arXiv subjects

Nguyen Thanh Son

Publications and source records attributed to Nguyen Thanh Son.

At least 19 recordsLinked to original sources

arXiv2026

Brockett cost function for symplectic eigenvalues

The sum of symplectic eigenvalues and corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson's theorem can be computed via minimization of a trace cost function under the symplecticity constraint. Optimal solutions to this problem only offer a symplectic basis for the symplectic eigenspace corresponding to the sought symplectic eigenvalues. In this note, we introduce a Brockett cost function and investigate its properties and the connection with the symplectic eigenvalues and eigenvectors of the considered matrix. Specifically, we prove that any stationary point consists of symplectic eigenvectors, characterize the saddle points and global minimizers based on which the trace minimization theorem for the symplectic eigenvalues is re-established, and the nonexistence of local nonglobal minimizers is justified.

math.OC↗

arXiv2026

A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold

We consider the optimization problem with a generally quadratic matrix constraint of the form $X^TAX = J$, where $A$ is a given nonsingular, symmetric $n\times n$ matrix and $J$ is a given $k\times k$ symmetric matrix, with $k\leq n$, satisfying $J^2 = I_k$. Since the feasible set constitutes a differentiable manifold, called the indefinite Stiefel manifold, we approach this problem within the framework of Riemannian optimization. Namely, we first equip the manifold with a Riemannian metric and construct the associated geometric structure, then propose a retraction based on the Cayley transform, and finally suggest a Riemannian gradient descent method using the attained materials, whose global convergence is guaranteed. Our results not only cover the known cases, the orthogonal and generalized Stiefel manifolds, but also provide a Riemannian optimization solution for other constrained problems which has not been investigated. As applications, we consider, via trace minimization, several eigenvalue problems of symmetric positive definite matrix pencils, including the linear response eigenvalue problem, and a matrix least square problem, a general framework for the Procrustes problem and constrained matrix equations. The presented numerical results justify the theoretical findings.

math.OC↗

arXiv2026

A generalized canonical metric for optimization on the indefinite Stiefel manifold

Various tasks in scientific computing can be modeled as an optimization problem on the indefinite Stiefel manifold. We address this using the Riemannian approach, which basically consists of equipping the feasible set with a Riemannian metric, preparing geometric tools such as orthogonal projections, formulae for Riemannian gradient, retraction and then extending an unconstrained optimization algorithm on the Euclidean space to the established manifold. The choice for the metric undoubtedly has a great influence on the method. In the previous work [D.V. Tiep and N.T. Son, A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold, arXiv:2410.22068v2[math.OC]], a tractable metric, which is indeed a family of Riemannian metrics defined by a symmetric positive-definite matrix depending on the contact point, has been used. In general, it requires solving a Lyapunov matrix equation every time when the gradient of the cost function is needed, which might significantly contribute to the computational cost. To address this issue, we propose a new Riemannian metric for the indefinite Stiefel manifold. Furthermore, we construct the associated geometric structure, including a so-called quasi-geodesic and propose a retraction based on this curve. We then numerically verify the performance of the Riemannian gradient descent method associated with the new geometry and compare it with the previous work.

math.OC↗

arXiv2024

Symplectic Stiefel manifold: tractable metrics, second-order geometry and Newton's methods

Optimization under the symplecticity constraint is an approach for solving various problems in quantum physics and scientific computing. Building on the results that this optimization problem can be transformed into an unconstrained problem on the symplectic Stiefel manifold, we construct geometric ingredients for Riemannian optimization with a new family of Riemannian metrics called tractable metrics and develop Riemannian Newton schemes. The newly obtained ingredients do not only generalize several existing results but also provide us with freedom to choose a suitable metric for each problem. To the best of our knowledge, this is the first try to develop the explicit second-order geometry and Newton's methods on the symplectic Stiefel manifold. For the Riemannian Newton method, we first consider novel operator-valued formulas for computing the Riemannian Hessian of a~cost function, which further allows the manifold to be endowed with a weighted Euclidean metric that can provide a preconditioning effect. We then solve the resulting Newton equation, as the central step of Newton's methods, directly via transforming it into a~saddle point problem followed by vectorization, or iteratively via applying any matrix-free iterative method either to the operator Newton equation or its saddle point formulation. Finally, we propose a hybrid Riemannian Newton optimization algorithm that enjoys both global convergence and quadratic/superlinear local convergence at the final stage. Various numerical experiments are presented to validate the proposed methods.

math.OC↗

arXiv2022

Optimization on the symplectic Stiefel manifold: SR decomposition-based retraction and applications

Numerous problems in optics, quantum physics, stability analysis, and control of dynamical systems can be brought to an optimization problem with matrix variable subjected to the symplecticity constraint. As this constraint nicely forms a so-called symplectic Stiefel manifold, Riemannian optimization is preferred, because one can borrow ideas from unconstrained optimization methods after preparing necessary geometric tools. Retraction is arguably the most important one which decides the way iterates are updated given a search direction. Two retractions have been constructed so far: one relies on the Cayley transform and the other is designed using quasi-geodesic curves. In this paper, we propose a new retraction which is based on an SR matrix decomposition. We prove that its domain contains the open unit ball which is essential in proving the global convergence of the associated gradient-based optimization algorithm. Moreover, we consider three applications--symplectic target matrix problem, symplectic eigenvalue computation, and symplectic model reduction of Hamiltonian systems--with various examples. The extensive numerical comparisons reveal the strengths of the proposed optimization algorithm.

