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Kurusch Ebrahimi-Fard

Publications and source records attributed to Kurusch Ebrahimi-Fard.

At least 19 recordsLinked to original sources
arXiv2026

Explicit and Effectively Symmetric Runge-Kutta Methods

Symmetry is a key property of numerical methods. The geometric properties of symmetric schemes make them an attractive option for integrating Hamiltonian systems, whilst their ability to exactly recover the initial condition without the need to store the entire solution trajectory makes them ideal for the efficient implementation of Neural ODEs. In this work, we present a Hopf algebraic approach to the study of symmetric B-series methods. We show that every B-series method can be written as the composition of a symmetric and "antisymmetric" component, and explore the structure of this decomposition for Runge-Kutta schemes. A major bottleneck of symmetric Runge-Kutta schemes is their implicit nature, which requires solving a nonlinear system at each step. By introducing a new set of order conditions which minimise the antisymmetric component of a scheme, we derive what we call Explicit and Effectively Symmetric (EES) schemes -- a new class of explicit Runge-Kutta schemes with near-symmetric properties. We present examples of second-order EES schemes and demonstrate that, despite their low order, these schemes readily outperform higher-order explicit schemes such as RK4 and RK5, and achieve results comparable to implicit symmetric schemes at a significantly lower computational cost.

math.NA↗
arXiv2025

Noncrossing arithmetic

Higher-order notions of Kreweras complementation have appeared in the literature in the works of Krawczyk, Speicher, Mastnak, Nica, Arizmendi, Vargas, and others. While the theory has been developed primarily for specific applications in free probability, it also possesses an elegant, purely combinatorial core that is of independent interest. The present article aims at offering a simple account of various aspects of higher-order Kreweras complementation on the basis of elementary arithmetic, (co)algebraic, categorical and simplicial properties of noncrossing partitions. The main idea is to consider noncrossing partitions as providing an interesting noncommutative analogue of the interplay between the divisibility poset and the multiplicative monoid of positive integers. Just as the divisibility poset can be regarded as the decalage of the multiplicative monoid, we exhibit the lattice of noncrossing partitions as the decalage of a partial monoid structure on noncrossing partitions encoding higher-order Kreweras complements. While our results may be considered familiar, several of the viewpoints can be regarded as novel, offering an efficient approach both conceptually and computationally.

math.CO↗
arXiv2025

A multiplicative surface signature through its Magnus expansion

In the last decade, the concept of path signature has achieved significant success in data science applications. It offers a powerful set of features that effectively capture and describe the characteristics of paths or sequential data. This is partly explained by the fact that the signature of a path can be computed in linear time, using a dynamic programming principle based on Chen's identity. The path signature can be viewed as a specific example of a product or time-/path-ordered integral. In other words, it represents a one-parameter object built on iterated integrals over a path. Defining a signature over surfaces requires considering iterated integrals over these surfaces, effectively introducing an additional parameter, resulting in a two-parameter signature. This extended signature is intrinsically connected to a non-commutative generalization of Stokes' theorem, which is fundamentally connected to the concept of crossed modules of groups. The latter provides a well-established framework in higher gauge theory, where crossed modules with feedback maps exhibiting non-trivial kernels, combined with multiparameter iterated integrals, play a pivotal role. Building on Kapranov's work, we explore the surface analog of the log-signature for paths by introducing a Magnus-type formula for the logarithm of the surface signature. This expression takes values in a free crossed module of Lie algebras, defined over a free Lie algebra. We furthermore prove a non-commutative sewing lemma applicable to the crossed module setting and give a definition of rough surface in the so-called Young-Hölder regularity regime along with a corresponding continuous extension theorem. This approach enables the analysis and computation of surface features that go beyond what can be expressed by computing line integrals along the boundary of a surface.

math.RA↗
arXiv2024

Multi-indice B-series

We propose a novel way to study numerical methods for ordinary differential equations in one dimension via the notion of multi-indice. The main idea is to replace rooted trees in Butcher's B-series by multi-indices. The latter were introduced recently in the context of describing solutions of singular stochastic partial differential equations. The combinatorial shift away from rooted trees allows for a compressed description of numerical schemes. Furthermore, such multi-indices B-series uniquely characterize the Taylor expansion of one-dimensional local and affine equivariant maps.

