Workshop: Numerical Analysis of Stochastic Partial Differential Equations (NASPDE), in Network Eurandom at Eindhoven University of Technology, May 16, 2023, about my recent works (i) "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" (link: https://doi.org/10.1080/14697688.2022.2135455), and (ii) "Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities" (link: https://arxiv.org/abs/2003.05708).
Numerical Smoothing and Hierarchical Approximations for E cient Option Pricin...Chiheb Ben Hammouda
My talk at the "Stochastic Numerics and Statistical Learning: Theory and Applications" Workshop at KAUST (King Abdullah University of Science and Technology), May 23, 2022, about my recent works "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" and "Multilevel Monte Carlo combined with numerical smoothing for robust and efficient option pricing and density estimation".
My talk entitled "Numerical Smoothing and Hierarchical Approximations for Efficient Option Pricing and Density Estimation", that I gave at the "International Conference on Computational Finance (ICCF)", Wuppertal June 6-10, 2022. The talk is related to our recent works "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" (link: https://arxiv.org/abs/2111.01874) and "Multilevel Monte Carlo combined with numerical smoothing for robust and efficient option pricing and density estimation" (link: https://arxiv.org/abs/2003.05708). In these two works, we introduce the numerical smoothing technique that improves the regularity of observables when approximating expectations (or the related integration problems). We provide a smoothness analysis and we show how this technique leads to better performance for the different methods that we used (i) adaptive sparse grids, (ii) Quasi-Monte Carlo, and (iii) multilevel Monte Carlo. Our applications are option pricing and density estimation. Our approach is generic and can be applied to solve a broad class of problems, particularly for approximating distribution functions, financial Greeks computation, and risk estimation.
My talk at the International Conference on Monte Carlo Methods and Applications (MCM2032) related to advances in mathematical aspects of stochastic simulation and Monte Carlo methods at Sorbonne Université June 28, 2023, about my recent works (i) "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" (link: https://doi.org/10.1080/14697688.2022.2135455), and (ii) "Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities" (link: https://arxiv.org/abs/2003.05708).
My talk at the "15th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing " MCQMC conference at Johannes Kepler Universität Linz, July 20, 2022, about my recent works "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" and "Multilevel Monte Carlo combined with numerical smoothing for robust and efficient option pricing and density estimation."
Hierarchical Deterministic Quadrature Methods for Option Pricing under the Ro...Chiheb Ben Hammouda
Conference talk at the SIAM Conference on Financial Mathematics and Engineering, held in virtual format, June 1-4 2021, about our recently published work "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model".
- Link of the paper: https://www.tandfonline.com/doi/abs/10.1080/14697688.2020.1744700
Numerical smoothing and hierarchical approximations for efficient option pric...Chiheb Ben Hammouda
1. The document presents a numerical smoothing technique to improve the efficiency of option pricing and density estimation when analytic smoothing is not possible.
2. The technique involves numerically determining discontinuities in the integrand and computing the integral only over the smooth regions. It also uses hierarchical representations and Brownian bridges to reduce the effective dimension of the problem.
3. The numerical smoothing approach outperforms Monte Carlo methods for high dimensional cases and improves the complexity of multilevel Monte Carlo from O(TOL^-2.5) to O(TOL^-2 log(TOL)^2).
This document discusses two techniques for improving the efficiency of option pricing methods: numerical smoothing and optimal damping. Numerical smoothing involves approximating non-smooth option payoffs with smooth functions, allowing for faster convergence of quadrature and multilevel Monte Carlo methods. Optimal damping adds a regularization term when pricing options under Lévy models using Fourier methods to improve stability and accuracy. The document outlines the theoretical underpinnings of how smoothness affects integration error and complexity, and presents numerical results demonstrating the effectiveness of these techniques.
This document summarizes a research paper about using hierarchical deterministic quadrature methods for option pricing under the rough Bergomi model. It discusses the rough Bergomi model and challenges in pricing options under this model numerically. It then describes the methodology used, which involves analytic smoothing, adaptive sparse grids quadrature, quasi Monte Carlo, and coupling these with hierarchical representations and Richardson extrapolation. Several figures are included to illustrate the adaptive construction of sparse grids and simulation of the rough Bergomi dynamics.
