The paper applies deep learning to solve complex control problems.
problem Solving stochastic control problems on finite horizons.
method Deep neural networks algorithms for control learning.
result Performance of algorithms on various control problems demonstrated.
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation …
TMM method improves risk measure computation in finance.
problem Computing risk measures in finance efficiently and accurately.
method Transport-based Mesh-free Method (TMM) using transportation and reproducing kernels.
result Sharp convergence rates and optimal computational times for risk measures.
New method estimates Schrödinger bridges using ML techniques.
problem Finding most likely stochastic evolution between two distributions.
method Equivalence with maximum likelihood estimation, numerical Gaussian process approach.
result Direct application of ML techniques for SBP estimation.
A new scheme for FBSDEs simplifies computation without Monte Carlo.
problem Numerical solution for decoupled FBSDEs with reduced complexity.
method Recursive marginal quantization for fully quantization-based scheme.
result Effective numerical procedure for financial applications.
Paper introduces efficient orthonormal transformations using Householder reflectors.
problem Fast numerical procedures for orthonormal transformations are computationally expensive.
method Develops orthonormal matrices using Householder reflectors to approximate any orthonormal or symmetric transform.
result Approximations of orthonormal or symmetric transforms using a few Householder reflectors are accurate and computationally efficient.
We introduce an additive stochastic mortality model which allows joint modelling and forecasting of underlying death causes. Parameter families for mortality trends can be chosen freely. As model settings become high dimensional, Markov chain Monte Carlo (MCMC) is used for parameter estimation. We then link our propose…
We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a di…
Paper introduces a new method for solving complex stochastic equations.
problem Solving forward-backward stochastic differential equations with jumps.
method Linear basis function regression technique.
result The proposed method is convergent and effective as shown by numerical experiments.
Deep learning improves air pollution forecasting and monitoring.
problem Limited applicability of deep learning in air pollution forecasting due to traditional PDE solvers.
method Combines deep-learning and domain-decomposition techniques for air pollution monitoring and forecasting.
result Reduces run-time by two orders of magnitude and extends model deployment beyond trained domains.
New method models dewetting of anisotropic particles using numerical techniques.
problem Modeling dewetting dynamics of particles with varying surface energies.
method Level set numerical approach with convolution kernels to handle anisotropic interfacial energies.
result Validated numerical scheme supports merging and splitting of interfaces.
New method combines ODE filters and numerical quadrature to propagate model uncertainty.
problem Propagation of model uncertainty in ODE solutions with uncertain parameters.
method Combining ODE filters with numerical quadrature.
result Effective propagation of both numerical and parametric uncertainty.
Probabilistic numerics expands numerical tasks with black box methods.
problem Difficult conditioning of random variables in numerical tasks.
method Construct probabilistic numerical methods based on final outputs, extrapolating limiting quantities.
result Higher orders of convergence achieved in various numerical tasks.
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…
Article reviews the history and development of probabilistic numerics.
problem Understanding the historical context and modern treatment of probabilistic numerics.
method Historical review and analysis of contributions from Sul'din and Larkin.
result Early ideas have matured and are now part of modern formal treatment.
New algorithm improves on existing methods for solving transport problems.
problem Finding a map to transport one distribution to another.
method Iterative Markovian Fitting (IMF) and Diffusion Schrödinger Bridge Matching (DSBM).
result DSBM significantly improves over previous SB numerics and recovers various transport methods.
The paper analyzes numerical instability in variational flows and proposes a diagnostic method.
problem Numerical instability in variational flows affects sampling, density evaluation, and ELBO estimation.
method Treated variational flows as dynamical systems, used shadowing theory for theoretical guarantees, and developed a diagnostic procedure.
result Despite numerical instability, results from variational flows can be accurate enough for practical applications.
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 spread option pricing with a new approximation method.
problem Inaccuracies in the original Kirk's formula for high correlation cases.
method Developed a new approximation method for spread option pricing.
result The Modified Kirk's Approximation method is extremely accurate and improves upon Kirk's approach.
A non-parametric method for evaluation of the aggregate loss distribution (ALD) by combining and numerically inverting the empirical characteristic functions (CFs) is presented and illustrated. This approach to evaluate ALD is based on purely non-parametric considerations, i.e., based on the empirical CFs of frequency …
New numerical method for pricing barrier options with continuous monitoring.
problem Pricing barrier options with continuous monitoring of underlying asset.
method Developed a numerical scheme to calculate fluctuation identities for exponential Lévy processes.
result Error analysis shows continuous monitoring limits discretely monitored scheme's accuracy.
A knot's thickness is measured by its β invariant, a new numerical invariant.
problem Measuring the thickness of knots.
method Introduced a new invariant β(K) and proved an inequality between β(K) and knot Floer thickness.
result All Montesinos knots have thickness at most one.
Paper adapts numerical scheme for parallel transport on diffeomorphism manifolds.
problem Efficient computation of parallel transport on high-dimensional manifold-valued data.
method Adapts a numerical scheme for parallel transport to finite-dimensional manifolds of diffeomorphisms.
result Qualitative and quantitative analysis of scheme's behavior on high-dimensional manifolds.
Improved stability for large-scale Bayesian sampling.
problem Reducing instability in Langevin dynamics for large datasets.
method Introducing a modified CCAdL thermostat with a scaling and squaring method and a truncated Taylor series approximation.
result Significantly improved numerical stability and accuracy over existing methods.