math.NA↗
arXiv2024

The Exponential Lie Series and a Chen-Strichartz Formula for Levy Processes

In this paper, we derive a Chen-Strichartz formula for stochastic differential equations driven by Levy processes, that is, we derive a series expansion of the logarithm of the flowmap of the stochastic differential equation in terms of commutators of vector fields with stochastic coefficients, and we provide an explicit formula for the components in this series. The stochastic components are generated by the Levy processes that drive the stochastic differential equation and their quadratic variation and power jumps; the vector fields are given as linear combinations of commutators of elements in the pre-Lie Magnus expansion generated by the original vector fields governing our stochastic differential equation. In particular, we show the logarithm of the flowmap is a Lie series. These results extend previous results for deterministic differential equations and continuous stochastic differential equations. For these, the Chen-Strichartz series has shown to play a pivotal role in the design of numerical integration schemes that preserve qualitative properties of the solution such as the construction of geometric numerical schemes and in the context of efficient numerical schemes.

math.PR↗
arXiv2024

Decompositions of Nonlinear Input-Output Systems to Zero the Output

Consider an input-output system where the output is the tracking error given some desired reference signal. It is natural to consider under what conditions the problem has an exact solution, that is, the tracking error is exactly the zero function. If the system has a well defined relative degree and the zero function is in the range of the input-output map, then it is well known that the system is locally left invertible, and thus, the problem has a unique exact solution. A system will fail to have relative degree when more than one exact solution exists. The general goal of this paper is to describe a decomposition of an input-output system having a Chen-Fliess series representation into a parallel product of subsystems in order to identify possible solutions to the problem of zeroing the output. For computational purposes, the focus is on systems whose generating series are polynomials. It is shown that the shuffle algebra on the set of generating polynomials is a unique factorization domain so that any polynomial can be uniquely factored modulo a permutation into its irreducible elements for the purpose of identifying the subsystems in a parallel product decomposition. This is achieved using the fact that this shuffle algebra is isomorphic to the symmetric algebra over the vector space spanned by Lyndon words. A specific algorithm for factoring generating polynomials into its irreducible factors is presented based on the Chen-Fox-Lyndon factorization of words.

eess.SY↗
arXiv2024

What is the Magnus Expansion?

The Magnus expansion, introduced by Wilhelm Magnus in 1954, is an infinite Lie series employed to express solutions for first-order homogeneous linear differential equations involving a linear operator. Since its discovery it has evolved into a pivotal tool used across diverse disciplines, including physics, chemistry, and engineering. Over the past 25 years, the Magnus expansion has undergone significant mathematical developments which revealed an intricate interplay between algebra, combinatorics, and geometry. By emphasizing a modern perspectives based on the use of pre- and post-Lie algebras, we discuss the Magnus expansion from the viewpoint of the notion of crossed morphism.

math-ph↗
arXiv2024

Tubings, chord diagrams, and Dyson--Schwinger equations

We give series solutions to single insertion place propagator-type systems of Dyson--Schwinger equations using binary tubings of rooted trees. These solutions are combinatorially transparent in the sense that each tubing has a straightforward contribution. The Dyson--Schwinger equations solved here are more general than those previously solved by chord diagram techniques, including systems and non-integer values of the insertion parameter $s$. We remark on interesting combinatorial connections and properties.

math.CO↗
arXiv2024

A Survey on the Munthe-Kaas-Wright Hopf Algebra

We survey the Munthe-Kaas--Wright Hopf algebra defined on planar rooted trees. This algebra serves a role akin to that of the Butcher--Connes--Kreimer Hopf algebra on non-planar rooted trees within the domain of numerical methods for ordinary differential equations. In the course of our presentation, we revisit Foissy's work on finite-dimensional comodules over the Butcher--Connes--Kreimer Hopf algebra and expand on his findings to include the Munthe-Kaas--Wright Hopf algebra. This involves detailing its endomorphisms and recursively constructing its primitive elements. These results are applied within the context of rough paths, where we describe an isomorphism between planarly branched and geometric rough paths. Our approach hinges on the extension of the Guin--Oudom construction to post-Lie algebras. In the case of the free post-Lie algebra defined on planar rooted trees, it yields the dual of the Munthe-Kaas--Wright Hopf algebra. Surprisingly, we uncover a natural connection between the concept of bialgebras in cointeraction and the Guin--Oudom construction. Prompted by the rough path perspective, we explore this finding through the lens of translations on rough paths. Additionally, we investigate the geometric embedding for planar regularity structures.

