Affine Manifolds Research Articles

Unified LSPG Model Reduction Framework and Assessment for Hypersonic Computational Fluid Dynamics

A unified least-squares Petrov–Galerkin (LSPG) framework for projection-based model order reduction featuring three different approximation manifolds [affine manifold, quadratic manifold, and nonlinear manifold built using a deep artificial neural network (ANN)] is presented. Its performance was assessed for a variable-speed version of the double-cone hypersonic benchmark problem. First, a high-dimensional viscous computational fluid dynamics model (HDM) was constructed, verified, and validated. The dimensionality of the HDM was then reduced using LSPG, each of the aforementioned approximation manifolds, and a global right reduced-order basis trained in the range 8≤M∞≤13. Each resulting global projection-based reduced-order model (PROM) was hyper-reduced and transformed into a hyper-reduced PROM (HPROM). The accuracy of each constructed HPROM was assessed for various quantities of interest and contrasted with that of snapshot interpolation. For this purpose, three different error measures were considered and discussed in the context of shock-dominated problems. Wall-clock and CPU time speedup factors are reported. Overall, it was shown that using a relatively small set of training data, all constructed LSPG HPROMs were nonlinearly stable, real-time capable, and highly predictive. The LSPG HPROM constructed using a nonlinear approximation manifold and an ANN was the most computationally efficient.

Automorphisms of projective manifolds

Let $(M,P\nabla_M)$ be a compact projective manifold and $Aut(M,P\nabla_M)$ its group of automorphisms. The purpose of this paper is to study the topological properties of $(M,P\nabla_M)$ if $Aut(M,P\nabla_M))$ is not discrete by applying the results of [13] and the Benzekri's functor which associates to a projective manifold a radiant affine manifold. This enables us to show that the orbits of the connected component of $Aut(M,P\nabla_M)$ are immersed projective submanifolds. We also classify $3$-dimensional compact projective manifolds such that $dim(Aut(M,P\nabla_M))\geq 2$.

Neural-network-augmented projection-based model order reduction for mitigating the Kolmogorov barrier to reducibility

Inspired by our previous work on a quadratic approximation manifold [1], we propose in this paper a computationally tractable approach for combining a projection-based reduced-order model (PROM) and an artificial neural network (ANN) to mitigate the Kolmogorov barrier to reducibility of parametric and/or highly nonlinear, high-dimensional, physics-based models. The main objective of our PROM-ANN concept is to reduce the dimensionality of the online approximation of the solution beyond what is achievable using affine and quadratic approximation manifolds, while maintaining accuracy. In contrast to previous approaches that exploited one form or another of an ANN, the training of the ANN part of our PROM-ANN does not involve data whose dimension scales with that of the high-dimensional model; and the resulting PROM-ANN can be efficiently hyperreduced using any well-established hyperreduction method. Hence, unlike many other ANN-based model order reduction approaches, the PROM-ANN concept we propose in this paper should be practical for large-scale and industry-relevant computational problems. We demonstrate the computational tractability of its offline stage and the superior wall clock time performance of its online stage for a large-scale, parametric, two-dimensional, model problem that is representative of shock-dominated unsteady flow problems.

Normal Cones Intersection Rule and Optimality Analysis for Low-Rank Matrix Optimization with Affine Manifolds

Normal Cones Intersection Rule and Optimality Analysis for Low-Rank Matrix Optimization with Affine Manifolds

Affine deformations of quasi‐divisible convex cones

AbstractWe study subgroups of obtained by adding a translation part to the holonomy of a finite‐volume convex projective surface. Under a natural condition on the translations added to the peripheral parabolic elements, we show that the affine action of the group on has convex domains of discontinuity which are regular, generalizing a result of Mess for globally hyperbolic flat spacetimes. We then classify all such domains arising from a fixed group and show that the quotient of each of them is an affine manifold foliated by convex surfaces with constant affine Gaussian curvature (CAGC). The proof is based on the analysis of CAGC surfaces developed in a previous work, along with a correspondence between the geometry of an affine space endowed with a convex cone and the geometry of a convex tube domain. We also show that the moduli space of such groups is a vector bundle over the moduli space of finite‐volume convex projective structures, with rank equal to the dimension of the Teichmüller space.

