Gradient flow structure and exponential decay of the sandwiched R\'enyi divergence for primitive Lindblad equations with GNS-detailed balance

We study the entropy production of the sandwiched R\'enyi divergence under the primitive Lindblad equation with GNS-detailed balance. We prove that the Lindblad equation can be identified as the gradient flow of the sandwiched R\'enyi divergence of any order ${\alpha} \in (0, \infty)$. This extends a previous result by Carlen and Maas [Journal of Functional Analysis, 273(5), 1810-1869] for the quantum relative entropy (i.e., ${\alpha} = 1$). Moreover, we show that the sandwiched R\'enyi divergence of any order ${\alpha} \in (0, \infty)$ decays exponentially fast under the time-evolution of such a Lindblad equation.Comment: 43 pages; 2 figures; add a new section about the necessary condition of having a gradient flow structur

Tensorization of the strong data processing inequality for quantum chi-square divergences

It is well-known that any quantum channel $\mathcal{E}$ satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum $\chi^2_{\kappa}$divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states $\rho$ and $\sigma$ does not increase under the action of any quantum channel $\mathcal{E}$. For a fixed channel $\mathcal{E}$ and a state $\sigma$, the divergence between output states $\mathcal{E}(\rho)$ and $\mathcal{E}(\sigma)$ might be strictly smaller than the divergence between input states $\rho$ and $\sigma$, which is characterized by the strong data processing inequality (SDPI). Among various input states $\rho$, the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum $\chi^2_{\kappa_{1/2}}$ divergence for arbitrary quantum channels and also for a family of $\chi^2_{\kappa}$ divergences (with $\kappa \ge \kappa_{1/2}$) for arbitrary quantum-classical channels.Comment: Accepted by Quantu

Stochastic dynamical low-rank approximation method

In this paper, we extend the dynamical low-rank approximation method to the space of finite signed measures. Under this framework, we derive stochastic low-rank dynamics for stochastic differential equations (SDEs) coming from classical stochastic dynamics or unraveling of Lindblad quantum master equations. We justify the proposed method by error analysis and also numerical examples for applications in solving high-dimensional SDE, stochastic Burgers' equation, and high-dimensional Lindblad equation.Comment: 27 pages, 8 figure

Further results on the Hamilton-Waterloo problem

In this paper, we almost completely solve the existence of an almost resolvable cycle system with odd cycle length. We also use almost resolvable cycle systems as well as other combinatorial structures to give some new solutions to the Hamilton-Waterloo problem

Host galaxy properties of mergers of stellar binary black holes and their implications for advanced LIGO gravitational wave sources

Understanding the host galaxy properties of stellar binary black hole (SBBH) mergers is important for revealing the origin of the SBBH gravitational-wave sources detected by advanced LIGO and helpful for identifying their electromagnetic counterparts. Here we present a comprehensive analysis of the host galaxy properties of SBBHs by implementing semi-analytical recipes for SBBH formation and merger into cosmological galaxy formation model. If the time delay between SBBH formation and merger ranges from \la\,Gyr to the Hubble time, SBBH mergers at redshift z\la0.3 occur preferentially in big galaxies with stellar mass M_*\ga2\times10^{10}\msun and metallicities $Z$ peaking at $\sim0.6Z_\odot$. However, the host galaxy stellar mass distribution of heavy SBBH mergers (M_{\bullet\bullet}\ga50\msun) is bimodal with one peak at \sim10^9\msun and the other peak at \sim2\times10^{10}\msun. The contribution fraction from host galaxies with Z\la0.2Z_\odot to heavy mergers is much larger than that to less heavy mergers. If SBBHs were formed in the early universe (e.g., $z>6$), their mergers detected at z\la0.3 occur preferentially in even more massive galaxies with M_*>3\times10^{10}\msun and in galaxies with metallicities mostly \ga0.2Z_\odot and peaking at $Z\sim0.6Z_\odot$, due to later cosmic assembly and enrichment of their host galaxies. SBBH mergers at z\la0.3 mainly occur in spiral galaxies, but the fraction of SBBH mergers occur in elliptical galaxies can be significant if those SBBHs were formed in the early universe; and about two thirds of those mergers occur in the central galaxies of dark matter halos. We also present results on the host galaxy properties of SBBH mergers at higher redshift.Comment: 12 pages, 9 figures, MNRAS accepte

