We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that pullbacks of hypergeometric functions by rational Belyi functions with restricted …
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48 results for “Heun's Method”
New method calculates geometric Brownian motion with affine drift and its integral.
problem Calculating the distribution of geometric Brownian motion with affine drift and its integral.
method Laplace transform approach and Heun differential equation.
result Joint distribution of geometric Brownian motion with affine drift and its integral can be determined.
New integration method improves BSDE-based PDE solvers.
problem Discretization bias in standard BSDE-based solvers.
method Proposed Stratonovich-based BSDE formulation with stochastic Heun integration.
result Eliminates bias issues and outperforms EM-based variants.
Investigates the relationship between ResNets and Neural ODEs, quantifying their closeness and providing training methods.
problem Quantifying the distance between ResNet dynamics and Neural ODE solutions.
method Bounding the distance between hidden state trajectories and Neural ODE solutions, using gradient descent and Heun's method.
result Gradient descent and Heun's method can implicitly regularize ResNets towards Neural ODEs, especially for smooth residual functions.
Enhances neural network solvers for PDEs with complex boundary conditions.
problem Challenges in solving PDEs with high accuracy and complex boundary conditions.
method Integrates natural gradient optimization with numerical time-stepping schemes to enforce Dirichlet boundary conditions.
result Superior accuracy and computational efficiency of the proposed methods for solving PDEs.
Explicit solutions to the Riemann-Hilbert problem will be found realising some irreducible non-rigid local systems. The relation to isomonodromy and the sixth Painleve equation will be described. Keywords: Riemann-Hilbert problem, Painleve equations, algebraic solutions, Heun equations, tetrahedral/octahedral group, tr…
Study on higher-dimensional black holes, focusing on retractions and scalar quasibound states.
problem Examining the physics of a five-dimensional non-extremal Reissner-Nordström black hole.
method Analyzing the line element and scalar field perturbations using polynomial conditions of Heun functions.
result Obtained analytical expressions for quasibound state frequencies and discussed system stability.
We consider the eigenvalue equation for the Laplace-Beltrami operator acting on scalar functions on the non-compact Eguchi-Hanson space. The corresponding differential equation is reducible to a confluent Heun equation with Ince symbol [0,2,1_2]. We construct approximations for the eigenfunctions and their asymptotic s…
We developed efficient methods to compute gradients for Neural SDEs, improving training speed and accuracy.
problem Training Neural SDEs requires accurate and efficient computation of gradients, which is challenging due to the complexity of SDEs.
method We introduced a reversible Heun method for solving backwards-in-time SDEs and a Brownian Interval for sampling and reconstructing Brownian motion.
result Our methods significantly improve training speed and accuracy for Neural SDEs, outperforming state-of-the-art techniques.
New analysis improves convergence guarantees for diffusion-based samplers in Wasserstein distance.
problem Improving convergence guarantees for diffusion-based generative models.
method Simple framework to analyze discretization, initialization, and score estimation errors.
result First Wasserstein convergence bound for the Heun sampler and improved results for Euler sampler.
Improved sampling for Diffusion Models by accounting for covariance.
problem Sampling quality degradation in few-step Diffusion Models.
method Covariance-aware sampler using Tweedie's formula and Fourier-space decomposition.
result Consistently superior samples compared to state-of-the-art samplers.
New estimators reduce variance in training variational autoencoders with discrete latent variables.
problem Training variational autoencoders with discrete latent variables requires efficient gradient estimation.
method Introduce ReinMax-Rao and ReinMax-CV estimators using Rao-Blackwellisation and control variates.
result Demonstrate superior performance on training variational autoencoders with discrete latent spaces.
In this paper we study a proposal of Nekrasov, Rosly and Shatashvili that describes the effective twisted superpotential obtained from a class S theory geometrically as a generating function in terms of certain complexified length-twist coordinates, and extend it to higher rank. First, we introduce a higher rank analog…
Investigation of the Estimation Accuracy of 5 Different Numerical ODE Solvers on 3 Case Studiesmath.NA
Study compares 5 ODE solvers on 3 case studies, finding varying accuracy.
problem Comparing estimation accuracy of 5 ODE solvers on 3 case studies.
method Used 5 different numerical ODE solvers (Euler's, Heun's, Midpoint, Runge-Kutta 4th order, ODE45) on 3 case studies and compared their results.
result Different solvers have varying accuracy depending on the case study.