English

In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes the form of a differential graded Lie algebra of graphs, denoted $\mathsf{fGC}_2$, together with an injective morphism towards the Chevalley-Eilenberg complex associated with the Schouten algebra. The latter morphism is given by explicit local formulas making implicit use of the supergeometric interpretation of the Schouten algebra as the algebra of functions on a graded symplectic manifold of degree $1$. The ambition of the present work is to generalise Kontsevich's construction to graded symplectic manifolds of arbitrary degree $n\geq1$. The corresponding graph model is given by the full Kontsevich graph complex $\mathsf{fGC}_d$ where $d=n+1$ stands for the dimension of the associated AKSZ type $\sigma$-model. This generalisation is instrumental to classify universal structures on graded symplectic manifolds. In particular, the zeroth cohomology of the full graph complex $\mathsf{fGC}_{d}$ is shown to act via $\mathsf{Lie}_\infty$-automorphisms on the algebra of functions on graded symplectic manifolds of degree $n$. This generalises the known action of the Grothendieck-Teichm\"{u}ller algebra $\mathfrak{grt}_1\simeq H^0(\mathsf{fGC}_2)$ on the space of polyvector fields. This extended action can in turn be used to generate new universal deformations of Hamiltonian functions, generalising Kontsevich flows on the space of Poisson manifolds to differential graded manifolds of higher degrees. As an application of the general formalism, universal deformations of Courant algebroids via trivalent graphs are presented.

Lagrangian and Hamiltonian dynamics for probabilities on the statistical bundle

We provide an Information-Geometric formulation of accelerated natural gradient on the Riemannian manifold of probability distributions, which is an affine manifold endowed with a dually-flat connection. In a non-parametric formalism, we consider the full set of positive probability functions on a finite sample space, and we provide a specific expression for the tangent and cotangent spaces over the statistical manifold, in terms of a Hilbert bundle structure that we call the Statistical Bundle. In this setting, we compute velocities and accelerations of a one-dimensional statistical model using the canonical dual pair of parallel transports and define a coherent formalism for Lagrangian and Hamiltonian mechanics on the bundle. We show how our formalism provides a consistent framework for accelerated natural gradient dynamics on the probability simplex, paving the way for direct applications in optimization.

Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerate Calabi-Yau surfaces

We introduce the notion of tropical Lagrangian multi-sections over a 2-dimensional integral affine manifold B with singularities, and use them to study the reconstruction problem for higher rank locally free sheaves over Calabi-Yau surfaces. To certain tropical Lagrangian multi-sections L over B, which are explicitly constructed by prescribing local models around the ramification points, we construct locally free sheaves E0(L,ks) over the singular projective scheme X0(B,P,s) associated to B equipped with a polyhedral decomposition P and a gluing data s. We then find combinatorial conditions on such an L under which the sheaf E0(L,ks) is simple. This produces explicit examples of smoothable pairs (X0(B,P,s),E0(L,ks)) in dimension 2.

On the Hermitian structures of the sequence of tangent bundles of an affine manifold endowed with a Riemannian metric

