Topological Expansion Research Articles

Adjoint method in PDE-based image compression

We consider a shape optimization based method for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in an L p -norm, for 1 ⩽ p < + ∞, between original images and their reconstructed counterparts using linear diffusion PDE-based inpainting. Reformulating the problem as a constrained optimization over sets (shapes), we derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small ball (a single pixel) using the adjoint method. Based on the achieved distributed topological shape derivatives, we propose a numerical approach to determine the optimal set and present numerical experiments showing the efficiency of our method. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.

Complete topological asymptotic expansion for L2 and H1 tracking-type cost functionals in dimension two and three

In this paper, we study the topological asymptotic expansion of a topology optimisation problem that is constrained by the Poisson equation with the design/shape variable entering through the right hand side. Using an averaged adjoint approach, we give explicit formulas for topological derivatives of arbitrary order for both an L2 and H1 tracking-type cost function in both dimension two and three and thereby derive the complete asymptotic expansion. As the asymptotic behaviour of the fundamental solution of the Laplacian differs in dimension two and three, also the derivation of the topological expansion significantly differs in dimension two and three. The complete expansion for the H1 cost functional directly follows from the analysis of the variation of the state equation. However, the proof of the asymptotics of the L2 tracking-type cost functional is significantly more involved and, surprisingly, the asymptotic behaviour of the bi-harmonic equation plays a crucial role in our proof.

Baryon-number — flavor separation in the topological expansion of QCD

Gauge invariance of QCD dictates the presence of string junctions in the wave functions of baryons. In high-energy inclusive processes, these baryon junctions have been predicted to induce the separation of the flows of baryon number and flavor. In this paper we describe this phenomenon using the analog-gas model of multiparticle production proposed long time ago by Feynman and Wilson and adapted here to accommodate the topological expansion in QCD. In this framework, duality arguments suggest the existence of two degenerate junction-antijunction glueball Regge trajectories of opposite C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{C} $$\\end{document}-parity with intercept close to 1/2. The corresponding results for the energy and rapidity dependence of baryon stopping are in reasonably good agreement with recent experimental findings from STAR and ALICE experiments. We show that accounting for correlations between the fragmenting strings further improves agreement with the data, and outline additional experimental tests of our picture at the existing (RHIC, LHC, JLab) and future (EIC) facilities.

Open-closed string duality, branes, and topological recursion

We consider matrix models exhibiting open-closed string duality in two-dimensional string theories with various amounts of supersymmetry. In particular, a relationship between matrix models in the β = 2 Wigner-Dyson class and models in the (1 + 2Γ, 2) Altland-Zirnbauer class relates the perturbative solutions of the two systems’ string equations. Point-like operator insertions in the closed string theory are mapped to the topological expansion of the free energy in the open string theory. We compute correlation functions of macroscopic loop operators and FZZT branes in a general topological gravity background. The relationship between the topological recursion of moduli space volumes and branes is discussed by analyzing the Virasoro conditions in the matrix models.

A noniterative reconstruction method for the inverse potential problem for a time-fractional diffusion equation

This paper is concerned with the reconstruction of the support of the potential term for a time-fractional diffusion equation from the final measured data. The aim of this paper is to propose an accurate approach based on the topological derivative method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing a given cost function. We derive a topological asymptotic expansion for the fractional model. The unknown support is reconstructed using the level-set curve of the topological gradient. We finally make some numerical examples proving the efficiency and accuracy of the proposed algorithm.

Topological sensitivity analysis for the 3D nonlinear Navier–Stokes equations

This work is devoted to a topological asymptotic expansion for the nonlinear Navier–Stokes operator. We consider the 3D Navier–Stokes equations as a model problem and we derive a topological sensitivity analysis for a design function with respect to the insertion of a small obstacle inside the fluid flow domain. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is examined. The performed mathematical framework can be applied for a large class of design functions and arbitrarily shaped geometric perturbations. The obtained asymptotic formula can serve as a useful tool for solving a variety of topology optimization problems in fluid mechanics.