math.RA↗
arXiv2024

Algebraic groups in non-commutative probability theory revisited

The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Schürmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie calculus, and results in another construction of groups of characters encoding the behaviour of states. Comparing the two, the first approach, recast recently in a general categorical language by Manzel and Schürmann, can be seen as largely driven by the theory of universal products, whereas the second construction builds on Hopf algebras and a suitable algebraization of the combinatorics of noncrossing set partitions. Although both address the same phenomena, moving between the two viewpoints is not obvious. We present here an attempt to unify the two approaches by making explicit the Hopf algebraic connections between them. Our presentation, although relying largely on classical ideas as well as results closely related to Manzel and Schürmann's aforementioned work, is nevertheless original on several points and fills a gap in the non-commutative probability literature. In particular, we systematically use the language and techniques of algebraic groups together with shuffle group techniques to prove that two notions of algebraic groups naturally associated with free, respectively Boolean and monotone, probability theories identify. We also obtain explicit formulas for various Hopf algebraic structures and detail arguments that had been left implicit in the literature.

math.RA↗
arXiv2024

On the Post-Lie Structure in SISO Affine Feedback Control Systems

The main objective of this work is to show that the single-input, single-output (SISO) affine feedback group, a transformation group in the context of the affine feedback interconnection of Chen-Fliess series, is a post-group in the sense of Bai, Guo, Sheng and Tang.

math.OC↗
arXiv2024

On the signature of an image

Over the past decade, the importance of the 1D signature which can be seen as a functional defined along a path, has been pivotal in both path-wise stochastic calculus and the analysis of time series data. By considering an image as a two-parameter function that takes values in a $d$-dimensional space, we introduce an extension of the path signature to images. We address numerous challenges associated with this extension and demonstrate that the 2D signature satisfies a version of Chen's relation in addition to a shuffle-type product. Furthermore, we show that specific variations of the 2D signature can be recursively defined, thereby satisfying an integral-type equation. We analyze the properties of the proposed signature, such as continuity, invariance to stretching, translation and rotation of the underlying image. Additionally, we establish that the proposed 2D signature over an image satisfies a universal approximation property.

math.CA↗
arXiv2024

Free post-groups, post-groups from group actions, and post-Lie algebras

After providing a short review on the recently introduced notion of post-group by Bai, Guo, Sheng and Tang, we exhibit post-group counterparts of important post-Lie algebras in the literature, including the infinite-dimensional post-Lie algebra of Lie group integrators. The notion of free post-group is examined, and a group isomorphism between the two group structures associated to a free post-group is explicitly constructed.

math.QA↗
arXiv2023

Conditionally monotone cumulants via shuffle algebra

In this work we study conditional monotone cumulants and additive convolution in the shuffle-algebraic approach to non-commutative probability. We describe c-monotone cumulants as an infinitesimal character and identify the c-monotone additive convolution as an associative operation in the set of pairs of characters in the dual of a double tensor Hopf algebra. In this algebraic framework, we understand previous results on c-monotone cumulants and prove a combinatorial formula that relates c-free and c-monotone cumulants. We also identify the notion of $t$-Boolean cumulants in the shuffle-algebraic approach and introduce the corresponding notion of $t$-monotone cumulants as a particular case of c-monotone cumulants.

math.OA↗
arXiv2023

A Formal Power Series Approach to Multiplicative Dynamic Feedback Interconnection

The goal of the paper is multi-fold. First, an explicit formula is derived to compute the non-commutative generating series of a closed-loop system when a (multi-input, multi-output) plant, given in Chen--Fliess series description is in multiplicative output feedback interconnection with another system, also given as Chen--Fliess series. Furthermore, it is shown that the multiplicative dynamic output feedback connection has a natural interpretation as a transformation group acting on the plant. A computational framework for computing the generating series for multiplicative dynamic output feedback is devised utilizing the Hopf algebras of the coordinate functions corresponding to the shuffle group and the multiplicative feedback group. The pre--Lie algebra in multiplicative feedback is shown to be an example of Foissy's com-pre-Lie algebras indexed by matrices with certain structure.

math.OC↗
arXiv2023

Generalized iterated-sums signatures

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F.~Király and H.~Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.

math.RA↗