New boundary treatment improves accuracy for complex PDEs.
problem Order reduction in high-order IMEX schemes for multidimensional PDEs.
method Novel boundary treatment algorithms for Cartesian meshes, treating implicit-explicit stages similarly to interior points.
result Recovery of designed order of convergence by numerical verification.
This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundar…
DGNet solves complex dynamical systems with neural networks and constraints.
problem Real-time accurate solutions for large-scale complex systems.
method Model-constrained discontinuous Galerkin Network (DGNet) for compressible Euler equations.
result DGNet achieves out-of-distribution generalization and improved stability.
Develops numerical methods for PDEs on hypergraphs and networks.
problem Solving PDEs on complex geometric structures like hypergraphs and networks.
method Hybrid finite element methods, focusing on hybrid discontinuous Galerkin methods.
result Derives numerical approximations for PDEs on hypergraphs and networks.
Neural Galerkin schemes use active learning to solve high-dimensional equations.
problem Inaccurate function approximations in high dimensions with limited training data.
method Neural Galerkin schemes based on deep learning with active learning for high-dimensional PDEs.
result Active data collection improves the numerical solution of high-dimensional equations.
Galerkin method outperforms graph-based methods in spectral decompositions.
problem Improving spectral decomposition methods in machine learning.
method Restricting study to a small set of test functions using the Galerkin method.
result Statistical and computational superiority of Galerkin method over graph-based approaches.
Paper explores solving HJB equations using neural networks.
problem Solving high-dimensional time-dependent HJB equations.
method Neural Galerkin methods with nonlinearly parametrized trial functions.
result Closed-form solutions for trial functions.
New ARIMA framework improves forecast accuracy for economic and financial time series.
problem Improving forecast accuracy for nonlinear dynamics in time series data.
method Projection-based ARIMA framework using Galerkin basis expansions.
result Galerkin-SARIMA matches or improves forecast accuracy compared to classical ARIMA/SARIMA.
Review and compare model order reduction methods for process engineering.
problem Creating computationally efficient yet accurate models for real-time applications.
method Nonlinear model order reduction methods, including general-purpose and tailored approaches for chemical processes.
result Comparison of eight model order reduction methods applied to an air separation process model.
One popular approach to option pricing in Lévy models is through solving the related partial integro differential equation (PIDE). For the numerical solution of such equations powerful Galerkin methods have been put forward e.g. by Hilber et al. (2013). As in practice large classes of models are maintained simultaneous…
A new method uses neural networks to improve POD-Galerkin models for complex systems.
problem Improving computational efficiency and accuracy in solving non-linear high-dimensional systems.
method Deep learning-based closure modeling using neural networks to approximate POD-Galerkin operators.
result The CD-ROM approach produces more accurate and stable models for complex systems.
Deep Galerkin Method estimates value function for mean-field control problem.
problem Optimal control of agents with average welfare as the objective.
method Apply DGM to estimate value function and distribution evolution.
result Neural network approximations converge to analytical solution.
Study methods to recover unknown processes in PDEs from data.
problem Identifying unknown processes in time-dependent PDEs using observational data.
method Theoretical analysis and numerical approaches including Galerkin and collocation algorithms.
result The Galerkin algorithm is more suitable for practical situations with noisy data.
The most recent update of financial option models is American options under stochastic volatility models with jumps in returns (SVJ) and stochastic volatility models with jumps in returns and volatility (SVCJ). To evaluate these options, mesh-based methods are applied in a number of papers but it is well-known that the…
Paper establishes convergence rates for learning elliptic pseudo-differential operators.
problem Learning elliptic pseudo-differential operators in partial differential equations.
method Wavelet-Galerkin framework, structured infinite-dimensional regression problem, sparse estimator, matrix compression, nested-support strategy.
result Obtained convergence rates for the estimator and efficient Galerkin solver.
New method learns diffusion transition density for Bayesian inference.
problem Bayesian inference on diffusions with inaccessible boundaries.
method Neural Galerkin framework to solve FP equation with Dirac mass.
result Approximates likelihood function for efficient posterior sampling.
Proves regularity for quasilinear elliptic equations in metric spaces.
problem Regularity of quasilinear elliptic equations in metric measure spaces.
method Galerkin's method as an alternative to difference quotients.
result Second-order and Lipschitz regularity for a wide class of elliptic equations.
This paper is concerned with the following Markovian stochastic differential equation of mean-reversion type \[ dR_t= (θ+σα(R_t, t))R_t dt +σR_t dB_t \] with an initial value R0=r0∈R, where θ∈R and σ>0 are constants, and the mean correction function $α:\mathbb{R}\times[0,\infty)\to α(x,t)\…
Proves nonemptyness of domains for specific group actions.
problem Nonemptyness of domains of proper discontinuity for Anosov groups of affine Lorentzian transformations.
method Proof of nonemptyness of domains of proper discontinuity.
result Proves nonemptyness of domains for Anosov groups of affine Lorentzian transformations.
New methods for clustering graphs using spectral analysis.
problem Graph clustering for complex systems.
method Transfer operators and spectral properties.
result Spectral clustering can be interpreted using Koopman operators.
Deep learning method proves convergence for high-dimensional PDEs.
problem Solving high-dimensional nonlinear PDEs for mean field control problems.
method Deep Galerkin method (DGM) for Hamilton-Jacobi-Bellman (HJB) equations.
result DGM converges to the true value function of mean field control problems.
We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a `small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of disc…
Cut-DeepONet handles discontinuities and sharp transitions in neural operators.
problem Neural operators struggle with discontinuities and sharp transitions in PDEs.
method Two-stage training framework that explicitly models discontinuities via a lifting strategy and input-dependent discontinuity prediction.
result Cut-DeepONet outperforms state-of-the-art methods on benchmark PDEs with low-resolution datasets.
