159 research outputs found
An inverse-free ADI algorithm for computing Lagrangian invariant subspaces
Summary: The numerical computation of Lagrangian invariant subspaces of large-scale Hamiltonian matrices is discussed in the context of the solution of Lyapunov equations. A new version of the low-rank alternating direction implicit method is introduced, which, in order to avoid numerical difficulties with solutions that are of very large norm, uses an inverse-free representation of the subspace and avoids inverses of ill-conditioned matrices. It is shown that this prevents large growth of the elements of the solution that may destroy a low-rank approximation of the solution. A partial error analysis is presented, and the behavior of the method is demonstrated via several numerical examples. Copyrigh
An inverse-free ADI algorithm for computing Lagrangian invariant subspaces
The numerical computation of Lagrangian invariant subspaces of large scale Hamiltonian matrices is discussed in the context of the solution of Lyapunov and Riccati equations. A new version of the low-rank alternating direction implicit method is introduced, which in order to avoid numerical difficulties with solutions that are of very large norm, uses an inverse-free representation of the subspace and avoids inverses of ill-conditioned matrices. It is shown that this prevents large growth of the elements of the solution which may destroy a low-rank approximation of the solution. A partial error analysis is presented and the behavior of the method is demonstrated via several numerical examples
Stoquasticity in circuit QED
We analyze whether circuit-QED Hamiltonians are stoquastic focusing on
systems of coupled flux qubits: we show that scalable sign-problem free path
integral Monte Carlo simulations can typically be performed for such systems.
Despite this, we corroborate the recent finding [arXiv:1903.06139] that an
effective, non-stoquastic qubit Hamiltonian can emerge in a system of
capacitively coupled flux qubits. We find that if the capacitive coupling is
sufficiently small, this non-stoquasticity of the effective qubit Hamiltonian
can be avoided if we perform a canonical transformation prior to projecting
onto an effective qubit Hamiltonian. Our results shed light on the power of
circuit-QED Hamiltonians for the use of quantum adiabatic computation and the
subtlety of finding a representation which cures the sign problem in these
system
Reduced order modeling of convection-dominated flows, dimensionality reduction and stabilization
We present methodologies for reduced order modeling of convection dominated flows. Accordingly, three main problems are addressed.
Firstly, an optimal manifold is realized to enhance reducibility of convection dominated flows. We design a low-rank auto-encoder to specifically reduce the dimensionality of solution arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Also, considering that the latent variables are often hard to interpret, many of these methods are dismissed in the reduced order modeling of dynamical systems governed by partial differential equations (PDEs). This deficiency is of importance to the extent that linear methods, such as principle component analysis (PCA) and Koopman operators, are still prevalent. Accordingly, we propose an interpretable nonlinear dimensionality reduction algorithm. An unsupervised learning problem is constructed that learns a diffeomorphic spatio-temporal grid which registers the output sequence of the PDEs on a non-uniform time-varying grid. The Kolmogorov n-width of the mapped data on the learned grid is minimized.
Secondly, the reduced order models are constructed on the realized manifolds. We project the high fidelity models on the learned manifold, leading to a time-varying system of equations. Moreover, as a data-driven model free architecture, recurrent neural networks on the learned manifold are trained, showing versatility of the proposed framework.
Finally, a stabilization method is developed to maintain stability and accuracy of the projection based ROMs on the learned manifold a posteriori. We extend the eigenvalue reassignment method of stabilization of linear time-invariant ROMs, to the more general case of linear time-varying systems. Through a post-processing step, the ROMs are controlled using a constrained nonlinear lease-square minimization problem. The controller and the input signals are defined at the algebraic level, using left and right singular vectors of the reduced system matrices. The proposed stabilization method is general and applicable to a large variety of linear time-varying ROMs
Krylov subspace techniques for model reduction and the solution of linear matrix equations
This thesis focuses on the model reduction of linear systems and the solution of large
scale linear matrix equations using computationally efficient Krylov subspace techniques.
Most approaches for model reduction involve the computation and factorization of large
matrices. However Krylov subspace techniques have the advantage that they involve only
matrix-vector multiplications in the large dimension, which makes them a better choice
for model reduction of large scale systems. The standard Arnoldi/Lanczos algorithms are
well-used Krylov techniques that compute orthogonal bases to Krylov subspaces and, by
using a projection process on to the Krylov subspace, produce a reduced order model that
interpolates the actual system and its derivatives at infinity. An extension is the rational
Arnoldi/Lanczos algorithm which computes orthogonal bases to the union of Krylov
subspaces and results in a reduced order model that interpolates the actual system and
its derivatives at a predefined set of interpolation points. This thesis concentrates on the
rational Krylov method for model reduction.
