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Large-scale optimization problems arising from the discretization of problems involving PDEs sometimes admit solutions that can be well approximated by low-rank matrices. In this paper, we will exploit this low-rank approximation property…

We present a prescription for choosing orthogonal bases of differential $n$-forms belonging to quadratic twisted period integrals, with respect to the intersection number inner product. To evaluate these inner products, we additionally…

Functions of chemical composition are complex and discrete in nature making it impossible to optimize them with gradient methods. Genetic algorithms, which do not use derivative information, are used to maximize the thermal conductivity of…

Generalized unitarity cut of a Feynman diagram generates an algebraic system of polynomial equations. At high-loop levels, these equations may define a complex curve or a (hyper-)surface with complicated topology. We study the curve cases,…

We present a new method for the decomposition of multi-loop Euclidean Feynman integrals into quasi-finite Feynman integrals. These are defined in shifted dimensions with higher powers of the propagators, make explicit both infrared and…

The Einstein equations have proven surprisingly difficult to solve numerically. A standard diagnostic of the problems which plague the field is the failure of computational schemes to satisfy the constraints, which are known to be…

In this paper, we introduce some new iterative optimisation algorithms on Riemannian manifolds and Hilbert spaces which have good global convergence guarantees to local minima. More precisely, these algorithms have the following properties:…

We present the integrand reduction via multivariate polynomial division as a natural technique to encode the unitarity conditions of Feynman amplitudes. We derive a recursive formula for the integrand reduction, valid for arbitrary…

We show how a large class of Feynman integrals can be efficiently reduced to master integrals by suitable covariant differentiation on the vector space dual to the one spanned by the master integrals. The connections needed in the covariant…

We define linearly reducible elliptic Feynman integrals, and we show that they can be algorithmically solved up to arbitrary order of the dimensional regulator in terms of a 1-dimensional integral over a polylogarithmic integrand, which we…

Cut-elimination is the bedrock of proof theory with a multitude of applications from computational interpretations to proof analysis. It is also the starting point for important meta-theoretical investigations including decidability,…

Discretizations of the Feynman-Kac path integral representation of the quantum mechanical density matrix are investigated. Each infinite-dimensional path integral is approximated by a Riemann integral over a finite-dimensional function…

A method of functional reduction for the dimensionally regularized one-loop Feynman integrals with massive propagators is described in detail. The method is based on a repeated application of the functional relations proposed by the author.…

The standard procedure when evaluating integrals of a given family of Feynman integrals, corresponding to some Feynman graph, is to construct an algorithm which provides the possibility to write any particular integral as a linear…

Magic relations are a class of integration-by-parts identities where all integrals in the generating sector drop out. Since their presence causes several otherwise successful methods in the Feynman-integral computational pipeline to break…

We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications of…

Interacting particle systems undergoing repeated mutation and selection steps model genetic evolution, and also describe a broad class of sequential Monte Carlo methods. The genealogical tree embedded into the system is important in both…

Constraints are found on the spatial variation of finite-time Lyapunov exponents of two and three-dimensional systems of ordinary differential equations. In a chaotic system, finite-time Lyapunov exponents describe the average rate of…

Higher-order diagrams required for radiative corrections to mixed electroweak and QCD processes at the LHC and anticipated future colliders will require numerically stable representations of the associated Feynman diagrams. The…

We present an algorithm based on continuation techniques that can be applied to solve numerically minimization problems with equality constraints. We focus on problems with a great number of local minima which are hard to obtain by local…