Phase diagram and topological expansion in the complex quartic random matrix model

AbstractWe use the Riemann–Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers of 4‐valent connected graphs with j vertices on a compact Riemann surface of genus g. We explicitly evaluate these numbers for Riemann surfaces of genus 0,1,2, and 3. Also, for a Riemann surface of an arbitrary genus g, we calculate the leading term in the asymptotics of as the number of vertices tends to infinity. Using the theory of quadratic differentials, we characterize the critical contours in the complex parameter plane where phase transitions in the quartic model take place, thereby proving a result of David. These phase transitions are of the following four types: (a) one‐cut to two‐cut through the splitting of the cut at the origin, (b) two‐cut to three‐cut through the birth of a new cut at the origin, (c) one‐cut to three‐cut through the splitting of the cut at two symmetric points, and (d) one‐cut to three‐cut through the birth of two symmetric cuts.

Mini-Workshop: Topological and Differential Expansions of o-minimal Structures

The workshop brought together researchers with expertise in areas of mathematics where model theory has had interesting applications. The areas of expertise spanned from expansions of o-minimal structures preserving tame geometric properties to expansions of specified fields by classical operators that preserve neo-stability properties. There were presentations and discussions on recent developments in definable groups and decompositions in relatively tame setups, the interplay of different notions of dimension and closure operators, and applications of the model theory of differential fields to diophantine geometry.

Autonomous topology planning for distribution network expansion: A learning-based decoupled optimization method

The distribution of urban electricity load is highly uneven in both space and time, posing significant challenges to the problem of distribution network expansion planning (DNEP). An autonomous topology planning method for distribution network expansion is proposed to address that, driven by a learning-based decoupled optimization approach. The planning problem is decoupled from the impact of distribution network operation and used as reinforcement cost prediction for autonomous planning after the generation of topology schemes. First, a reinforcement model of the original distribution network is built that considers the actual distance calculation method of load nodes. Then an effective method for generating historical operation samples is designed, followed by the development of a learning-based reinforcement cost prediction model for the original distribution network based on operation state learning using XGBoost, which enables the prediction of said costs under different scenarios. Utilizing a distribution network topology generating method driven by the adjustable weight minimum spanning tree (AWMST) algorithm, an autonomous topology expansion planning strategy generation method employing the learning-based decoupled optimization approach is established. The feasibility and superior analysis of the proposed method is verified and analyzed on an actual urban distribution network case with simulation scenarios. The results show that the proposed method is faster and more flexible to deal with the distribution network planning problem brought about by rapid load growth in urban distribution networks.

Transition Amplitudes in 3D Quantum Gravity: Boundaries and Holography in the Coloured Boulatov Model

We consider transition amplitudes in the coloured simplicial Boulatov model for three-dimensional Riemannian quantum gravity. First, we discuss aspects of the topology of coloured graphs with non-empty boundaries. Using a modification of the standard rooting procedure of coloured tensor models, we then write transition amplitudes systematically as topological expansions. We analyse the transition amplitudes for the simplest boundary topology, the 2-sphere, and prove that they factorize into a sum entirely given by the combinatorics of the boundary spin network state and that the leading order is given by graphs representing the closed 3-ball in the large N limit. This is the first step towards a more detailed study of the holographic nature of coloured Boulatov-type GFT models for topological field theories and quantum gravity.

Formal topological description of three-dimensional topological entities based on k-dimensional pseudo-manifold

This paper clarifies the necessity of studying three-dimensional spatial data models, analyzes the research progress and existing problems of topological spatial relationship description methods, and proposes a complete and formal description framework of three-dimensional topological spatial relations based on point set topological theory and dimension expansion method. On this basis, the topological spatial relations existing in three-dimensional spatial objects are classified, and five fundamental topological spatial relations are defined. The mutual exclusivity and completeness of the minimal set of 3D topological space relations are proved. Based on point set topology and k-dimensional pseudo-manifold, this paper proposes the formal description of three-dimensional topological properties and spatial relations between spatial entities. The simplicial homology is applied to manifolds, the topological properties of three-dimensional space entities are revealed, and the formal description method of three-dimensional space entities is given through the geometric structural relationship between simplex and simplicial k-complex.