Study proves existence of equilibrium in incomplete economies with discontinuous volatility.
problem Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
method Established existence of solution for Markovian quadratic BSDEs with discontinuous generators using unique continuation and backward uniqueness.
result Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
New domains of discontinuity found for Anosov representations.
problem Understanding Anosov representations acting on homogeneous spaces.
method Constructing open domains of discontinuity for Anosov representations acting on specific homogeneous spaces.
result Describes the largest possible open domains of discontinuity for Zariski dense Anosov representations.
As early as 1972, Penrose - in a purely formal way - introduced a "discontinuous coordinate transformation", which relates a continuous representation of the metric of impulsive pp-waves to a discontinuous one. On the basis of the invertibility concept for generalized functions developed recently by the first author, w…
A machine learning approach to compute Black-Scholes prices with uncertain volatility.
problem Approximating financial markets with continuous-time models like Black-Scholes when data is discrete.
method Generalized Polynomial Chaos (gPC) method combined with a machine learning technique called Bi-Fidelity.
result Efficient numerical method to quantify uncertainty in derivative pricing.
Study the discontinuity of functions not embeddable in Euclidean space.
problem Understanding discontinuity of non-embeddable functions.
method Define a modulus of discontinuity and establish lower bounds.
result Quantified nonembeddability results and topological Tverberg theorem.
A new model captures forward curve dynamics with stochastic volatility.
problem Modeling continuous-time evolution of forward curves in financial markets.
method Affine stochastic volatility model with modulated dynamics.
result Model allows for maturity-specific risk and volatility clustering.
Study new symmetries in non-symmetric spaces and discontinuous groups.
problem Analyze symmetries in non-symmetric homogeneous spaces and discontinuous groups.
method Investigate discrete series, discontinuous groups, and analysis on pseudo-Riemannian spaces.
result New insights into symmetries of non-symmetric homogeneous spaces and discontinuous groups.
Study optimal semi-static hedging for illiquid markets using dynamic cash and static quoted derivatives.
problem Optimal pricing of exotic derivatives in illiquid markets with bid-ask spreads.
method Use Galerkin method and integration quadratures to approximate hedging problem as convex optimization, solved by interior point method.
result Semi-static hedging improves pricing and reduces transaction costs compared to static or dynamic trading alone.
Paper finds surface groups can deform in reductive symmetric spaces.
problem Finding deformations of discontinuous groups in reductive symmetric spaces.
method Analyzing Zariski dense surface subgroups and their deformations.
result Surface groups of high genus can deform in reductive symmetric spaces.
A new method for unsupervised disentanglement using axis-aligned cliffs.
problem Unsupervised disentanglement of latent factors under nonlinear maps.
method Encouraging axis-aligned discontinuities (cliffs) in the estimated density of factors.
result Cliff method outperforms baselines on disentanglement benchmarks.
New method estimates active subspaces for jump-discontinuous functions.
problem Estimating active subspaces for discontinuous functions like ABMs.
method Extending active subspaces to discontinuous functions, using Gaussian process.
result Identifies important parameters in ABM simulations of refugee movement.
This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, we begin by reviewing the current knowledge of possible global forms of pseudo-Riemannian manifolds with constant curvatures, and discuss what kind of …
We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we consider PDEs where the function is constrained to be positive and integrate to uni…
The first known example of a complete Riemannian manifold whose isoperimetric profile is discontinuous is given.
Paper analyzes error in stochastic approximation for discontinuous functions.
problem Estimating expected error in discontinuous stochastic approximation.
method Uses finite differences and O(n−1/5) error estimate for discontinuous functions. result Achieves error estimate of O(n−1/5) for discontinuous stochastic representation. Bayesian emulator tackles models with discontinuities efficiently.
problem Uncertainty quantification of models with multiple discontinuities.
method TENSE framework, designed correlation structures, single emulator object.
result Efficient emulation of complex models with multiple discontinuities.
High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since mesh…
We characterise completely when limit sets, as parametrised by Cannon-Thurston maps, move discontinuously for a sequence of algebraically convergent quasi-Fuchsian groups.
We show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu.
Volunteer labor can temporarily yield lower benefits to charities than its costs. In such instances, organizations may wish to defer volunteer donations to a later date. Exploiting a discontinuity in blood donations' eligibility criteria, we show that deferring donors reduces their future volunteerism. In our setting, …
We develop a general term structure framework taking stochastic discontinuities explicitly into account. Stochastic discontinuities are a key feature in interest rate markets, as for example the jumps of the term structures in correspondence to monetary policy meetings of the ECB show. We provide a general analysis of …
Study proves rigid spectral properties of planets with metric discontinuities.
problem Establishing spectral rigidity for spherically symmetric planets with discontinuities.
method Novel trace formula applied to two wave types in spherically symmetric manifolds with boundary and interior interfaces.
result Spectral rigidity of spherically symmetric planets with discontinuities is proven.
We study the properly discontinuous and isometric actions on the unit sphere of infinite dimensional Hilbert spaces and we get some new examples of Hilbert manifold with costant positive sectional curvature. We prove some necessary conditions for a group to act isometrically and properly discontinuously and in the case…
In this paper, we present a method for the accurate estimation of the derivative (aka.~sensitivity) of expectations of functions involving an indicator function by combining a stochastic algorithmic differentiation and a regression. The method is an improvement of the approach presented in [Risk Magazine April 2018]. T…
This work models overnight rates with jumps and discontinuities, extending classical short-rate models.
problem Capturing the jump behavior and discontinuities in overnight rates for accurate modeling.
method Developed a term structure modeling framework based on overnight rates, accommodating stochastic discontinuities.
result Simple specifications can capture the jump behavior of overnight rates, and explicit valuation formulas are provided.