OPINS: An Orthogonally Projected Implicit Null-space Method for Singular and Nonsingular Saddle-point Systems

Saddle-point systems appear in many scientific and engineering applications. The systems can be sparse, symmetric or nonsymmetric, and possibly singular. In many of these applications, the number of constraints is relatively small compared to the number of unknowns. The traditional null-space method is inefficient for these systems, because it is expensive to find the null space explicitly. Some alternatives, notably constraint-preconditioned or projected Krylov methods, are relatively efficient, but they can suffer from numerical instability and even nonconvergence. In addition, most existing methods require the system to be nonsingular or be reducible to a nonsingular system. In this paper, we propose a new method, called OPINS, for singular and nonsingular saddle-point systems. OPINS is equivalent to the null-space method with an orthogonal projector, without forming the orthogonal basis of the null space explicitly. OPINS can not only solve for the unique solution for nonsingular saddle-point problems, but also find the minimum-norm solution in terms of the solution variables for singular systems. The method is efficient and easy to implement using existing Krylov solvers for singular systems. At the same time, it is more stable than the other alternatives, such as projected Krylov methods. We present some preconditioners to accelerate the convergence of OPINS for nonsingular systems, and compare OPINS against some present state-of-the-art methods for various types of singular and nonsingular systems

Some results on generalized strong external difference families

A generalized strong external difference family (briefly $(v, m; k_1,\dots,k_m; \lambda_1,\dots,\lambda_m)$-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we construct some new GSEDFs for $m=2$ and use them to obtain some results on graph decomposition. We also give some nonexistence results for GSEDFs. Especially, we prove that a $(v, 3;k_1,k_2,k_3; \lambda_1,\lambda_2,\lambda_3)$-GSEDF does not exist when $k_1+k_2+k_3< v$.Comment: generalized strong external difference family; difference set; character theory; graph decomposition; nonexistenc

Summary of the 2018 CKM working group on semileptonic and leptonic bb-hadron decays

A summary of WG II of the CKM 2018 conference on semileptonic and leptonic $b$-hadron decays is presented. This includes discussions on the CKM matrix element magitudes $|V_{ub}|$ and $|V_{cb}|$, lepton universality tests such as $R(D^{*})$ and leptonic decays. As is usual for semileptonic and leptonic decays, much discussion is devoted towards the interplay between theoretical QCD calculations and the experimental measurements.Comment: Proceedings of the 10th International Workshop on the CKM Unitarity Triangle (CKM 2018), Heidelberg, Germany, September 17-21, 201

CariGANs: Unpaired Photo-to-Caricature Translation

Facial caricature is an art form of drawing faces in an exaggerated way to convey humor or sarcasm. In this paper, we propose the first Generative Adversarial Network (GAN) for unpaired photo-to-caricature translation, which we call "CariGANs". It explicitly models geometric exaggeration and appearance stylization using two components: CariGeoGAN, which only models the geometry-to-geometry transformation from face photos to caricatures, and CariStyGAN, which transfers the style appearance from caricatures to face photos without any geometry deformation. In this way, a difficult cross-domain translation problem is decoupled into two easier tasks. The perceptual study shows that caricatures generated by our CariGANs are closer to the hand-drawn ones, and at the same time better persevere the identity, compared to state-of-the-art methods. Moreover, our CariGANs allow users to control the shape exaggeration degree and change the color/texture style by tuning the parameters or giving an example caricature.Comment: To appear at SIGGRAPH Asia 201

A Comparison of Preconditioned Krylov Subspace Methods for Large-Scale Nonsymmetric Linear Systems

Preconditioned Krylov subspace (KSP) methods are widely used for solving large-scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely Gauss-Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in 2D and 3D with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts. BoomerAMG with proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill-conditioned or saddle-point problems while multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.Comment: Numerical Linear Algebra with Applications, 201