math.OC↗

arXiv2022

Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem

Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, are extended to the case of symplectic eigenvalues. In this note, we will generalize Williamson's diagonal form for symmetric positive-definite matrices to the case of symmetric positive-semidefinite matrices, which allows us to define symplectic eigenvalues, and prove the trace minimization theorem in the new setting.

math.OC↗

arXiv2021

Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric

The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space $\mathbb{R}^{2n\times 2p}$. The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel manifold. We obtain the expression of the Riemannian gradient with respect to the Euclidean metric, which then used in optimization algorithms. Numerical experiments on the nearest symplectic matrix problem and the symplectic eigenvalue problem illustrate the effectiveness of Euclidean-based algorithms.

math.OC↗

arXiv2021

Symplectic eigenvalue problem via trace minimization and Riemannian optimization

We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson's theorem. It is formulated as minimizing a trace cost function over the symplectic Stiefel manifold. We first investigate various theoretical aspects of this optimization problem such as characterizing the sets of critical points, saddle points, and global minimizers as well as proving that non-global local minimizers do not exist. Based on our recent results on constructing Riemannian structures on the symplectic Stiefel manifold and the associated optimization algorithms, we then propose solving the symplectic eigenvalue problem in the framework of Riemannian optimization. Moreover, a connection of the sought solution with the eigenvalues of a special class of Hamiltonian matrices is discussed. Numerical examples are presented.

math.OC↗

arXiv2020

A property of the spherical derivative of an entire curve in complex projective space

We establish a type of the Picard's theorem for entire curves in $P^n(\mathbb C)$ whose spherical derivative vanishes on the inverse images of hypersurface targets. Then, as a corollary, we prove that there is an union $D$ of finite number of hypersurfaces in the complex projective space $P^n(\mathbb C)$ such that for every entire curve $f$ in $P^n(\mathbb C)$, if the spherical derivative $f^{\#}$ of $f$ is bounded on $ f^{-1}(D)$, then $f^{\#}$ is bounded on the entire complex plane, and hence, $f$ is a Brody curve.

math.CV↗

arXiv2020

Convergence analysis of a Crank-Nicolson Galerkin method for an inverse source problem for parabolic equations with boundary observations

This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method is applied to the least squares functional with an quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach to zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and corresponding convergence rates are proved. Finally, several numerical experiments are presented to illustrate the theoretical findings.

math.NA↗

arXiv2020

Riemannian Optimization on the Symplectic Stiefel Manifold

The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. Optimization problems on $\mathrm{Sp}(2p,2n)$ find applications in various areas, such as optics, quantum physics, numerical linear algebra and model order reduction of dynamical systems. The purpose of this paper is to propose and analyze gradient-descent methods on $\mathrm{Sp}(2p,2n)$, where the notion of gradient stems from a Riemannian metric. We consider a novel Riemannian metric on $\mathrm{Sp}(2p,2n)$ akin to the canonical metric of the (standard) Stiefel manifold. In order to perform a feasible step along the antigradient, we develop two types of search strategies: one is based on quasi-geodesic curves, and the other one on the symplectic Cayley transform. The resulting optimization algorithms are proved to converge globally to critical points of the objective function. Numerical experiments illustrate the efficiency of the proposed methods.

math.OC↗

arXiv2020

Balanced truncation for parametric linear systems using interpolation of Gramians: a comparison of algebraic and geometric approaches

When balanced truncation is used for model order reduction, one has to solve a pair of Lyapunov equations for two Gramians and uses them to construct a reduced-order model. Although advances in solving such equations have been made, it is still the most expensive step of this reduction method. Parametric model order reduction aims to determine reduced-order models for parameter-dependent systems. Popular techniques for parametric model order reduction rely on interpolation. Nevertheless, the interpolation of Gramians is rarely mentioned, most probably due to the fact that Gramians are symmetric positive semidefinite matrices, a property that should be preserved by the interpolation method. In this contribution, we propose and compare two approaches for Gramian interpolation. In the first approach, the interpolated Gramian is computed as a linear combination of the data Gramians with positive coefficients. Even though positive semidefiniteness is guaranteed in this method, the rank of the interpolated Gramian can be significantly larger than that of the data Gramians. The second approach aims to tackle this issue by performing the interpolation on the manifold of fixed-rank positive semidefinite matrices. The results of the interpolation step are then used to construct parametric reduced-order models, which are compared numerically on two benchmark problems.

math.NA↗

arXiv2017

Schmidt's subspace theorem for moving hypersurface targets

It was discovered that there is a formal analogy between Nevanlinna theory and Diophantine approximation. Via Vojta's dictionary, the Second Main Theorem in Nevanlinna theory corresponds to Schmidt's Subspace Theorem in Diophantine approximation. Recently, Cherry, Dethloff, and Tan (arXiv:1503.08801v2 [math.CV]) obtained a Second Main Theorem for moving hypersurfaces intersecting projective varieites. In this paper, we shall give the counterpart of their Second Main Theorem in Diophantine approximation.

