arXiv subjects
Publications and source records attributed to Rakesh Halder.
Let $Y\to X$ be a proper map between proper hyperbolic metric spaces. A Cannon--Thurston map is a continuous extension $\partial Y \to \partial X$. We prove that in most known settings in which a Cannon--Thurston map exists it is uniformly finite-to-one. This answers a question due to Swarup from Bestvina's problem list and generalizes previous results of Cannon--Thurston, Kapovich--Lustig, Dowdall--Kapovich--Taylor and Ghosh.
Given a tree of hyperbolic metric spaces $\pi:X\to T$ a la Bestvina--Feighn (\cite{BF}), and a hyperbolic subspace $Y$ of $X$ with an induced tree of hyperbolic spaces structure over a subtree $S\subset T$, we address the question as to when the Cannon--Thurston (CT) map exists for the inclusion $Y\to X$. In this paper, we find additional sufficient conditions under which the CT map $\partial Y \to \partial X$ exists. However, we show with examples that this may fail to hold in general. These results about trees of spaces are then applied to graphs of hyperbolic groups to prove various existence results for CT maps. A very special instance of these results is the following: \emph{Suppose $G_1$ and $G_2$ are hyperbolic groups with a common quasiconvex subgroup $H$, and the free product with amalgamation $G = G_1 *_H G_2$ is hyperbolic. Suppose $K_i < G_i$, $i = 1,2$ are hyperbolic subgroups containing $H$ and $K=K_1*_H K_2$. Then $K$ (is hyperbolic and,) the inclusion $K\to G$ admits a CT map if the inclusions $K_i\to G_i$, $i=1,2$ admit CT maps.}
Reduced-order models (ROMs) are very popular for surrogate modeling of full-order computational fluid dynamics (CFD) simulations, allowing for real-time approximation of complex flow phenomena. However, their application to CFD models including large eddy simulation (LES) and direct numerical simulaton (DNS) is limited due to the highly chaotic and multi-scale nature of resolved turbulent flow. Due to the large amounts of noise present in small-scale turbulent structures, error accumulation becomes a major issue, making long-term prediction of unsteady flow infeasible. While linear subspace methods like dynamic mode decomposition (DMD) can be used to pre-process turbulent flow data to remove small-scale structures, this often requires a very large number of modes and a non-trivial mode selection process. In this work, a ROM framework using Koopman $\beta$-variational autoencoders ($\beta$-VAEs) is introduced for reduced-order modeling of large-scale turbulence. The Koopman operator captures the variation of a non-linear dynamical system through a linear representation of state observables. By constraining the latent space of a $\beta$-VAE to grow linearly using a Koopman-inspired loss function, small-scale turbulent structures are filtered out in reconstructions of input data and latent variables are denoised in an unsupervised manner so that they can be sufficiently modeled over time. Combined with a long short-term memory (LSTM) ensemble for time series prediction of latent variables, the model is tested on LES flow past a Windsor body at multiple yaw angles, showing that the Koopman $\beta$-VAE can effectively denoise latent variables and remove small-scale structures from reconstructions while acting globally over multiple cases.
Metric (graph) bundles generalize the notion of fiber bundles to the context of geometric group theory and were introduced by Mj and Sardar. Suppose $X$ is a metric (graph) bundle over $B$ such that the fibers are (uniformly) hyperbolic, and the total space $X$ is also hyperbolic. In this generality, Mj--Sardar proved that the inclusion of a fiber into $X$ admits a continuous extension to the (Gromov) boundary. In this article, we prove that such a continuous extension map between boundaries is surjective in the following two key settings. $(1)$ The fibers are uniformly quasiisometric to a nonelementary hyperbolic group. $(2)$ The fibers are one-ended hyperbolic metric spaces. Our result generalizes a theorem of Bowditch in which the fibers were assumed to be the hyperbolic plane, and it answers a question posed by Lazarovich, Margolis and Mj.
In this short note, we prove the existence of weakly malnormal, virtually free, quasiconvex subgroups in any nonelementary hyperbolic group. This extends a result of Ilya Kapovich, where he proved the existence of malnormal quasiconvex subgroups in torsion-free hyperbolic groups.
In this paper, we describe a procedure to construct pairs of hyperbolic groups $H<G$ with the following properties. 1) Every geodesic ray $\gamma$ in $H$ converges to a point $\xi_{\gamma}\in \partial G$. 2) The inclusion of $H$ into $G$ does not extend continuously to $\partial H \to \partial G$. In other words, a Cannon--Thurston map does not exist for this pair of hyperbolic groups. Jeon, Kapovich, Leininger and Ohshika gave a property of conical limit points in the presence of a Cannon--Thurston map. We convert this into a criterion for the existence of Cannon--Thurston maps and use it to prove the non-existence result in (2). We obtain, in particular, a geometric proof of Baker--Riley's counterexample.
The use of deep learning has become increasingly popular in reduced-order models (ROMs) to obtain low-dimensional representations of full-order models. Convolutional autoencoders (CAEs) are often used to this end as they are adept at handling data that are spatially distributed, including solutions to partial differential equations. When applied to unsteady physics problems, ROMs also require a model for time-series prediction of the low-dimensional latent variables. Long short-term memory (LSTM) networks, a type of recurrent neural network useful for modeling sequential data, are frequently employed in data-driven ROMs for autoregressive time-series prediction. When making predictions at unseen design points over long time horizons, error propagation is a frequently encountered issue, where errors made early on can compound over time and lead to large inaccuracies. In this work, we propose using bagging, a commonly used ensemble learning technique, to develop a fully data-driven ROM framework referred to as the CAE-eLSTM ROM that uses CAEs for spatial reconstruction of the full-order model and LSTM ensembles for time-series prediction. When applied to two unsteady fluid dynamics problems, our results show that the presented framework effectively reduces error propagation and leads to more accurate time-series prediction of latent variables at unseen points.
Motivated by the work of Bestvina-Feighn ([BF92]) and Mj-Sardar ([MS12]), we define trees of metric bundles subsuming both the trees of metric spaces and the metric bundles. Then we prove a combination theorem for these spaces. More precisely, we prove that the total space of a tree of metric bundles is hyperbolic if the following hold (see Theorem $1.5$). $(1)$ The fibers are uniformly hyperbolic metric spaces and the base is also hyperbolic metric space, $(2)$ barycenter maps for the fibers are uniformly coarsely surjective, $(3)$ the edge spaces are uniformly qi embedded in the corresponding fibers and $(4)$ the Bestvina-Feighn hallway flaring condition is satisfied. As an application, we provide a combination theorem for certain complexes of groups over finite simplicial complex (see Theorem $1.3$).
The use of reduced-order models (ROMs) in physics-based modeling and simulation almost always involves the use of linear reduced basis (RB) methods such as the proper orthogonal decomposition (POD). For some nonlinear problems, linear RB methods perform poorly, failing to provide an efficient subspace for the solution space. The use of nonlinear manifolds for ROMs has gained traction in recent years, showing increased performance for certain nonlinear problems over linear methods. Deep learning has been popular to this end through the use of autoencoders for providing a nonlinear trial manifold for the solution space. In this work, we present a non-intrusive ROM framework for steady-state parameterized partial differential equations (PDEs) that uses convolutional autoencoders (CAEs) to provide a nonlinear solution manifold and is augmented by Gaussian process regression (GPR) to approximate the expansion coefficients of the reduced model. When applied to a numerical example involving the steady incompressible Navier-Stokes equations solving a lid-driven cavity problem, it is shown that the proposed ROM offers greater performance in prediction of full-order states when compared to a popular method employing POD and GPR over a number of ROM dimensions.