95 research outputs found
Randomized Joint Diagonalization of Symmetric Matrices
Given a family of nearly commuting symmetric matrices, we consider the task
of computing an orthogonal matrix that nearly diagonalizes every matrix in the
family. In this paper, we propose and analyze randomized joint diagonalization
(RJD) for performing this task. RJD applies a standard eigenvalue solver to
random linear combinations of the matrices. Unlike existing optimization-based
methods, RJD is simple to implement and leverages existing high-quality linear
algebra software packages. Our main novel contribution is to prove robust
recovery: Given a family that is -near to a commuting family, RJD
jointly diagonalizes this family, with high probability, up to an error of norm
O(). No other existing method is known to enjoy such a universal
robust recovery guarantee. We also discuss how the algorithm can be further
improved by deflation techniques and demonstrate its state-of-the-art
performance by numerical experiments with synthetic and real-world data
Signatures of dissipative quantum chaos
Understanding the far-from-equilibrium dynamics of dissipative quantum
systems, where dissipation and decoherence coexist with unitary dynamics, is an
enormous challenge with immense rewards. Often, the only realistic approach is
to forgo a detailed microscopic description and search for signatures of
universal behavior shared by collections of many distinct, yet sufficiently
similar, complex systems. Quantum chaos provides a powerful statistical
framework for addressing this question, relying on symmetries to obtain
information not accessible otherwise. This thesis examines how to reconcile
chaos with dissipation, proceeding along two complementary lines. In Part I, we
apply non-Hermitian random matrix theory to open quantum systems with Markovian
dissipation and discuss the relaxation timescales and steady states of three
representative examples of increasing physical relevance: single-particle
Lindbladians and Kraus maps, open free fermions, and dissipative
Sachdev-Ye-Kitaev (SYK) models. In Part II, we investigate the symmetries,
correlations, and universality of many-body open quantum systems, classifying
several models of dissipative quantum matter. From a theoretical viewpoint,
this thesis lays out a generic framework for the study of the universal
properties of realistic, chaotic, and dissipative quantum systems. From a
practical viewpoint, it provides the concrete building blocks of dynamical
dissipative evolution constrained by symmetry, with potential technological
impact on the fabrication of complex quantum structures.
(Full abstract in the thesis.)Comment: PhD Thesis, University of Lisbon (2023). 264 pages, 54 figures.
Partial overlap with arXiv:1905.02155, arXiv:1910.12784, arXiv:2007.04326,
arXiv:2011.06565, arXiv:2104.07647, arXiv:2110.03444, arXiv:2112.12109,
arXiv:2210.07959, arXiv:2210.01695, arXiv:2211.01650, arXiv:2212.00474, and
arXiv:2305.0966
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Dynamics of the QR-flow for upper Hessenberg real matrices
We investigate the main phase space properties of the QR-flow when restricted to upper Hessenberg matrices. A complete description of the linear behavior of the equilibrium matrices is given. The main result classifies the possible - and -limits of the orbits for this system. Furthermore, we characterize the set of initial matrices for which there is convergence towards an equilibrium matrix. Several numerical examples show the different limit behavior of the orbits and illustrate the theory
Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022
The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing
Kernel PCA and the Nyström method
This thesis treats kernel PCA and the Nystrom method. We present a novel incre- ¨
mental algorithm for calculation of kernel PCA, which we extend to incremental
calculation of the Nystrom approximation. We suggest a new data-dependent ¨
method to select the number of data points to include in the Nystrom subset, ¨
and create a statistical hypothesis test for the same purpose. We further present
a cross-validation procedure for kernel PCA to select the number of principal
components to retain. Finally, we derive kernel PCA with the Nystrom method ¨
in line with linear PCA and study its statistical accuracy through a confidence
bound
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System
Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics
Sliding Mode Control
The main objective of this monograph is to present a broad range of well worked out, recent application studies as well as theoretical contributions in the field of sliding mode control system analysis and design. The contributions presented here include new theoretical developments as well as successful applications of variable structure controllers primarily in the field of power electronics, electric drives and motion steering systems. They enrich the current state of the art, and motivate and encourage new ideas and solutions in the sliding mode control area
Order reduction of semilinear differential matrix and tensor equations
In this thesis, we are interested in approximating, by model order reduction, the solution to large-scale matrix- or tensor-valued semilinear Ordinary Differential Equations (ODEs). Under specific hypotheses on the linear operators and the considered domain, these ODEs often stem from the space discretization on a tensor basis of semilinear Partial Differential Equations (PDEs) with a dimension greater than or equal to two.
The bulk of this thesis is devoted to the case where the discrete system is a matrix equation. We consider separately the cases of general Lipschitz continuous nonlinear functions and the Differential Riccati Equation (DRE) with a quadratic nonlinear term. In both settings, we construct a pair of left-right approximation spaces that leads to a reduced semilinear matrix differential equation with the same structure as the original problem, which can be more rapidly integrated with matrix-oriented integrators. For the DRE, under certain assumptions on the data, we show that a reduction process onto rational Krylov subspaces obtains significant computational and memory savings as opposed to current approaches.
In the more general setting, a challenging difference lies in selecting and constructing the two approximation bases to handle the nonlinear term effectively. In addition, the nonlinear term also needs to be approximated for efficiency. To this end, in the framework of the Proper Orthogonal Decomposition (POD) methodology and the Discrete Empirical Interpolation Method (DEIM), we derive a novel matrix-oriented reduction process leading to a practical, structure-aware low order approximation of the original problem.
In the final part of the thesis, we consider the multidimensional setting. Here we extend the matrix-oriented POD-DEIM algorithm to the tensor setting and illustrate how we can apply it to systems of such equations. Moreover, we discuss how to integrate the reduced-order model and, in particular, how to solve the resulting tensor-valued linear systems
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