math.NT↗

arXiv2014

Energy-Aware Lease Scheduling in Virtualized Data Centers

Energy efficiency has become an important measurement of scheduling algorithms in virtualized data centers. One of the challenges of energy-efficient scheduling algorithms, however, is the trade-off between minimizing energy consumption and satisfying quality of service (e.g. performance, resource availability on time for reservation requests). We consider resource needs in the context of virtualized data centers of a private cloud system, which provides resource leases in terms of virtual machines (VMs) for user applications. In this paper, we propose heuristics for scheduling VMs that address the above challenge. On performance evaluation, simulated results have shown a significant reduction on total energy consumption of our proposed algorithms compared with an existing First-Come-First-Serve (FCFS) scheduling algorithm with the same fulfillment of performance requirements. We also discuss the improvement of energy saving when additionally using migration policies to the above mentioned algorithms.

cs.DC↗

arXiv2014

EPOBF: Energy Efficient Allocation of Virtual Machines in High Performance Computing Cloud

Cloud computing has become more popular in provision of computing resources under virtual machine (VM) abstraction for high performance computing (HPC) users to run their applications. A HPC cloud is such cloud computing environment. One of challenges of energy efficient resource allocation for VMs in HPC cloud is tradeoff between minimizing total energy consumption of physical machines (PMs) and satisfying Quality of Service (e.g. performance). On one hand, cloud providers want to maximize their profit by reducing the power cost (e.g. using the smallest number of running PMs). On the other hand, cloud customers (users) want highest performance for their applications. In this paper, we focus on the scenario that scheduler does not know global information about user jobs and user applications in the future. Users will request shortterm resources at fixed start times and non interrupted durations. We then propose a new allocation heuristic (named Energy-aware and Performance per watt oriented Bestfit (EPOBF)) that uses metric of performance per watt to choose which most energy-efficient PM for mapping each VM (e.g. maximum of MIPS per Watt). Using information from Feitelson's Parallel Workload Archive to model HPC jobs, we compare the proposed EPOBF to state of the art heuristics on heterogeneous PMs (each PM has multicore CPU). Simulations show that the EPOBF can reduce significant total energy consumption in comparison with state of the art allocation heuristics.

cs.DC↗

arXiv2012

Performance Constraint and Power-Aware Allocation For User Requests In Virtual Computing Lab

Cloud computing is driven by economies of scale. A cloud system uses virtualization technology to provide cloud resources (e.g. CPU, memory) to users in form of virtual machines. Virtual machine (VM), which is a sandbox for user application, fits well in the education environment to provide computational resources for teaching and research needs. In resource management, they want to reduce costs in operations by reducing expensive cost of electronic bill of large-scale data center system. A lease-based model is suitable for our Virtual Computing Lab, in which users ask resources on a lease of virtual machines. This paper proposes two host selection policies, named MAP (minimum of active physical hosts) and MAP-H2L, and four algorithms solving the lease scheduling problem. FF-MAP, FF-MAP-H2L algorithms meet a trade-off between the energy consumption and Quality of Service (e.g. performance). The simulation on 7-day workload, which converted from LLNL Atlas log, showed the FF-MAP and FF-MAP-H2L algorithms reducing 7.24% and 7.42% energy consumption than existing greedy mapping algorithm in the leasing scheduler Haizea. In addition, we introduce a ratio θof consolidation in HalfPI-FF-MAP and PI-FF-MAP algorithms, in which θis π/2 and π, and results on their simulations show that energy consumption decreased by 34.87% and 63.12% respectively.

cs.DC↗

arXiv2005

Electron and Spin Correlations in Semiconductor Heterostructures: Quantum Singwi-Tosi-Land-Sjolander Theory

We apply the quantum Singwi-Tosi-Land-Sjolander (QSTLS) theory for a study of many-body effects in the quasi-two-dimensional (Q2D) electron liquid (EL) in GaAs/AlxGa1-xAs heterojunctions. The effect of the layer thickness is included through a variational approach. We have calculated the density, spin-density static structure factors, spin-dependent pair distribution functions (PDF) and compared our results with those of two-dimensional (2D) EL given in earlier papers. Using the static structure factors (SSF) we have calculated various dynamic correlation functions such as spin-dependent local-field factors (LFF) and effective potentials of the Q2D EL. We have also calculated the inverse static dielectric function of the 2D and Q2D EL using different approximations. We find that the effect of finite thickness on the dielectric function is remarkable and at the intermediate values of wave number q there is a significant difference between the QSTLS and STLS results.

cond-mat.mes-hall↗

arXiv2005

Isospin Separation of Hidden Non-Abelian Monopole

The scheme of isospin separation is suggested for the equation describing the five-dimensional 'charge-dyon' system in a non-Abelian SU(2) model. As a result, we obtain the Schrodinger equation for 'bare' particle, moving in Coulomb potential plus potential of five-dimensional Dirac monopole.

hep-th↗