The Topological State Derivative: An Optimal Control Perspective on Topology Optimisation

In this paper, we introduce the topological state derivative for general topological dilatations and explore its relation to standard optimal control theory. We show that for a class of partial differential equations, the shape-dependent state variable can be differentiated with respect to the topology, thus leading to a linearised system resembling those occurring in standard optimal control problems. However, a lot of care has to be taken when handling the regularity of the solutions of this linearised system. In fact, we should expect different notions of (very) weak solutions, depending on whether the main part of the operator or its lower order terms are being perturbed. We also study the relationship with the topological state derivative, usually obtained through classical topological expansions involving boundary layer correctors. A feature of the topological state derivative is that it can either be derived via Stampacchia-type regularity estimates or alternately with classical asymptotic expansions. It should be noted that our approach is flexible enough to cover more than the usual case of point perturbations of the domain. In particular, and in the line of (Delfour in SIAM J Control Optim 60(1):22–47, 2022; J Convex Anal 25(3):957–982, 2018), we deal with more general dilatations of shapes, thereby yielding topological derivatives with respect to curves, surfaces or hypersurfaces. To draw the connection to usual topological derivatives, which are typically expressed with an adjoint equation, we show how usual first-order topological derivatives of shape functionals can be easily computed using the topological state derivative.

Topological asymptotic expansion for the full Navier–Stokes equations

This paper is concerned with a topological sensitivity analysis for the two dimensional incompressible Navier–Stokes equations. We derive a topological asymptotic expansion for a shape functional with respect to the creation of a small geometric perturbation inside the fluid flow domain. The geometric perturbation is modeled as a small obstacle. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is discussed. The obtained results are valid for a large class of shape fonctions and arbitrarily shaped geometric perturbations. The established topological asymptotic expansion provides a useful tool for shape and topology optimization in fluid mechanics.

A Matrix Model for Flat Space Quantum Gravity

We take a step towards the non-perturbative description of a two-dimensional dilaton-gravity theory which has a vanishing cosmological constant and contains black holes. This is done in terms of a double-scaled Hermitian random matrix model which non-perturbatively computes the partition function for the asymptotic Bondi Hamiltonian. To arrive at this connection we first construct the gauge-invariant asymptotic phase space of the theory and determine the relevant asymptotic boundary conditions, compute the classical S-matrix and, finally, shed light on the interpretation of the Euclidean path integral defined in previous works. We then construct a matrix model that matches the topological expansion of the latter, to all orders. This allows us to compute the fine-grained Bondi spectrum and other late time observables and to construct asymptotic Hilbert spaces. We further study aspects of the semi-classical dynamics of the finite cut-off theory coupled to probe matter and find evidence of maximally chaotic behavior in out-of-time-order correlators. We conclude with a strategy for constructing the non-perturbative S-matrix for our model coupled to probe matter and comment on the treatment of black holes in celestial holography.

Finite free cumulants: Multiplicative convolutions, genus expansion and infinitesimal distributions

Given two polynomials p ( x ) , q ( x ) p(x), q(x) of degree d d , we give a combinatorial formula for the finite free cumulants of p ( x ) ⊠ d q ( x ) p(x)\boxtimes _d q(x) . We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that ⊠ d \boxtimes _d converges to ⊠ \boxtimes as d d goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building on our results we give a new short and conceptual proof of a recent result (see J. Hoskins and Z. Kabluchko [Exp. Math. (2021), pp. 1–27]; S. Steinerberger [Exp. Math. (2021), pp. 1–6]) that connects root distributions of polynomial derivatives with free fractional convolution powers.

