arXiv research

Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.

We present a stochastic numerical method for solving fully non-linear free boundary problems of parabolic type and provide a rate of convergence under reasonable conditions on the non-linearity.

Study $p$-parabolicity on graphs using various energy functionals.

Paper solves Musielak-Orlicz-Gauss image problem using parabolic flows.

A major challenge in current optimization research for deep learning is to automatically find optimal step sizes for each update step. The optimal step size is closely related to the shape of the loss in the update step direction. However, this shape has not yet been examined in detail. This work shows empirically that…

The paper proves smoothness of transition layers in the Allen-Cahn equation.

We consider the Cauchy problem associated with a general parabolic partial differential equation in $d$ dimensions. We find a family of closed-form asymptotic approximations for the unique classical solution of this equation as well as rigorous short-time error estimates. Using a boot-strapping technique, we also provi…

We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and an…

New method uses tensor trains for efficient PDE approximation.

Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of ga…

Market illiquidity, feedback effects, presence of transaction costs, risk from unprotected portfolio and other nonlinear effects in PDE based option pricing models can be described by solutions to the generalized Black-Scholes parabolic equation with a diffusion term nonlinearly depending on the option price itself. Di…

In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handl…

The stability and robustness of compact schemes for parabolic PDEs are analyzed.

New learning scheme solves high-dimensional semi-linear PDEs using sparse grids and Picard approximations.

In this paper we analyze a nonlinear Black--Scholes model for option pricing under variable transaction costs. The diffusion coefficient of the nonlinear parabolic equation for the price $V$ is assumed to be a function of the underlying asset price and the Gamma of the option. We show that the generalizations of the cl…

Tensor trains simplify solving complex PDEs efficiently.

Novel filter uses deep BSDE for nonlinear density approximation.

Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.

Quantum machine learning solves high-dimensional PDEs with lower variance and improved accuracy.

In the present work, we propose a new multifactor stochastic volatility model in which slow factor of volatility is approximated by a parabolic arc. We retain ourselves to the perturbation technique to obtain approximate expression for European option prices. We introduce the notion of modified Black-Scholes price. We …

Deep neural nets approximate high-dimensional HJB equations efficiently.

Defines connections on parabolic vector bundles for Lie algebroids.

Study describes metrics on moduli space of Higgs bundles, proving exponential decay rate.

Criterion found for Lie algebroid connections on parabolic bundles.

Computes deformations of parabolic structures on Riemann surfaces.

D2SRM solves complex PDEs using deep learning.

Paper analyzes and proves convergence of a new method for solving complex PDEs.

Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary fiel…

We provide some criteria to $p$-parabolicity of Riemannian submersions. In particular, if $N$ is $p$-parabolic and $π:M\to N$ is a Riemannian submersion with uniformly bounded volume of fibers, then $M$ is also $p$-parabolic. In the case of warped manifolds we characterize $p$-parabolicity in terms of a volume growth c…

Classifies holomorphic parabolic geometries on complex manifolds.

Alternative proof classifies Kleinian groups with two parabolics.

Defines non-parabolic curves in spatial hybrid space with applications.

We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Gri…

Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its word problem, and generating sets of the parabolic subgroups. We also give an a…

I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate $G$-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quan…

Maximal regularity for nonuniformly parabolic problems with normal degeneration.

New estimates for nodal and singular sets of parabolic inequalities.

$Sol_3$ is shown not to be parabolic.

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss functi…

Study gauge theory of real and quaternionic parabolic bundles over real curves.

Proves monotonicity of parabolic frequency on all manifolds without curvature assumptions.

The goal of this paper is to classify parametrically parabolic submanifolds in any codimension. First, we describe the ones that are ruled and show that they are the only parabolic submanifolds that admit an isometric immersion as a hypersurface. Then, we classify the nonruled ones by two different means. In fact, we p…

The aim of this paper is to construct the parabolic version of the Donaldson--Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non--emptiness of the moduli space of parabolic sta…

The paper defines parabolic subgroups for complex braid groups and proves they form a lattice.

We consider automorphisms of homogeneous parabolic geometries with a fixed point. Parabolic geometries carry the distinguished distributions and we study those automorphisms which enjoy natural actions on the distributions at the fixed points. We describe the sets of such automorphisms on homogeneous parabolic geometri…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

New method to determine parabolic surfaces invariant under Killing fields.

We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for bra…