ERDM integrates rolling forecasts with diffusion models for complex dynamics.
problem Forecasting complex dynamics with rolling forecasts and diffusion models.
method Adapting EDM components for rolling forecasts, introducing novel loss weighting, efficient initialization, and hybrid architecture.
result ERDM outperforms diffusion-based baselines in 2D Navier-Stokes simulations and ERA5 weather forecasting.
Improved iterative methods for risk parity portfolio weights.
problem Solving for portfolio weights in risk parity allocation.
method Enhanced CCD and Newton methods, including a rescaling step and improved initial guess.
result Improved CCD method is the best, three times faster with 40% fewer iterations.
We describe a novel optimization method for finite sums (such as empirical risk minimization problems) building on the recently introduced SAGA method. Our method achieves an accelerated convergence rate on strongly convex smooth problems. Our method has only one parameter (a step size), and is radically simpler than o…
A new method combines Laplace and Variational Bayes for scalable inference.
problem Complex models and large datasets make exact inference infeasible.
method Low-Rank Variational Bayes Correction (VBC) using Laplace method and Variational Bayes correction in a lower dimension.
result The method ensures scalability in both model complexity and data size.
Unified framework for model explanation methods based on feature removal.
problem Unclear relationships and preferences among various model explanation methods.
method Characterizes removal-based explanations along three dimensions.
result Unified 26 existing methods, including widely used approaches.
This work reviews and evaluates methods for predicting prediction intervals in regression problems.
problem Calibration of prediction intervals in regression problems.
method Four classes of methods: Bayesian, ensemble, direct interval estimation, and conformal prediction.
result Conformal prediction can be used as a general calibration procedure.
Derives kernel PCA with Nyström method for scalability.
problem Scalability of kernel PCA.
method Nyström method for kernel PCA.
result Provides scalable alternative to full kernel PCA.
In this paper, the author considers the numerical computation of CVA for large systems by Mote Carlo methods. He introduces two types of stochastic mesh methods for the computations of CVA. In the first method, stochastic mesh method is used to obtain the future value of the derivative contracts. In the second method, …
New method combines spectral and sparse methods for Gaussian processes.
problem Efficiently fitting Gaussian processes to large datasets.
method Orthogonally decoupled variational Fourier features.
result Competitive performance on synthetic and real-world data.
A Large-Scale Comparative Analysis of Imputation Methods for Single-Cell RNA Sequencing Dataq-bio.GN
A comprehensive benchmark of 15 scRNA-seq imputation methods across various datasets and analyses.
problem Imputation of single-cell RNA sequencing data to recover latent transcriptional signals.
method Evaluation of 15 imputation methods across 30 datasets and 6 downstream analyses.
result Traditional methods generally outperform DL-based methods in scRNA-seq data analysis.
New methods using natural gradient for structured optimization.
problem Structured optimization problems.
method Structured second-order methods via natural gradient descent.
result Efficiency demonstrated on non-convex and deep learning problems.
Unified Convergence Analysis of Stochastic Momentum Methods for Convex and Non-convex Optimizationmath.OC
Recently, {\it stochastic momentum} methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of t…
We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference me…
A new method speeds up deep neural network training.
problem Nonconvex optimization in deep neural networks.
method Scaled conjugate gradient method for nonconvex optimization.
result The method converges faster and achieves lower scores in practical applications.
We propose a new stochastic dual coordinate ascent technique that can be applied to a wide range of regularized learning problems. Our method is based on Alternating Direction Multiplier Method (ADMM) to deal with complex regularization functions such as structured regularizations. Although the original ADMM is a batch…
NCG methods improve shape optimization efficiency.
problem Shape optimization problems
method Nonlinear conjugate gradient methods
result NCG methods are efficient for shape optimization
Pricing Derivatives under Multiple Stochastic Factors by Localized Radial Basis Function Methodsq-fin.CP
We propose two localized Radial Basis Function (RBF) methods, the Radial Basis Function Partition of Unity method (RBF-PUM) and the Radial Basis Function generated Finite Differences method (RBF-FD), for solving financial derivative pricing problems arising from market models with multiple stochastic factors. We demons…
Uncertainty Aware Trader-Company Method: Interpretable Stock Price Prediction Capturing Uncertaintyq-fin.CP
Proposes UTC method for stock price prediction with uncertainty quantification.
problem Lack of uncertainty estimates in stock prediction methods.
method Combines TC method with probabilistic modeling for point and uncertainty predictions.
result UTC method achieves higher returns and lower risks than baselines.
