Hessian Tensor Research Articles

Curvature analysis of concircular trajectories in doubly warped product manifolds

<p>The aim of this research paper was to explore the various characteristics of the doubly warped product manifold, focusing particularly on aspects such as the Hessian, Riemannian curvature, Ricci curvature, and concircular curvature tensor components. By examining the necessary conditions that would classify the manifold as Riemann-flat, Ricci-flat, and concircularly-flat, the study aimed to expand our understanding of these concepts. To achieve this, the research incorporated the application of these findings to a generalized Robertson-Walker doubly warped product manifold scenario. This approach allowed us to identify and analyze the specific circumstances under which the manifold displayed concircular flatness.</p>

An augmented invariant-based model of the pressure Hessian tensor using a combination of physics-assisted neural networks

Modeling the velocity gradient dynamics in incompressible turbulence requires modeling two unclosed quantities: the pressure Hessian tensor and the viscous Laplacian tensor. In this work, we model the pressure Hessian tensor using a combination of two different physics-embedded deep neural networks. The first neural network is trained specifically to predict the alignment tendencies of the eigen-vectors of the pressure Hessian tensor, whereas the second neural network is trained only to predict the magnitude of the tensor. This separation of tasks allows us to define mathematically optimal and physics-informed customized loss functions separately for the two aspects (alignment and magnitude) of the tensor. Both neural networks take invariants of the velocity gradient tensor as inputs. Even though the training of the two networks is performed using direct numerical simulation database of an incompressible stationary isotropic turbulence at a particular Reynolds number, we extensively evaluate the model at different Reynolds numbers and in different kinds of flow fields. In incompressible flows, the proposed model shows significant improvements over the existing phenomenological model (the recent fluid deformation closure model or the RFD model) of the pressure Hessian tensor. While the improvements in the alignment tendencies are convincingly evident in the shapes of the probability density functions of the cosines of various angles between eigenvectors, the improvements in the prediction of the magnitude of the pressure Hessian tensor using the new model are quantifiable in the range of 28%–89% (depending on the type of the flow field) compared to the RFD model.

A physics-informed deep learning closure for Lagrangian velocity gradient evolution

The pressure Hessian tensor is entangled with the inherent nonlinearity and nonlocality of turbulence; thus, it is of crucial importance in modeling the Lagrangian evolution of the velocity gradient tensor (VGT). In the present study, we introduce the functional modeling strategy into the classic structural modeling strategy to model the pressure Hessian tensor based on deep neural networks (DNNs). The pressure Hessian tensor and its contributions to the VGT evolution are set as, respectively, the structural and functional learning targets. An a priori test shows that the present DNN-based model accurately establishes the mapping from the VGT to the pressure Hessian tensor and adequately models the physical effect of the pressure Hessian tensor on VGT invariants. An a posteriori test verifies that the present model reproduces well the principal features of turbulence-like skewness and vorticity strain-rate alignments obtained via direct numerical simulations. Importantly, the flow topology is accurately predicted, particularly for the strain-production-dominant regions in the invariant space. Moreover, an extrapolation test shows the generalization ability of the present model to higher Reynolds number flows that have not been trained.

Role of pressure in the dynamics of intense velocity gradients in turbulent flows

We investigate the role of pressure, via its Hessian tensor ${\boldsymbol {H}}$ , on amplification of vorticity and strain-rate and contrast it with other inviscid nonlinear mechanisms. Results are obtained from direct numerical simulations of isotropic turbulence with Taylor-scale Reynolds number in the range 140–1300. Decomposing ${\boldsymbol {H}}$ into local isotropic ( ${\boldsymbol {H}}^{I}$ ) and non-local deviatoric ( ${\boldsymbol {H}}^{D}$ ) components reveals that ${\boldsymbol {H}}^{I}$ depletes vortex stretching, whereas ${\boldsymbol {H}}^{D}$ enables it, with the former slightly stronger. The resulting inhibition is significantly weaker than the nonlinear mechanism which always enables vortex stretching. However, in regions of intense vorticity, identified using conditional statistics, contribution from ${\boldsymbol {H}}$ prevails over nonlinearity, leading to overall depletion of vortex stretching. We also observe near-perfect alignment between vorticity and the eigenvector of ${\boldsymbol {H}}$ corresponding to the smallest eigenvalue, which conforms with well-known vortex-tubes. We discuss the connection between this depletion, essentially due to (local) ${\boldsymbol {H}}^{I}$ , and recently identified self-attenuation mechanism (Buaria et al., Nat. Commun., vol. 11, 2020, p. 5852), whereby intense vorticity is locally attenuated through inviscid effects. In contrast, the influence of ${\boldsymbol {H}}$ on strain-amplification is weak. It opposes strain self-amplification, together with vortex stretching, but its effect is much weaker than vortex stretching. Correspondingly, the eigenvectors of strain and ${\boldsymbol {H}}$ do not exhibit any strong alignments. For all results, the dependence on Reynolds number is very weak. In addition to the fundamental insights, our work provides useful data and validation benchmarks for future modelling endeavours, for instance in Lagrangian modelling of velocity gradient dynamics, where conditional ${\boldsymbol {H}}$ is explicitly modelled.