Numerical Smoothing and Hierarchical Approximations for E cient Option Pricin...Chiheb Ben Hammouda
My talk at the "Stochastic Numerics and Statistical Learning: Theory and Applications" Workshop at KAUST (King Abdullah University of Science and Technology), May 23, 2022, about my recent works "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" and "Multilevel Monte Carlo combined with numerical smoothing for robust and efficient option pricing and density estimation".
My talk entitled "Numerical Smoothing and Hierarchical Approximations for Efficient Option Pricing and Density Estimation", that I gave at the "International Conference on Computational Finance (ICCF)", Wuppertal June 6-10, 2022. The talk is related to our recent works "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" (link: https://arxiv.org/abs/2111.01874) and "Multilevel Monte Carlo combined with numerical smoothing for robust and efficient option pricing and density estimation" (link: https://arxiv.org/abs/2003.05708). In these two works, we introduce the numerical smoothing technique that improves the regularity of observables when approximating expectations (or the related integration problems). We provide a smoothness analysis and we show how this technique leads to better performance for the different methods that we used (i) adaptive sparse grids, (ii) Quasi-Monte Carlo, and (iii) multilevel Monte Carlo. Our applications are option pricing and density estimation. Our approach is generic and can be applied to solve a broad class of problems, particularly for approximating distribution functions, financial Greeks computation, and risk estimation.
My talk at the International Conference on Monte Carlo Methods and Applications (MCM2032) related to advances in mathematical aspects of stochastic simulation and Monte Carlo methods at Sorbonne Université June 28, 2023, about my recent works (i) "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" (link: https://doi.org/10.1080/14697688.2022.2135455), and (ii) "Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities" (link: https://arxiv.org/abs/2003.05708).
My talk at the "15th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing " MCQMC conference at Johannes Kepler Universität Linz, July 20, 2022, about my recent works "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" and "Multilevel Monte Carlo combined with numerical smoothing for robust and efficient option pricing and density estimation."
Hierarchical Deterministic Quadrature Methods for Option Pricing under the Ro...Chiheb Ben Hammouda
Conference talk at the SIAM Conference on Financial Mathematics and Engineering, held in virtual format, June 1-4 2021, about our recently published work "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model".
- Link of the paper: https://www.tandfonline.com/doi/abs/10.1080/14697688.2020.1744700
Numerical smoothing and hierarchical approximations for efficient option pric...Chiheb Ben Hammouda
1. The document presents a numerical smoothing technique to improve the efficiency of option pricing and density estimation when analytic smoothing is not possible.
2. The technique involves numerically determining discontinuities in the integrand and computing the integral only over the smooth regions. It also uses hierarchical representations and Brownian bridges to reduce the effective dimension of the problem.
3. The numerical smoothing approach outperforms Monte Carlo methods for high dimensional cases and improves the complexity of multilevel Monte Carlo from O(TOL^-2.5) to O(TOL^-2 log(TOL)^2).
This document discusses two techniques for improving the efficiency of option pricing methods: numerical smoothing and optimal damping. Numerical smoothing involves approximating non-smooth option payoffs with smooth functions, allowing for faster convergence of quadrature and multilevel Monte Carlo methods. Optimal damping adds a regularization term when pricing options under Lévy models using Fourier methods to improve stability and accuracy. The document outlines the theoretical underpinnings of how smoothness affects integration error and complexity, and presents numerical results demonstrating the effectiveness of these techniques.
This document summarizes a research paper about using hierarchical deterministic quadrature methods for option pricing under the rough Bergomi model. It discusses the rough Bergomi model and challenges in pricing options under this model numerically. It then describes the methodology used, which involves analytic smoothing, adaptive sparse grids quadrature, quasi Monte Carlo, and coupling these with hierarchical representations and Richardson extrapolation. Several figures are included to illustrate the adaptive construction of sparse grids and simulation of the rough Bergomi dynamics.
