Study disproves a generalized numerical criterion for certain pairs.
problem Generalized numerical criterion for pairs
method Provided counterexamples
result Negative answer to the generalized numerical criterion problem
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48 results for “Numerical Analysis”
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
FiNCAT tool automatically identifies financial numerals in documents.
problem Differentiating between in-claim and out-of-claim numerals in financial documents.
method Extracts context embeddings of numerals using BERT, then uses Logistic Regression to classify.
result Achieved a Macro F1 score of 0.8223 on validation set.
Study uses neural networks to solve complex equations efficiently.
problem Solving parametric partial differential equations.
method Machine learning and deep neural networks.
result Performance of the model is independent of parameter space dimension.
Two new PCA variants improve financial data analysis.
problem Numerical instability and nonstationarity in PCA for finance.
method Iterated and exponentially weighted moving PCA variants using Ogita-Aishima iteration.
result Improved stability and adaptability in financial data analysis.
This paper performs the numerical analysis and the computation of a Spread option in a market with imperfect liquidity. The number of shares traded in the stock market has a direct impact on the stock's price. Thus, we consider a full-feedback model in which price impact is fully incorporated into the model. The price …
Improved financial sentiment analysis using simple instruction tuning of LLMs.
problem Lack of accurate financial sentiment analysis by large language models.
method Instruction tuning of general-purpose LLMs with a small portion of financial sentiment data.
result Significant improvement in financial sentiment analysis, especially in complex scenarios.
Study identifies numerical signs of blow-up in hydrodynamic equations.
problem Determining if numerical results of blow-up are genuine or artifacts.
method Geometrically consistent spatiotemporal discretization of complexified Euler equations.
result Identification of a signature based on supremum norm growth rates of vorticity.
Datasets with a mixture of numerical and categorical attributes are routinely encountered in many application domains. In this work we examine an approach to clustering such datasets using homogeneity analysis. Homogeneity analysis determines a euclidean representation of the data. This can be analyzed by leveraging th…
Conditions of Stability for explicit finite difference scheme and some results of numerical analysis for a unified 2 factor model of structural and reduced form types for corporate bonds with fixed discrete coupon are provided. It seems to be difficult to get solution formula for PDE model which generalizes Agliardi's …
Many complex systems generate multifractal time series which are long-range cross-correlated. Numerous methods have been proposed to characterize the multifractal nature of these long-range cross correlations. However, several important issues about these methods are not well understood and most methods consider only o…
Two-layer networks struggle with high frequencies due to numerical and computational limitations.
problem High frequency approximation and learning in shallow networks.
method Mathematical and computational analysis focusing on numerical error, computational cost, and stability.
result Explicit answers to fundamental computational issues in shallow networks' high frequency handling.
Lie Symmetry Classification and Numerical Analysis of KdV Equation with Power-law Nonlinearitymath.DG
In this paper, a complete Lie symmetry analysis of the damped wave equation with time-dependent coefficients is investigated. Then the invariant solutions and the exact solutions generated from the symmetries are presented. Moreover, a Lie algebraic classifications and the optimal system are discussed. Finally, using C…
We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast F…
The study analyzes numerical stability in large language models using mixed-precision arithmetic.
problem Numerical stability of large language models using low-precision arithmetic.
method Developed a mixed-precision analysis of transformer inference, deriving bounds for condition numbers and forward error.
result Established that numerical stability is determined by the interplay between weight magnitude and the growth of the residual stream.
The analysis of manifold-valued data requires efficient tools from Riemannian geometry to cope with the computational complexity at stake. This complexity arises from the always-increasing dimension of the data, and the absence of closed-form expressions to basic operations such as the Riemannian logarithm. In this pap…
FBSJNN solves PIDEs and FBSDEJs with deep learning, offering theoretical and numerical efficiency.
problem Solving Partial Integro-Differential Equations and Forward-Backward Stochastic Differential Equations with Jumps.
method FBSJNN framework using a single neural network for both solution approximation and non-local integral.
result FBSJNN achieves numerical solutions with a relative error of , demonstrating efficiency.
Research covers geometry, analysis, and integration on infinite-dimensional spaces.
problem Exploring geometric and analytical structures in infinite-dimensional settings.
method Analyzes numerical schemes, Lie groups, connections, and integration theory.
result Developed new methods for integration and analysis on infinite-dimensional manifolds.
Numerically estimates Colding-Minicozzi entropies of self-shrinkers.
problem Estimating Colding-Minicozzi entropies of self-shrinkers.
method Numerical estimation of entropies for specific self-shrinkers.
result Colding-Minicozzi entropies of -dimensional Angenent torus are decreasing with dimension.
In applications like computer aided design, geometric models are often represented numerically as polynomial splines or NURBS, even when they originate from primitive geometry. For purposes such as redesign and isogeometric analysis, it is of interest to extract information about the underlying geometry through reverse…
We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of genera…
Proves solution uniqueness for biomembrane shape prediction.
problem Proving solution uniqueness for the genus one Canham variational problem.
method Combining numeric analytic continuation and singularity analysis to prove non-negativity of a sequence.
result Proves positivity of the sequence, leading to solution uniqueness.
We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estima…
Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.
problem Analyzing stability of noncompact hypersurfaces with curvature blowup.
method Numerical overlap method to construct global solutions.
result Existence of near and far classes of initial data leading to distinct behaviors.
This paper compares analytical and numerical solutions of the Black-Scholes model.
problem Comparing analytical and numerical methods for solving the Black-Scholes model.
method Analytical solution (variable separation) and numerical solution (finite differences) of the Black-Scholes equation.
result Numerical solutions provide more accurate results for complex scenarios.