Ricci Soliton of CR-Warped Product Manifolds and Their Classifications

In this article, we derived an equality for CR-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a CR-warped product submanifolds in Ka¨hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of CR-warped product submanifolds in the Ka¨hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor.

Bounds on Ricci curvature for doubly warped products pointwise bi-slant submanifolds and applications to physics

In this article, we obtain bounds for Ricci curvature for doubly warped products pointwise bi-slant submanifolds in generalized complex space forms and discuss the equality case of the inequality. We also derive the non-existence of such immersions. Finally, we construct some applications of the result in terms of the Harmonic function, Hessian tensor, and Dirchilet energy function.

A model problem for the evolution of strain structure at the crossing of a flame front

The local thermal effect of a flame front is simulated by a model for a mass density front by specifying a likely expansion rate. This model problem includes two independent parameters, namely the heat release parameter and a parameter akin to a Karlovitz number. The analysis is focused on the influence of the Karlovitz number on the evolution of strain properties at the crossing of the front. The latter are derived from an equation system for the velocity gradient tensor and the pressure Hessian tensor undergoing the forcing of the expansion rate. Strain eigenvalues, orientation of strain principal axes, and stretching in the direction of forcing are especially scrutinized. Furthermore, the model shows that, when approaching a flame front, the special alignments of strain are mostly caused by anisotropy of pressure Hessian resulting from forcing by expansion.

Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold Ω p + q in the hyperbolic space ℍ m − 1 satisfy various extrinsic restrictions, then Ω p + q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π 1 Ω p + q is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.

Null Homology Groups and Stable Currents in Warped Product Submanifolds of Euclidean Spaces

In this paper, we prove that, for compact warped product submanifolds Mn in an Euclidean space En+k, there are no stable p-currents, homology groups are vanishing, and M3 is homotopic to the Euclidean sphere S3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature, and the Hessian tensor. The results in this paper can be considered an extension of Xin’s work in the framework of a compact warped product submanifold, when the base manifold is minimal in ambient manifolds.

Classification of primary constraints for new general relativity in the premetric approach

We introduce a novel procedure for studying the Hamiltonian formalism of new general relativity (NGR) based on the mathematical properties encoded in the constitutive tensor defined by the premetric approach. We derive the canonical momenta conjugate to the tetrad field and study the eigenvalues of the Hessian tensor, which is mapped to a Hessian matrix with the help of indexation formulas. The properties of the Hessian matrix heavily rely on the possible values of the free coefficients [Formula: see text] appearing in the NGR Lagrangian. We find four null eigenvalues associated with trivial primary constraints in the temporal part of the momenta. The remaining eigenvalues are grouped in four sets, which have multiplicity 3, 1, 5 and 3, and can be set to zero depending on different choices of the coefficients [Formula: see text]. There are nine possible different cases when one, two, or three sets of eigenvalues are imposed to vanish simultaneously. All cases lead to a different number of primary constraints, which are consistent with previous work on the Hamiltonian analysis of NGR by Blixt et al. (2018).

Principal strain rate evolution within turbulent premixed flames for different combustion regimes