Abstract Let (M, ∇, 〈, 〉) be a manifold endowed with a flat torsionless connection r and a Riemannian metric 〈, 〉 and (TkM) k ≥1 the sequence of tangent bundles given by TkM = T(Tk −1 M) and T 1 M = TM. We show that, for any k ≥ 1, TkM carries a Hermitian structure (Jk , gk ) and a flat torsionless connection ∇k and when M is a Lie group and (∇, 〈, 〉) are left invariant there is a Lie group structure on each TkM such that (Jk , gk , ∇k ) are left invariant. It is well-known that (TM, J 1, g 1) is Kähler if and only if 〈, 〉 is Hessian, i.e, in each system of affine coordinates (x 1, . . ., xn ), 〈 ∂ x i , ∂ x j 〉 = ∂ 2 φ ∂ x i ∂ x j \left\langle {{\partial _x}_{_i},{\partial _{{x_j}}}} \right\rangle = {{{\partial ^2}\phi } \over {{\partial _x}_{_i}{\partial _x}_j}} . Having in mind many generalizations of the Kähler condition introduced recently, we give the conditions on (∇, 〈, 〉) so that (TM, J 1, g 1) is balanced, locally conformally balanced, locally conformally Kähler, pluriclosed, Gauduchon, Vaisman or Calabi-Yau with torsion. Moreover, we can control at the level of (∇, 〈, 〉) the conditions insuring that some (TkM, Jk , gk ) or all of them satisfy a generalized Kähler condition. For instance, we show that there are some classes of (M, ∇, 〈, 〉) such that, for any k ≥ 1, (TkM, Jk , gk ) is balanced non-Kähler and Calabi-Yau with torsion. By carefully studying the geometry of (M, ∇, 〈, 〉), we develop a powerful machinery to build a large classes of generalized Kähler manifolds.

A homology theory for tropical cycles on integral affine manifolds and a perfect pairing

We introduce a cap product pairing for homology and cohomology of tropical cycles on integral affine manifolds with singularities. We show the pairing is perfect over $\mathbb{Q}$ in degree one when the manifold has at worst symple singularities. By joint work with Siebert, the pairing computes period integrals and its perfectness implies the versality of canonical Calabi-Yau degenerations. We also give an intersection theoretic application for Strominger-Yau-Zaslow fibrations. The treatment of the cap product and Poincar\'e-Lefschetz by simplicial methods for constructible sheaves might be of independent interest.

Integral Affine 3-Manifolds

Affine manifolds are called integral if there is an atlas such that all transition maps are affine transformations with integer matrices of linear parts. In this paper, we describe all complete integral affine structures on compact three-dimensional manifolds up to a finite-sheeted covering. Also a complete list of integral affine structures on the three-dimensional torus and compact three-dimensional nilmanifolds was obtained.

On the Markus conjecture in convex case

In this paper, we show that any convex affine domain with a nonempty limit set on the boundary under the action of the identity component of the automorphism group cannot cover a compact affine manifold with a parallel volume, which is a positive answer to the Markus conjecture for convex case. Consequently, we show that the Markus conjecture is true for convex affine manifolds of dimension ≤ 5.

Linear foliations on affine manifolds

In this paper, we study affine manifolds endowed with linear foliations. These are foliations defined by vector subspaces invariant by the linear holonomy. We show that an n-dimensional compact, complete, and oriented affine manifold endowed with a codimension 1 linear foliation F is homeomorphic to the n-dimensional torus if the leaves of F are simply connected. Let (M,∇M) be a 3-dimensional compact affine manifold endowed with a codimension 1 linear foliation. We prove that (M,∇M) has a finite cover which is homeomorphic to the total space of a bundle over the circle if its developing map is injective, and has a convex image.

The Existence of Affine Structures on the Borel Subalgebra of Dimension 6

The notion of affine structures arises in many fields of mathematics, including convex homogeneous cones, vertex algebras, and affine manifolds. On the other hand, it is well known that Frobenius Lie algebras correspond to the research of homogeneous domains. Moreover, there are 16 isomorphism classes of 6-dimensional Frobenius Lie algebras over an algebraically closed field. The research studied the affine structures for the 6-dimensional Borel subalgebra of a simple Lie algebra. The Borel subalgebra was isomorphic to the first class of Csikós and Verhóczki’s classification of the Frobenius Lie algebras of dimension 6 over an algebraically closed field. The main purpose was to prove that the Borel subalgebra of dimension 6 was equipped with incomplete affine structures. To achieve the purpose, the axiomatic method was considered by studying some important notions corresponding to affine structures and their completeness, Borel subalgebras, and Frobenius Lie algebras. A chosen Frobenius functional of the Borel subalgebra helped to determine the affine structure formulas well. The result shows that the Borel subalgebra of dimension 6 has affine structures which are not complete. Furthermore, the research also gives explicit formulas of affine structures. For future research, another isomorphism class of 6-dimensional Frobenius Lie algebra still needs to be investigated whether it has complete affine structures or not.