Talk of Michael Samet, entitled "Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models" at the International Conference on Computational Finance (ICCF)", Wuppertal June 6-10, 2022
This document introduces a generalized method for constructing sub-quadratic complexity multipliers for finite fields of characteristic 2. It begins by reintroducing the Winograd short convolution algorithm in the context of polynomial multiplication. It then presents a recursive construction technique that extends any d-point multiplier into an n=dk-point multiplier with sub-quadratic area and logarithmic delay complexity. Several new constructions are obtained using this technique, one of which is identical to the Karatsuba multiplier. The techniques aim to develop bit-parallel multipliers with better time and/or space complexity than the traditional quadratic complexity approaches.
This document presents a framework for efficiently pricing multi-asset options using Fourier methods. It discusses using the Fourier transform to map option pricing problems to frequency space, where the integrand may have better regularity. A damping parameter is introduced to ensure the transformed functions have sufficient decay at infinity. However, literature provides no guidance on choosing optimal damping parameters. The document proposes a method called Optimal Damping with Hierarchical Adaptive Quadrature to select damping parameters that improve the convergence rate of quadrature pricing methods in Fourier space. It applies this method to price options under various multi-dimensional models in numerical experiments.
Efficient Fourier Pricing of Multi-Asset Options: Quasi-Monte Carlo & Domain ...Chiheb Ben Hammouda
My talk at ICCF24 with abstract: Efficiently pricing multi-asset options poses a significant challenge in quantitative finance. While the Monte Carlo (MC) method remains a prevalent choice, its slow convergence rate can impede practical applications. Fourier methods, leveraging the knowledge of the characteristic function, have shown promise in valuing single-asset options but face hurdles in the high-dimensional context. This work advocates using the randomized quasi-MC (RQMC) quadrature to improve the scalability of Fourier methods with high dimensions. The RQMC technique benefits from the smoothness of the integrand and alleviates the curse of dimensionality while providing practical error estimates. Nonetheless, the applicability of RQMC on the unbounded domain, $\mathbb{R}^d$, requires a domain transformation to $[0,1]^d$, which may result in singularities of the transformed integrand at the corners of the hypercube, and deteriorate the rate of convergence of RQMC. To circumvent this difficulty, we design an efficient domain transformation procedure based on the derived boundary growth conditions of the integrand. This transformation preserves the sufficient regularity of the integrand and hence improves the rate of convergence of RQMC. To validate this analysis, we demonstrate the efficiency of employing RQMC with an appropriate transformation to evaluate options in the Fourier space for various pricing models, payoffs, and dimensions. Finally, we highlight the computational advantage of applying RQMC over quadrature methods in the Fourier domain, and over the MC method in the physical domain for options with up to 15 assets.
This document discusses automatic Bayesian cubature for numerical integration. It begins with an introduction to multivariate integration and the challenges it poses. It then describes an automatic cubature algorithm that generates sample points and computes error bounds iteratively until a tolerance threshold is met. Next, it covers Bayesian cubature, which treats integrands as random functions to obtain probabilistic error bounds. It defines a Bayesian trio identity relating the integration error to discrepancies, variations, and alignments. The document concludes with discussions of future work.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
1) The document discusses wavelet transforms as a recent algorithm for image compression. Wavelet transforms can capture variations at different scales in an image, making them well-suited for reducing spatial redundancy.
2) A typical lossy image compression system uses four main components - source encoding, thresholding, quantization, and entropy encoding - to achieve compression by removing different types of redundancy in images.
3) Experimental results on the Lena test image showed that soft thresholding followed by quantization achieved higher peak signal-to-noise ratios than hard thresholding and quantization, demonstrating the effectiveness of wavelet transforms for image compression.
This document summarizes research on computing stochastic partial differential equations (SPDEs) using an adaptive multi-element polynomial chaos method (MEPCM) with discrete measures. Key points include:
1) MEPCM uses polynomial chaos expansions and numerical integration to compute SPDEs with parametric uncertainty.
2) Orthogonal polynomials are generated for discrete measures using various methods like Vandermonde, Stieltjes, and Lanczos.
3) Numerical integration is tested on discrete measures using Genz functions in 1D and sparse grids in higher dimensions.