A solvable model of flat space holography

We propose an explicit realization of flat space holography in two dimensions where both sides of the duality are independently defined and the boundary theory is completely solvable. In the bulk, we define a novel \U0001d4a9 = 1 flat space supergravity theory and exactly compute the full topological expansion of its Euclidean partition function with an arbitrary number of boundaries. On the boundary, we consider a double scaled Hermitian random matrix model with Gaussian potential and use the loop equations to show it independently reproduces the bulk partition function to all orders in the topological expansion. The non-perturbative completion of the supergravity theory provided by the solvable Gaussian matrix model allows for the exact, and in many cases analytic, computation of observables in flat space quantum gravity.

Coupling Shape Optimization and Topological Derivative for Maxwell Equations

The paper deals with a coupling algorithm using shape and topological derivatives of a given cost functional and a problem governed by nonstationary Maxwell’s equations in 3D. To establish the shape and topological derivatives, an adjoint method is used. For the topological asymptotic expansion, two examples of cost functionals are considered with the perturbation of the electric permittivity and magnetic permeability. We combine the shape derivative and topological one to propose an algorithm. The proposed algorithm allows to insert a small inhomogeneity (electric or magnetic) in a given shape.

Large- N expansion and string theory out of equilibrium

We analyze the large-$N$ expansion of general nonequilibrium systems with fluctuating matrix degrees of freedom and $SU(N)$ symmetry, using the Schwinger-Keldysh formalism and its closed real-time contour with a forward and backward component. In equilibrium, the large-$N$ expansion of such systems leads to a sum over topologies of two-dimensional surfaces of increasing topological complexity, predicting the possibility of a dual description in terms of string theory. We extend this argument away from equilibrium and study the universal features of the topological expansion in the dual string theory. We conclude that in nonequilibrium string perturbation theory, the sum over world sheet topologies is further refined: Each world sheet surface $\mathrm{\ensuremath{\Sigma}}$ undergoes a triple decomposition into the part ${\mathrm{\ensuremath{\Sigma}}}^{+}$ corresponding to the forward branch of the time contour, the part ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{-}}$ on the backward branch, and the part ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{\wedge}}$ that corresponds to the instant in the far future where the two branches of the time contour meet. The sum over topologies becomes a sum over the triple decompositions. We generalize our findings to the Kadanoff-Baym time contour relevant for systems at finite temperature and to the case of closed and open, oriented or unoriented strings. Our results are universal and follow solely from the features of the large-$N$ expansion without any assumptions about the world sheet dynamics.

Celestial Matrix Model.

We construct a Hermitian random matrix model that provides a stable nonperturbative completion of Cangemi-Jackiw (CJ) gravity, a two-dimensional theory of flat spacetimes. The matrix model reproduces, to all orders in the topological expansion, the Euclidean partition function of CJ gravity with an arbitrary number of boundaries. The nonperturbative completion enables the exact computation of observables in flat space quantum gravity which we use to explicitly characterize the Bondi Hamiltonian spectrum. Wediscuss the implications of our results for the flat space S-matrix and black holes.

Singular geometry perturbation based method for shape-topology optimization in unsteady Stokes flow

This paper concerns the topological derivatives and its applications for solving shape and topology optimization problems in fluid mechanics. The fluid flow is governed by the unsteady incompressible Stokes equations in the two dimensional case. We derive a topological sensitivity analysis for this parabolic-type operator. The proposed approach is based on a preliminary estimate describing the variation of the velocity field caused by the presence of a small obstacle inside the fluid flow. We obtain a topological asymptotic expansion for the unsteady Stokes operator valid for a large class of shape functions and an arbitrarily shaped geometric perturbation. Then, the topological gradient is exploited for building an efficient and accurate topology optimization algorithm. Finally, we present some numerical investigations showing the efficiency of the proposed approach.