In the rational Krylov method an important issue is the selection of interpolation points
for which various techniques are available in the literature with different selection criteria.
One of these techniques selects the interpolation points such that the approximation
satisfies the necessary conditions for H2 optimal approximation. However it is possible
to have more than one approximation for which the necessary optimality conditions are
satisfied. In this thesis, some conditions on the interpolation points are derived, that
enable us to compute all approximations that satisfy the necessary optimality conditions
and hence identify the global minimizer to the H2 optimal model reduction problem.
It is shown that for an H2 optimal approximation that interpolates at m interpolation
points, the interpolation points are the simultaneous solution of m multivariate polynomial
equations in m unknowns. This condition reduces to the computation of zeros of a
linear system, for a first order approximation. In case of second order approximation the
condition is to compute the simultaneous solution of two bivariate polynomial equations.
These two cases are analyzed in detail and it is shown that a global minimizer to the
H2 optimal model reduction problem can be identified. Furthermore, a computationally
efficient iterative algorithm is also proposed for the H2 optimal model reduction problem
that converges to a local minimizer.
In addition to the effect of interpolation points on the accuracy of the rational interpolating
approximation, an ordinary choice of interpolation points may result in a reduced
order model that loses the useful properties such as stability, passivity, minimum-phase and bounded real character as well as structure of the actual system. Recently in the
literature it is shown that the rational interpolating approximations can be parameterized
in terms of a free low dimensional parameter in order to preserve the stability of the
actual system in the reduced order approximation. This idea is extended in this thesis
to preserve other properties and combinations of them. Also the concept of parameterization
is applied to the minimal residual method, two-sided rational Arnoldi method
and H2 optimal approximation in order to improve the accuracy of the interpolating
approximation.
The rational Krylov method has also been used in the literature to compute low rank
approximate solutions of the Sylvester and Lyapunov equations, which are useful for
model reduction. The approach involves the computation of two set of basis vectors in
which each vector is orthogonalized with all previous vectors. This orthogonalization
becomes computationally expensive and requires high storage capacity as the number of
basis vectors increases. In this thesis, a restart scheme is proposed which restarts without
requiring that the new vectors are orthogonal to the previous vectors. Instead, a set of
two new orthogonal basis vectors are computed. This reduces the computational burden
of orthogonalization and the requirement of storage capacity. It is shown that in case
of Lyapunov equations, the approximate solution obtained through the restart scheme
approaches monotonically to the actual solution
Geometry of variational methods: dynamics of closed quantum systems
We present a systematic geometric framework to study closed quantum systems
based on suitably chosen variational families. For the purpose of (A) real time
evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary
time evolution, we show how the geometric approach highlights the necessity to
distinguish between two classes of manifolds: K\"ahler and non-K\"ahler.
Traditional variational methods typically require the variational family to be
a K\"ahler manifold, where multiplication by the imaginary unit preserves the
tangent spaces. This covers the vast majority of cases studied in the
literature. However, recently proposed classes of generalized Gaussian states
make it necessary to also include the non-K\"ahler case, which has already been
encountered occasionally. We illustrate our approach in detail with a range of
concrete examples where the geometric structures of the considered manifolds
are particularly relevant. These go from Gaussian states and group theoretic
coherent states to generalized Gaussian states.Comment: Submission to SciPost, 47+10 pages, 8 figure
Dirac pairings, one-form symmetries and Seiberg-Witten geometries
The Coulomb phase of a quantum field theory, when present, illuminates the
analysis of its line operators and one-form symmetries. For 4d
field theories the low energy physics of this phase is encoded in the special
K\"ahler geometry of the moduli space of Coulomb vacua. We clarify how the
information on the allowed line operator charges and one-form symmetries is
encoded in the special K\"ahler structure. We point out the important
difference between the lattice of charged states and the homology lattice of
the abelian variety fibered over the moduli space, which, when principally
polarized, is naturally identified with a choice of the lattice of mutually
local line operators. This observation illuminates how the distinct S-duality
orbits of global forms of theories are encoded geometrically.Comment: v3 - with minor correction
Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures
The Drinfeld - Sokolov construction associates a hierarchy of bihamiltonian
integrable systems with every untwisted affine Lie algebra. We compute the
complete set of invariants of the related bihamiltonian structures with respect
to the group of Miura type transformations.Comment: 73 pages, no figure
- …