The statistical behaviors of the principal strain rates and its evolution in turbulent premixed flames have been analyzed using a three-dimensional direct numerical simulations dataset of statistically planar turbulent premixed flames with different turbulence intensities spanning from the corrugated flamelet regime to the thin reaction zone regime. It has been found that the scalar gradient predominantly aligns collinearly with the most extensive principal strain rate eigendirection within the flame for large values of Damköhler numbers and small values of turbulence intensities and Karlovitz numbers. However, this tendency weakens with the increasing turbulence intensity, which, for a given integral length scale, amounts to a decrease (an increase) in the Damköhler (Karlovitz) number. Moreover, it has been observed that the terms due to molecular diffusion, pressure Hessian, and the correlation between pressure and density gradients play key roles in the evolution of principal strain rates for flames with large values of Damköhler numbers and small values of Karlovitz numbers. However, the relative importance of the terms arising from the correlation between pressure and density gradients and the pressure Hessian relative to the strain rate and vorticity contributions of the principal strain rate transport diminishes with the increasing Karlovitz number and decreasing Damköhler number. The statistical behaviors of the mean values of the terms of the transport equation of the principal strain rate have been explained based on the relative alignments of principal strain rate eigenvectors with vorticity, pressure gradient, and the eigenvectors of the pressure Hessian tensor. The findings of the current analysis suggest that the pressure gradient and pressure Hessian tensor play key roles in the evolution of principal strain rates within premixed turbulent flames, and their influence needs to be accounted for high fidelity modeling of the tangential strain rate and scalar–turbulence interaction terms of the flame surface density and scalar dissipation rate transport equations, respectively. This provides possible explanations for the modification in the alignment of the reactive scalar gradient with local principal strain rates in premixed flames in comparison to that in non-reacting turbulent flows.

A single-step third-order temporal discretization with Jacobian-free and Hessian-free formulations for finite difference methods

In this paper, we present an algorithmic extension of the method called the Picard Integration Formulation (PIF) that belongs to temporal updates based on the Lax-Wendroff procedure. The new extension is called the system-free (SF) approach, which furnishes ease of calculating the Jacobian and the Hessian terms necessary for third-order temporal accuracy in the original PIF method. In contrast to the analytical calculations of the Jacobian and the Hessian tensor terms in the original PIF method, our new SF approach utilizes finite difference approximations that replace the analytical calculations of the Jacobian and Hessian terms with Jacobian-free and Hessian-free approximations in a way commonly adopted in the context of iterative methods. The resulting SF approach enables our new PIF method to be a computationally efficient single-step, third-order accurate temporal scheme, whose computational performance is twice faster than the three-stage SSP-RK3 method with the same accuracy.

Analyzing the effect of dilatation on the velocity gradient tensor using a model problem

The effect of variable mass density on the velocity gradient tensor is addressed by means of a model problem. An equation system for both the velocity gradient and the pressure Hessian tensor is solved assuming a realistic expansion rate. The model results show the evolution of the velocity gradient tensor as the density front is approached and are relevant to the physics of flame fronts.

Adaptive ensemble PTV

Ensemble particle tracking velocimetry (EPTV) is a method to extract high-resolution statistical information on flow fields from particle image velocimetry (PIV) images. The process is based on tracking particles and extracting the velocity probability distribution functions of the image ensemble in averaging-regions deemed to contain a sufficient number of particle pairs/tracks. The size of the averaging regions depends on the particle density and the number of snapshots. An automatic adaptive variation of the ensemble PTV is presented to further push the spatial resolution of the method. The proposed adaptive-EPTV is based on stretching and orienting the averaging regions along the direction of maximum curvature of the velocity fields. The process requires a predictor calculation with isotropic-window EPTV to compute the second derivatives of the mean velocity components. In a second step, the principal directions of the Hessian tensor are calculated to tune the optimal orientation and stretch of the averaging regions. The stretching and orientation are achieved using a Gaussian windowing with different standard deviation along the local principal direction of the Hessian tensor. The algorithm is first validated using three different synthetic datasets: a sinusoidal displacement field, a channel flow and the flow around a NACA 0012 airfoil. An experimental test case of an impinging jet equipped with a fractal grid at the nozzle outlet is also carried out.

Ricci curvature on warped product submanifolds of complex space forms and its applications

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality.

A closure for Lagrangian velocity gradient evolution in turbulence using recent-deformation mapping of initially Gaussian fields

The statistics of the velocity gradient tensor in turbulent flows is of both theoretical and practical importance. The Lagrangian view provides a privileged perspective for studying the dynamics of turbulence in general, and of the velocity gradient tensor in particular. Stochastic models for the Lagrangian evolution of velocity gradients in isotropic turbulence, with closure models for the pressure Hessian and viscous Laplacian, have been shown to reproduce important features such as non-Gaussian probability distributions, skewness and vorticity strain-rate alignments. The recent fluid deformation (RFD) closure introduced the idea of mapping an isotropic Lagrangian pressure Hessian as the upstream initial condition using the fluid deformation tensor. Recent work on a Gaussian fields closure, however, has shown that even Gaussian isotropic velocity fields contain significant anisotropy for the conditional pressure Hessian tensor due to the inherent velocity–pressure couplings, and that assuming an isotropic pressure Hessian as the upstream condition may not be realistic. In this paper, Gaussian isotropic field statistics is used to generate more physical upstream conditions for the recent fluid deformation mapping. In this new framework, known isotropy relations can be satisfied by tuning the free model parameters and the original Gaussian field coefficients can be directly used without direct numerical simulation (DNS)-based re-adjustment. A detailed comparison of results from the new model, referred to as the recent deformation of Gaussian fields (RDGF) closure, with existing models and DNS shows the improvements gained, especially in various single-time statistics of the velocity gradient tensor at moderate Reynolds numbers. Application to arbitrarily high Reynolds numbers remains an open challenge for this type of model, however.

