AMFR-W Numerical Methods for Solving High-Dimensional SABR/LIBOR PDE Models

2021 ◽  
Vol 43 (1) ◽  
pp. B30-B54
Author(s):  
J. G. López-Salas ◽  
S. Pérez-Rodríguez ◽  
C. Vázquez
Keyword(s):  
Numerical Methods ◽  
High Dimensional

scholarly journals A machine learning framework for solving high-dimensional mean field game and mean field control problems

2020 ◽  
Vol 117 (17) ◽  
pp. 9183-9193
Author(s):  
Lars Ruthotto ◽  
Stanley J. Osher ◽  
Wuchen Li ◽  
Levon Nurbekyan ◽  
Samy Wu Fung
Keyword(s):  
Machine Learning ◽  
Numerical Methods ◽  
Mean Field ◽  
Two Dimensions ◽  
High Dimensional ◽  
Mean Field Games ◽  
Spatial Discretization ◽  
Learning Framework ◽  
Field Control ◽  
Crowd Motion

Mean field games (MFG) and mean field control (MFC) are critical classes of multiagent models for the efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State-of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse of dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton–Jacobi–Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using a Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that are beyond reach with existing numerical methods.

Tensor Numerical Methods: Actual Theory and Recent Applications

2019 ◽  
Vol 19 (1) ◽  
pp. 1-4 ◽  
Author(s):  
Ivan Gavrilyuk ◽  
Boris N. Khoromskij
Keyword(s):  
Numerical Modeling ◽  
Big Data ◽  
Data Analysis ◽  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Solution ◽  
Large Data ◽  
High Dimensional ◽  
Data Sets ◽  
Integral Differential Equations

AbstractMost important computational problems nowadays are those related to processing of the large data sets and to numerical solution of the high-dimensional integral-differential equations. These problems arise in numerical modeling in quantum chemistry, material science, and multiparticle dynamics, as well as in machine learning, computer simulation of stochastic processes and many other applications related to big data analysis. Modern tensor numerical methods enable solution of the multidimensional partial differential equations (PDE) in {\mathbb{R}^{d}} by reducing them to one-dimensional calculations. Thus, they allow to avoid the so-called “curse of dimensionality”, i.e. exponential growth of the computational complexity in the dimension size d, in the course of numerical solution of high-dimensional problems. At present, both tensor numerical methods and multilinear algebra of big data continue to expand actively to further theoretical and applied research topics. This issue of CMAM is devoted to the recent developments in the theory of tensor numerical methods and their applications in scientific computing and data analysis. Current activities in this emerging field on the effective numerical modeling of temporal and stationary multidimensional PDEs and beyond are presented in the following ten articles, and some future trends are highlighted therein.

Stochastic Collocation vial1-Minimisation on Low Discrepancy Point Sets with Application to Uncertainty Quantification

2016 ◽  
Vol 6 (2) ◽  
pp. 171-191 ◽  
Author(s):  
Yongle Liu ◽  
Ling Guo
Keyword(s):  
Numerical Methods ◽  
Complex Systems ◽  
Collocation Method ◽  
Random Sampling ◽  
Target Function ◽  
High Dimensional ◽  
Stochastic Collocation ◽  
Point Sets ◽  
Ode Model ◽  
Better Than

AbstractVarious numerical methods have been developed in order to solve complex systems with uncertainties, and the stochastic collocation method usingl1-minimisation on low discrepancy point sets is investigated here. Halton and Sobol' sequences are considered, and low discrepancy point sets and random points are compared. The tests discussed involve a given target function in polynomial form, high-dimensional functions and a random ODE model. Our numerical results show that the low discrepancy point sets perform as well or better than random sampling for stochastic collocation vial1-minimisation.

The formulation and analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein–Gordon equations

2018 ◽  
Vol 39 (4) ◽  
pp. 2016-2044 ◽  
Author(s):  
Bin Wang ◽  
Xinyuan Wu
Keyword(s):  
Numerical Methods ◽  
Global Convergence ◽  
Nonlinear Stability ◽  
Numerical Experiments ◽  
High Dimensional ◽  
Local Error ◽  
Discrete Gradient Method ◽  
New Energy ◽  
Klein Gordon ◽  
Duhamel Principle

Abstract In this paper we focus on the analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein–Gordon equations. A novel energy-preserving scheme is developed based on the discrete gradient method and the Duhamel principle. The local error, global convergence and nonlinear stability of the new scheme are analysed in detail. Numerical experiments are implemented to compare with existing numerical methods in the literature, and the numerical results show the remarkable efficiency of the new energy-preserving scheme presented in this paper.

scholarly journals Variational principle for time-periodic quantum systems

2020 ◽  
Vol 75 (10) ◽  
pp. 855-861
Author(s):  
Nils Krüger
Keyword(s):  
Variational Principle ◽  
Numerical Methods ◽  
Solvable Model ◽  
Quantum Systems ◽  
Time Dependent ◽  
High Dimensional ◽  
Benchmark System ◽  
Time Periodic ◽  
Very High ◽  
Floquet States

AbstractA variational principle enabling one to compute individual Floquet states of a periodically time-dependent quantum system is formulated, and successfully tested against the benchmark system provided by the analytically solvable model of a linearly driven harmonic oscillator. The principle is particularly well suited for tracing individual Floquet states through parameter space, and may allow one to obtain Floquet states even for very high-dimensional systems which cannot be treated by the known standard numerical methods.

scholarly journals Solving High-Dimensional Problems in Statistical Modelling: A Comparative Study

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1806
Author(s):  
Stamatis Choudalakis ◽  
Marilena Mitrouli ◽  
Athanasios Polychronou ◽  
Paraskevi Roupa
Keyword(s):  
Parameter Estimation ◽  
Numerical Methods ◽  
Comparative Study ◽  
Real Data ◽  
Statistical Modelling ◽  
High Dimensional ◽  
Minimum Norm ◽  
Supersaturated Designs ◽  
Extensive Analysis ◽  
Selection For

In this work, we present numerical methods appropriate for parameter estimation in high-dimensional statistical modelling. The solution of these problems is not unique and a crucial question arises regarding the way that a solution can be found. A common choice is to keep the corresponding solution with the minimum norm. There are cases in which this solution is not adequate and regularisation techniques have to be considered. We classify specific cases for which regularisation is required or not. We present a thorough comparison among existing methods for both estimating the coefficients of the model which corresponds to design matrices with correlated covariates and for variable selection for supersaturated designs. An extensive analysis for the properties of design matrices with correlated covariates is given. Numerical results for simulated and real data are presented.

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