4) The method is demonstrated on the KdV equation with random initial conditions. Future work includes applying these techniques to SPDEs driven
MVPA with SpaceNet: sparse structured priorsElvis DOHMATOB
The GraphNet (aka S-Lasso), as well as other “sparsity + structure” priors like TV (Total-Variation), TV-L1, etc., are not easily applicable to brain data because of technical problems
relating to the selection of the regularization parameters. Also, in
their own right, such models lead to challenging high-dimensional optimization problems. In this manuscript, we present some heuristics for speeding up the overall optimization process: (a) Early-stopping, whereby one halts the optimization process when the test score (performance on leftout data) for the internal cross-validation for model-selection stops improving, and (b) univariate feature-screening, whereby irrelevant (non-predictive) voxels are detected and eliminated before the optimization problem is entered, thus reducing the size of the problem. Empirical results with GraphNet on real MRI (Magnetic Resonance Imaging) datasets indicate that these heuristics are a win-win strategy, as they add speed without sacrificing the quality of the predictions. We expect the proposed heuristics to work on other models like TV-L1, etc.
This document summarizes techniques for approximating marginal likelihoods and Bayes factors, which are important quantities in Bayesian inference. It discusses Geyer's 1994 logistic regression approach, links to bridge sampling, and how mixtures can be used as importance sampling proposals. Specifically, it shows how optimizing the logistic pseudo-likelihood relates to the bridge sampling optimal estimator. It also discusses non-parametric maximum likelihood estimation based on simulations.
This document presents an efficient convex hull algorithm for finding the convex hull of a planar set of points. The algorithm partitions the set of points into four regions based on the minimum and maximum x and y coordinates. It then finds the convex hull parts belonging to each region in parallel. Those parts are merged to derive the full convex hull. For each region, the algorithm uses a modified gradient concept to efficiently process the points and find the boundary points of the convex hull part. The algorithm achieves parallelism, data reduction, and has lower computational cost compared to traditional interior points algorithms. However, its drawbacks include difficulty extending it to higher dimensions and its static nature which requires the entire dataset from the beginning.
The document discusses uncertainty quantification (UQ) using quasi-Monte Carlo (QMC) integration methods. It introduces parametric operator equations for modeling input uncertainty in partial differential equations. Both forward and inverse UQ problems are considered. QMC methods like interlaced polynomial lattice rules are discussed for approximating high-dimensional integrals arising in UQ, with convergence rates superior to standard Monte Carlo. Algorithms for single-level and multilevel QMC are presented for solving forward and inverse UQ problems.
A generalized class of normalized distance functions called Q-Metrics is described in this presentation. The Q-Metrics approach relies on a unique functional, using a single bounded parameter (Lambda), which characterizes the conventional distance functions in a normalized per-unit metric space. In addition to this coverage property, a distinguishing and extremely attractive characteristic of the Q-Metric function is its low computational complexity. Q-Metrics satisfy the standard metric axioms. Novel networks for classification and regression tasks are defined and constructed using Q-Metrics. These new networks are shown to outperform conventional feed forward back propagation networks with the same size when tested on real data sets.
A generalized class of normalized distance functions called Q-Metrics is described in this presentation. The Q-Metrics approach relies on a unique functional, using a single bounded parameter Lambda, which characterizes the conventional distance functions in a normalized per-unit metric space. In addition to this coverage property, a distinguishing and extremely attractive characteristic of the Q-Metric function is its low computational complexity. Q-Metrics satisfy the standard metric axioms. Novel networks for classification and regression tasks are defined and constructed using Q-Metrics. These new networks are shown to outperform conventional feed forward back propagation networks with the same size when tested on real data sets.
This document discusses various methods for approximating marginal likelihoods and Bayes factors, including:
1. Geyer's 1994 logistic regression approach for approximating marginal likelihoods using importance sampling.
2. Bridge sampling and its connection to Geyer's approach. Optimal bridge sampling requires knowledge of unknown normalizing constants.
3. Using mixtures of importance distributions and the target distribution as proposals to estimate marginal likelihoods through Rao-Blackwellization. This connects to bridge sampling estimates.
4. The document discusses various methods for approximating marginal likelihoods and comparing hypotheses using Bayes factors. It outlines the historical development and connections between different approximation techniques.
We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.
International journal of engineering and mathematical modelling vol2 no1_2015_1IJEMM
Our efforts are mostly concentrated on improving the convergence rate of the numerical procedures both from the viewpoint of cost-efficiency and accuracy by handling the parametrization of the shape to be optimized. We employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The algorithms mimick classical multigrid strategies and are found very effective in terms of convergence acceleration. In this paper, we analyse and demonstrate the efficiency of the two-level correction algorithm which is the basic block of a more general miltilevel strategy.