Survey of spectral, probabilistic, and deep metric learning methods.
problem Developing effective distance metrics for various machine learning tasks.
method Divided into spectral, probabilistic, and deep approaches, covering various techniques and their applications.
result Comprehensive overview of metric learning methods, including new developments and applications.
Various approaches to gene selection for cancer classification based on microarray data can be found in the literature and they may be grouped into two categories: univariate methods and multivariate methods. Univariate methods look at each gene in the data in isolation from others. They measure the contribution of a p…
A novel weighted feature selection method using fuzzy sets improves classification accuracy and stability.
problem Improving feature selection accuracy and stability in machine learning models.
method Combination of four feature selection methods using fuzzy sets and bootstrap.
result Our method achieved significantly higher stability than individual methods.
New method improves accuracy in computing implied volatility.
problem Computing implied volatility from the Black-Scholes model.
method Adaptive gradient descent optimizers for numerical computation.
result More accurate results compared to close form approximation and Newton-Raphson method.
The paper examines Wiener process for LID estimation methods.
problem Estimating local intrinsic dimension in high-dimensional datasets.
method Investigates recent LID estimation methods from a Wiener process perspective.
result Explains how methods behave under non-ideal conditions.
We evaluate how modern outlier detection methods perform in identifying outliers in e-commerce conversion rate data. Based on the limitations identified, we then present a novel method to detect outliers in e-commerce conversion rate. This unsupervised method is made more business relevant by letting it automatically a…
Spectral methods of moments provide a powerful tool for learning the parameters of latent variable models. Despite their theoretical appeal, the applicability of these methods to real data is still limited due to a lack of robustness to model misspecification. In this paper we present a hierarchical approach to methods…
We propose an optimization method for minimizing the finite sums of smooth convex functions. Our method incorporates an accelerated gradient descent (AGD) and a stochastic variance reduction gradient (SVRG) in a mini-batch setting. Unlike SVRG, our method can be directly applied to non-strongly and strongly convex prob…
We discuss the relevance of the recent Machine Learning (ML) literature for economics and econometrics. First we discuss the differences in goals, methods and settings between the ML literature and the traditional econometrics and statistics literatures. Then we discuss some specific methods from the machine learning l…
R package for counterfactual explanation methods.
problem Lack of unified interfaces for counterfactual explanation methods.
method Developed a modular R6-based interface for three existing counterfactual methods and proposed extensions.
result Comparison of implemented methods' quality and runtime behavior.
We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type methods for minimizing s…
The field of medical image reconstruction has seen roughly four types of methods. The first type tended to be analytical methods, such as filtered back-projection (FBP) for X-ray computed tomography (CT) and the inverse Fourier transform for magnetic resonance imaging (MRI), based on simple mathematical models for the …
Derives new optimization methods using variational integrators.
problem Optimization methods in machine learning.
method Variational integrators and principles of Hamilton and Lagrange-d'Alembert.
result Derives two families of optimization methods, including Nesterov's accelerated gradient method.
Krylov Cubic Regularized Newton: A Subspace Second-Order Method with Dimension-Free Convergence Ratemath.OC
A new method for faster optimization in high dimensions.
problem Slow convergence in high-dimensional optimization problems.
method Subspace cubic regularized Newton method within Krylov subspace.
result Achieves a dimension-independent convergence rate of O(1/mk + 1/k^2).
Pricing options with VG model using FFTphysics.soc-ph
We discuss various analytic and numerical methods that have been used to get option prices within a framework of the VG model. We show that some popular methods, for instance, Carr-Madan's FFT method could blow up for certain values of the model parameters even for an European vanilla option. Alternative methods - one …
The article derives a formula for predicting claims uncertainty using the GCC method.
problem Claims reserving uncertainty in non-life insurance modeling.
method Generalized Cape Cod (GCC) method, derived an analytical formula for MSEP.
result Derives an analytical formula for the mean squared error of prediction (MSEP) of the GCC method.