Transformation groups of certain flat affine manifolds

In this paper we characterize the group of affine transformations of a flat affine simply connected manifold whose developing map is a diffeomorphism. This is proved by making use of some simple facts about homeomorphisms of $\mathbb{R}^n$ preserving open connected sets. We show some examples where the characterization is useful.

A Lattice Version of the Atiyah–Singer Index Theorem

We formulate and prove a lattice version of the Atiyah-Singer index theorem. The main theorem gives a $K$-theoretic formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds. We apply the main theorem to an index problem of Wilson-Dirac operators in lattice gauge theory.

A machine learning approach to optimal Tikhonov regularization I: Affine manifolds

Despite a variety of available techniques, such as discrepancy principle, generalized cross validation, and balancing principle, the issue of the proper regularization parameter choice for inverse problems still remains one of the relevant challenges in the field. The main difficulty lies in constructing an efficient rule, allowing to compute the parameter from given noisy data without relying either on any a priori knowledge of the solution, noise level or on the manual input. In this paper, we propose a novel method based on a statistical learning theory framework to approximate the high-dimensional function, which maps noisy data to the optimal Tikhonov regularization parameter. After an offline phase where we observe samples of the noisy data-to-optimal parameter mapping, an estimate of the optimal regularization parameter is computed directly from noisy data. Our assumptions are that ground truth solutions of the inverse problem are statistically distributed in a concentrated manner on (lower-dimensional) linear subspaces and the noise is sub-gaussian. We show that for our method to be efficient, the number of previously observed samples of the noisy data-to-optimal parameter mapping needs to scale at most linearly with the dimension of the solution subspace. We provide explicit error bounds on the approximation accuracy from noisy data of unobserved optimal regularization parameters and ground truth solutions. Even though the results are more of theoretical nature, we present a recipe for the practical implementation of the approach. We conclude with presenting numerical experiments verifying our theoretical results and illustrating the superiority of our method with respect to several state-of-the-art approaches in terms of accuracy or computational time for solving inverse problems of various types.

On a Metric Affine Manifold with Several Orthogonal Complementary Distributions

A Riemannian manifold endowed with k>2 orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs or nets composed of orthogonal foliations. In this article, we define the mixed scalar curvature of an almost multi-product structure endowed with a linear connection, and represent this kind of curvature using fundamental tensors of distributions and the divergence of a geometrically interesting vector field. Using this formula, we prove decomposition and non-existence theorems and integral formulas that generalize results (for k=2) on almost product manifolds with the Levi-Civita connection. Some of our results are illustrated by examples with statistical and semi-symmetric connections.

Closed Affine Manifolds with an Invariant Line

A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.

The Kontsevich graph orientation morphism revisited

The orientation morphism $Or(\cdot)(P)\colon \gamma\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ on finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $\gamma$ with ordered sets of edges and without multiple edges and one-cycles. It is known that $d$-cocycles $\boldsymbol{\gamma}\in\ker d$ with respect to the vertex-expanding differential $d=[{\bullet}\!\!{-}\!{-}\!\!{\bullet},\cdot]$ are mapped by $Or$ to Poisson cocycles $Q(P)\in\ker\,[\![ P,{\cdot}]\!]$, that is, to infinitesimal symmetries of Poisson bi-vectors $P$. The formula of orientation morphism $Or$ was expressed in terms of the edge orderings as well as parity-odd and parity-even derivations on the odd cotangent bundle $\Pi T^* N^d$ over any $d$-dimensional affine real Poisson manifold $N^d$. We express this formula in terms of (un)oriented graphs themselves, i.e. without explicit reference to supermathematics on $\Pi T^* N^d$.