New method solves sparse PCA and CCA with guaranteed convergence.
problem Sparse PCA and CCA for large-scale data analysis.
method Alternating manifold proximal gradient method.
result Unified convergence analysis for the proposed method.
Efficient algorithm for orthogonal canonical correlation analysis (OCCA).
problem Solving the OCCA problem with orthogonality constraints.
method Sub-maximization problem with self-consistent-field (SCF) iteration for trace-fractional structure and orthogonal linear projections.
result Proposed algorithm converges globally to a KKT point and is more efficient.
Closed pricing formulas for Variance Gamma model payoffs.
problem Pricing path-independent payoffs in the Variance Gamma model.
method Mellin transform theory and multidimensional complex analysis.
result Closed-form pricing formulas with accelerated convergence for short-term options.
Local Neural Operators enable efficient system-level analysis of complex PDEs.
problem System-level analysis of large-scale dynamical systems using neural operators.
method Integrating local Neural Operators with Krylov subspace iterative methods for stability and bifurcation analysis.
result Demonstrated effectiveness of local Neural Operators in fixed-point, stability, and bifurcation analysis of nonlinear PDEs.
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.
A novel robust meta-analysis model using the distribution for outlier accommodation and detectionstat.ME
A new meta-analysis model detects and accommodates outliers.
problem Outliers in meta-analysis studies can skew results.
method Proposes a novel Meta model using the distribution for robustness.
result Demonstrates superior performance in detecting and accommodating outliers.
In this article, we propose a new numerical approach to high-dimensional partial differential equations (PDEs) arising in the valuation of exotic derivative securities. The proposed method is extended from Reisinger and Wittum (2007) and uses principal component analysis (PCA) of the underlying process in combination w…
The ratio of two probability densities can be used for solving various machine learning tasks such as covariate shift adaptation (importance sampling), outlier detection (likelihood-ratio test), and feature selection (mutual information). Recently, several methods of directly estimating the density ratio have been deve…
An analysis of Uniswap marketsq-fin.TR
Uniswap markets perform well and closely track reference prices.
problem Understanding and validating the performance of Uniswap markets.
method Formal analysis and numerical simulation of constant product markets.
result Uniswap markets closely track reference prices under common conditions.
Optimizes over flag manifolds for numerical PDE and statistics.
problem Optimizing over flag manifolds for numerical PDE and statistics.
method Develops tools for Riemannian optimization on flag manifolds, deriving analytic expressions and parameterizations.
result Closed-form analytic expressions and parameterizations for various geometric objects on flag manifolds.
Asymptotic Analysis of Sampling Estimators for Randomized Numerical Linear Algebra Algorithmsmath.ST
Develops asymptotic analysis for RandNLA sampling estimators in least-squares problems.
problem Lack of distributional information for RandNLA estimators in statistical inference.
method Asymptotic analysis of sampling estimators for least-squares problems in two settings.
result Sampling estimators are asymptotically normally distributed under mild conditions.
The authors aim to develop numerical schemes of the two representative quadratic hedging strategies: locally risk minimizing and mean-variance hedging strategies, for models whose asset price process is given by the exponential of a normal inverse Gaussian process, using the results of Arai et al. \cite{AIS}, and Arai …
This study connects Gaussian processes and RKHS, bridging two machine learning communities.
problem Understanding the relationship between Gaussian processes and RKHS.
method Examining connections and equivalences in regression, interpolation, and other topics.
result Established the equivalence between Gaussian Hilbert space and RKHS.
Study on queues with Hawkes arrivals, proving steady-state behavior and developing an efficient algorithm.
problem Analyzing the steady-state behavior of queues with Hawkes arrivals.
method Novel coupling techniques and exponential convergence results for workload and busy period processes.
result Exponential convergence of queueing processes to their stationary distribution.
In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints -- we recall…
We develop a new method to solve complex physics equations more accurately and efficiently.
problem Challenges in solving functional differential equations due to high computational costs and inaccurate approximations.
method Combining physics-informed neural networks (PINNs) with cylindrical approximation to handle functional derivatives.
result Our method achieves typical relative error orders of PINNs of on two FDEs.
A numerical algorithm for fully nonlinear HJB equations: an approach by control randomizationmath.PR
We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [9] for representing fully nonlinear HJB equations. In particular, this allows us to numerically solve stochastic control problems with controlled volatility,…
Study proposes Local Linear Encoding for better feature discretization.
problem Improving feature discretization for numeric data.
method Theoretical analysis and Local Linear Encoding (LLE) method.
result LLE outperforms conventional methods with fewer parameters.
Numerical valuation of American basket options via partial differential complementarity problemsmath.NA
Study efficient numerical methods for American basket options.
problem Valuation of American basket options.
method Partial differential complementarity problems (PDCPs) and efficient discretization.
result Approximations of American basket options are close and converge favourably.
Develops a new method for quantizing rough volatility for volatility derivatives pricing.
problem Pricing volatility derivatives in rough volatility models.
method Functional quantization of rough volatility using offline computable quantizers.
result Pricing VIX Futures in the rough Bergomi model shows competitive results.
Improved numerical solution for BSDEs with reduced boundary errors.
problem Boundary errors in numerical solution of BSDEs.
method Modified damping and shifting schemes to transform target function into a bounded periodic function, applying Fourier transforms.
result Significant reduction in boundary errors with improved accuracy and convergence.
In the recent years, Riemannian shape analysis of curves and surfaces has found several applications in medical image analysis. In this paper we present a numerical discretization of second order Sobolev metrics on the space of regular curves in Euclidean space. This class of metrics has several desirable mathematical …
A new PCA method using T-norm outperforms existing methods.
problem Outliers and noise sensitivity in classical PCA.
method PCA based on T-norm maximization.
result The method outperforms PCA-, SPCA, and PCA in numerical experiments.