Efficient Solution of Two-Stage Stochastic Linear Programs Using Interior Poi...SSA KPI
The document describes efficient solution methods for two-stage stochastic linear programs (SLPs) using interior point methods. Interior point methods require solving large, dense systems of linear equations at each iteration, which can be computationally difficult for SLPs due to their structure leading to dense matrices. The paper reviews methods for improving computational efficiency, including reformulating the problem, exploiting special structures like transpose products, and explicitly factorizing the matrices to solve smaller independent systems in parallel. Computational results show explicit factorizations generally require the least effort.
Talk of Michael Samet, entitled "Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models" at the International Conference on Computational Finance (ICCF)", Wuppertal June 6-10, 2022
This document introduces a generalized method for constructing sub-quadratic complexity multipliers for finite fields of characteristic 2. It begins by reintroducing the Winograd short convolution algorithm in the context of polynomial multiplication. It then presents a recursive construction technique that extends any d-point multiplier into an n=dk-point multiplier with sub-quadratic area and logarithmic delay complexity. Several new constructions are obtained using this technique, one of which is identical to the Karatsuba multiplier. The techniques aim to develop bit-parallel multipliers with better time and/or space complexity than the traditional quadratic complexity approaches.
This document presents a framework for efficiently pricing multi-asset options using Fourier methods. It discusses using the Fourier transform to map option pricing problems to frequency space, where the integrand may have better regularity. A damping parameter is introduced to ensure the transformed functions have sufficient decay at infinity. However, literature provides no guidance on choosing optimal damping parameters. The document proposes a method called Optimal Damping with Hierarchical Adaptive Quadrature to select damping parameters that improve the convergence rate of quadrature pricing methods in Fourier space. It applies this method to price options under various multi-dimensional models in numerical experiments.
Efficient Fourier Pricing of Multi-Asset Options: Quasi-Monte Carlo & Domain ...Chiheb Ben Hammouda
My talk at ICCF24 with abstract: Efficiently pricing multi-asset options poses a significant challenge in quantitative finance. While the Monte Carlo (MC) method remains a prevalent choice, its slow convergence rate can impede practical applications. Fourier methods, leveraging the knowledge of the characteristic function, have shown promise in valuing single-asset options but face hurdles in the high-dimensional context. This work advocates using the randomized quasi-MC (RQMC) quadrature to improve the scalability of Fourier methods with high dimensions. The RQMC technique benefits from the smoothness of the integrand and alleviates the curse of dimensionality while providing practical error estimates. Nonetheless, the applicability of RQMC on the unbounded domain, $\mathbb{R}^d$, requires a domain transformation to $[0,1]^d$, which may result in singularities of the transformed integrand at the corners of the hypercube, and deteriorate the rate of convergence of RQMC. To circumvent this difficulty, we design an efficient domain transformation procedure based on the derived boundary growth conditions of the integrand. This transformation preserves the sufficient regularity of the integrand and hence improves the rate of convergence of RQMC. To validate this analysis, we demonstrate the efficiency of employing RQMC with an appropriate transformation to evaluate options in the Fourier space for various pricing models, payoffs, and dimensions. Finally, we highlight the computational advantage of applying RQMC over quadrature methods in the Fourier domain, and over the MC method in the physical domain for options with up to 15 assets.
This document discusses automatic Bayesian cubature for numerical integration. It begins with an introduction to multivariate integration and the challenges it poses. It then describes an automatic cubature algorithm that generates sample points and computes error bounds iteratively until a tolerance threshold is met. Next, it covers Bayesian cubature, which treats integrands as random functions to obtain probabilistic error bounds. It defines a Bayesian trio identity relating the integration error to discrepancies, variations, and alignments. The document concludes with discussions of future work.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
1) The document discusses wavelet transforms as a recent algorithm for image compression. Wavelet transforms can capture variations at different scales in an image, making them well-suited for reducing spatial redundancy.
2) A typical lossy image compression system uses four main components - source encoding, thresholding, quantization, and entropy encoding - to achieve compression by removing different types of redundancy in images.
