapunov equation.The fourth paper, Paper D, connects the NEP to the two-parameter eigenvalue problem. The latter is a generalization of the linear eigenvalue problem in the sense that there are two eigenvalue-eigenvector equations, both depending on two scalar variables. If we fix one of the variables, then we can use one of the equations, which is then a generalized eigenvalue problem, to solve for the other variable. In that sense, the solved-for variable can be understood as a family of functions of the first variable. Hence, it is a variable elimination technique where the second equation can be understood as a family of NEPs. Methods for NEPs can thus be adapted and exploited to solve the original problem. The idea can also be reversed, providing linearizations for certain NEPs.
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| 9. |
- Ringh, Emil, et al.
(författare)
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Sylvester-based preconditioning for the waveguide eigenvalue problem
- 2018
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 542:1, s. 441-463
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Tidskriftsartikel (refereegranskat)abstract
- We consider a nonlinear eigenvalue problem (NEP) arising from absorbing boundary conditions in the study of a partial differential equation (PDE) describing a waveguide. We propose a new computational approach for this large-scale NEP based on residual inverse iteration (Resinv) with preconditioned iterative solves. Similar to many preconditioned iterative methods for discretized PDEs, this approach requires the construction of an accurate and efficient preconditioner. For the waveguide eigenvalue problem, the associated linear system can be formulated as a generalized Sylvester equation AX+XB+A1XB1+A2XB2+K(ring operator)X=C, where (ring operator) denotes the Hadamard product. The equation is approximated by a low-rank correction of a Sylvester equation, which we use as a preconditioner. The action of the preconditioner is efficiently computed by using the matrix equation version of the Sherman-Morrison-Woodbury (SMW) formula. We show how the preconditioner can be integrated into Resinv. The results are illustrated by applying the method to large-scale problems.
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