Asymptotic normality and optimalities in estimation of large Gaussian graphical models

The Gaussian graphical model, a popular paradigm for studying relationship among variables in a wide range of applications, has attracted great attention in recent years. This paper considers a fundamental question: When is it possible to estimate low-dimensional parameters at parametric square-root rate in a large Gaussian graphical model? A novel regression approach is proposed to obtain asymptotically efficient estimation of each entry of a precision matrix under a sparseness condition relative to the sample size. When the precision matrix is not sufficiently sparse, or equivalently the sample size is not sufficiently large, a lower bound is established to show that it is no longer possible to achieve the parametric rate in the estimation of each entry. This lower bound result, which provides an answer to the delicate sample size question, is established with a novel construction of a subset of sparse precision matrices in an application of Le Cam's lemma. Moreover, the proposed estimator is proven to have optimal convergence rate when the parametric rate cannot be achieved, under a minimal sample requirement. The proposed estimator is applied to test the presence of an edge in the Gaussian graphical model or to recover the support of the entire model, to obtain adaptive rate-optimal estimation of the entire precision matrix as measured by the matrix $\ell_q$ operator norm and to make inference in latent variables in the graphical model. All of this is achieved under a sparsity condition on the precision matrix and a side condition on the range of its spectrum. This significantly relaxes the commonly imposed uniform signal strength condition on the precision matrix, irrepresentability condition on the Hessian tensor operator of the covariance matrix or the $\ell_1$ constraint on the precision matrix. Numerical results confirm our theoretical findings. The ROC curve of the proposed algorithm, Asymptotic Normal Thresholding (ANT), for support recovery significantly outperforms that of the popular GLasso algorithm.

A direct numerical simulation-based investigation and modeling of pressure Hessian effects on compressible velocity gradient dynamics

The pressure Hessian tensor plays a key role in shaping the behavior of the velocity gradient tensor, and in turn, that of many incumbent non-linear processes in a turbulent flow field. In compressible flows, the role of pressure Hessian is even more important because it represents the level of fluid-thermodynamic coupling existing in the flow field. In this work, we first perform a direct numerical simulation-based study to clearly identify, isolate, and understand various important inviscid mechanisms that govern the evolution of the pressure Hessian tensor in compressible turbulence. The ensuing understanding is then employed to introduce major improvements to the existing Lagrangian model of the pressure Hessian tensor (the enhanced Homogenized Euler equation or EHEE) in terms of (i) non-symmetric, non-isentropic effects and (ii) improved representation of the anisotropic portion of the pressure Hessian tensor. Finally, we evaluate the new model extensively by comparing the new model results against known turbulence behavior over a range of Reynolds and Mach numbers. Indeed, the new model shows much improved performance as compared to the EHEE model.

Two new regularity criteria for the Navier–Stokes equations via two entries of the velocity Hessian tensor

We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\R^3$, and provide a sufficient condition to ensure the smoothness of the solution. It involves only two entries of the velocity Hessian.

Branching and bounding improvements for global optimization algorithms with Lipschitz continuity properties

We present improvements to branch and bound techniques for globally optimizing functions with Lipschitz continuity properties by developing novel bounding procedures and parallelisation strategies. The bounding procedures involve nonconvex quadratic or cubic lower bounds on the objective and use estimates of the spectrum of the Hessian or derivative tensor, respectively. As the nonconvex lower bounds are only tractable if solved over Euclidean balls, we implement them in the context of a recent branch and bound algorithm (Fowkes et al. in J Glob Optim 56:1791---1815, 2013) that uses overlapping balls. Compared to the rectangular tessellations of traditional branch and bound, overlapping ball coverings result in an increased number of subproblems that need to be solved and hence makes the need for their parallelization even more stringent and challenging. We develop parallel variants based on both data- and task-parallel paradigms, which we test on an HPC cluster on standard test problems with promising results.