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 …
RandNLA uses randomness for matrix problems in machine learning.
problem Matrix problems in machine learning.
method Randomized Numerical Linear Algebra.
result New challenges in RandNLA due to hardware trends and advances in ML.
The paper analyzes contraction rates for GP regression approximations.
problem Computational infeasibility of exact GP posterior in large-scale applications.
method Lanczos and conjugate gradient approximations of the posterior mean.
result Minimax contraction rates for these approximations in large-scale applications.
The main results of A. Zorich and I. Dynnikov about plane sections of periodic surfaces are extended to the PL case. As an application, the Stereographic Map of a truncated octahedron, extended to the whole $\Rt$ by periodicity, is analyzed numerically.
Develops efficient methods for approximating densities of financial models with jumps.
problem Approximating densities of affine jump diffusions with state-independent jump intensities.
method Recursive approach for deriving closed-form solutions to moments, constructing density approximations via moment matching.
result Superior computational efficiency and precision in option pricing and simulation compared to existing techniques.
High order splitting schemes with complex timesteps are applied to Kolmogorov backward equations stemming from stochastic differential equations in Stratonovich form. In the setting of weighted spaces, the necessary analyticity of the split semigroups can be easily proved. A numerical example from interest rate theory,…
In numerous applicative contexts, data are too rich and too complex to be represented by numerical vectors. A general approach to extend machine learning and data mining techniques to such data is to really on a dissimilarity or on a kernel that measures how different or similar two objects are. This approach has been …
Solves complex Plateau problem in higher dimensions using numerical invariants.
problem Complex Plateau problem in higher dimensions.
method Generalized Yau's conjecture to higher dimensions and proved it for local complete intersection singularities.
result Solved complex Plateau problem for any dimension n≥3. Clustering is one of the most common unsupervised learning tasks in machine learning and data mining. Clustering algorithms have been used in a plethora of applications across several scientific fields. However, there has been limited research in the clustering of point patterns - sets or multi-sets of unordered elemen…
This paper tackles Bayesian system identification with probabilistic numerical methods.
problem Accurately modeling nonlinear dynamic systems from noisy data.
method Probabilistic Sequential Monte Carlo (SMC) combined with probabilistic numerical integration.
result Efficient identification of latent states and system parameters from noisy measurements.
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 …
We study numerical restricted volumes of (1,1) classes on compact Kahler manifolds, as introduced by Boucksom. Inspired by work of Ein-Lazarsfeld-Mustata-Nakamaye-Popa on restricted volumes of line bundles on projective manifolds, we pose a natural conjecture to the effect that irreducible components of the non-Kahler …
Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss them have varied considerably from one area to another. Over the last five year…
The paper stabilizes invertible neural networks by using Gaussian mixture models.
problem Invertible neural networks can have exploding Lipschitz constants, leading to numerical errors.
method The authors use Gaussian mixture models to stabilize the latent distribution of invertible neural networks.
result Numerical simulations confirm that this modification improves sampling quality in multimodal applications.
Combines neural networks with splitting-up method for filtering equations.
problem Approximating the solution of filtering equations for signal processes.
method Combines splitting-up method with neural networks.
result Produces an approximation of the unnormalised conditional distribution.
A hybrid method combines model-based and data-driven approaches for multiscale constitutive responses.
problem High computational costs and inaccuracies in nonlinear multiscale methods.
method Hybrid methodology combining model-based constitutive laws, data-driven corrections, and computational multiscale approaches.
result Model-data-driven approach improves macroscale simulations with similar accuracy and computational cost.
Machine learning enhances ocean studies by analyzing sparse, multi-scale data.
problem Sparse, multi-scale ocean data limits traditional methods.
method Machine learning techniques applied to ocean observations, theory, and numerical simulations.
result ML can advance theoretical oceanographic exploration and numerical simulations.
latrend simplifies longitudinal clustering for numeric measurements.
problem Clustering of longitudinal data to identify common trends over time.
method Unified framework for applying various clustering methods.
result Facilitates comparison and rapid prototyping of new methods.
xVal tokenizes numbers continuously for better scientific model training.
problem Lack of continuous numerical tokenization for scientific datasets in LLMs.
method xVal: Continuous numerical tokenization strategy.
result xVal outperforms other numerical tokenization methods on scientific datasets.
Paper tackles missing data in medical records using advanced optimization.
problem Missing data in medical records hinders machine learning applications.
method Applied multi-level stochastic optimization for imputation.
result The method significantly outperforms current approaches and reduces error up to 75%.
New model combines physics and machine learning for ocean dynamics.
problem Discovering hidden laws governing ocean dynamics.
method Develops Deep Neural Numerical Models (DNNMs) to learn hidden variables of physical laws.
result Illustrates DNNMs applied to Sea Surface Height dynamics, connecting to QG model.
A research frontier has emerged in scientific computation, wherein numerical error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly dete…
This paper develops numerical methods for finding optimal dividend pay-out and reinsurance policies. A generalized singular control formulation of surplus and discounted payoff function are introduced, where the surplus is modeled by a regime-switching process subject to both regular and singular controls. To approxima…
New efficient method for inverse Z-transform reduces complexity significantly.
problem Efficient numerical realization of inverse Z-transform for large n.
method Derives sufficient conditions for new scheme, applies to option pricing.
result Significant reduction in complexity for large n, especially for European options.