3) Experimental results on the Lena test image showed that soft thresholding followed by quantization achieved higher peak signal-to-noise ratios than hard thresholding and quantization, demonstrating the effectiveness of wavelet transforms for image compression.
This document summarizes research on computing stochastic partial differential equations (SPDEs) using an adaptive multi-element polynomial chaos method (MEPCM) with discrete measures. Key points include:
1) MEPCM uses polynomial chaos expansions and numerical integration to compute SPDEs with parametric uncertainty.
2) Orthogonal polynomials are generated for discrete measures using various methods like Vandermonde, Stieltjes, and Lanczos.
3) Numerical integration is tested on discrete measures using Genz functions in 1D and sparse grids in higher dimensions.
4) The method is demonstrated on the KdV equation with random initial conditions. Future work includes applying these techniques to SPDEs driven
MVPA with SpaceNet: sparse structured priorsElvis DOHMATOB
The GraphNet (aka S-Lasso), as well as other “sparsity + structure” priors like TV (Total-Variation), TV-L1, etc., are not easily applicable to brain data because of technical problems
relating to the selection of the regularization parameters. Also, in
their own right, such models lead to challenging high-dimensional optimization problems. In this manuscript, we present some heuristics for speeding up the overall optimization process: (a) Early-stopping, whereby one halts the optimization process when the test score (performance on leftout data) for the internal cross-validation for model-selection stops improving, and (b) univariate feature-screening, whereby irrelevant (non-predictive) voxels are detected and eliminated before the optimization problem is entered, thus reducing the size of the problem. Empirical results with GraphNet on real MRI (Magnetic Resonance Imaging) datasets indicate that these heuristics are a win-win strategy, as they add speed without sacrificing the quality of the predictions. We expect the proposed heuristics to work on other models like TV-L1, etc.
This document summarizes techniques for approximating marginal likelihoods and Bayes factors, which are important quantities in Bayesian inference. It discusses Geyer's 1994 logistic regression approach, links to bridge sampling, and how mixtures can be used as importance sampling proposals. Specifically, it shows how optimizing the logistic pseudo-likelihood relates to the bridge sampling optimal estimator. It also discusses non-parametric maximum likelihood estimation based on simulations.
This document presents an efficient convex hull algorithm for finding the convex hull of a planar set of points. The algorithm partitions the set of points into four regions based on the minimum and maximum x and y coordinates. It then finds the convex hull parts belonging to each region in parallel. Those parts are merged to derive the full convex hull. For each region, the algorithm uses a modified gradient concept to efficiently process the points and find the boundary points of the convex hull part. The algorithm achieves parallelism, data reduction, and has lower computational cost compared to traditional interior points algorithms. However, its drawbacks include difficulty extending it to higher dimensions and its static nature which requires the entire dataset from the beginning.
The document discusses uncertainty quantification (UQ) using quasi-Monte Carlo (QMC) integration methods. It introduces parametric operator equations for modeling input uncertainty in partial differential equations. Both forward and inverse UQ problems are considered. QMC methods like interlaced polynomial lattice rules are discussed for approximating high-dimensional integrals arising in UQ, with convergence rates superior to standard Monte Carlo. Algorithms for single-level and multilevel QMC are presented for solving forward and inverse UQ problems.
A generalized class of normalized distance functions called Q-Metrics is described in this presentation. The Q-Metrics approach relies on a unique functional, using a single bounded parameter (Lambda), which characterizes the conventional distance functions in a normalized per-unit metric space. In addition to this coverage property, a distinguishing and extremely attractive characteristic of the Q-Metric function is its low computational complexity. Q-Metrics satisfy the standard metric axioms. Novel networks for classification and regression tasks are defined and constructed using Q-Metrics. These new networks are shown to outperform conventional feed forward back propagation networks with the same size when tested on real data sets.
A generalized class of normalized distance functions called Q-Metrics is described in this presentation. The Q-Metrics approach relies on a unique functional, using a single bounded parameter Lambda, which characterizes the conventional distance functions in a normalized per-unit metric space. In addition to this coverage property, a distinguishing and extremely attractive characteristic of the Q-Metric function is its low computational complexity. Q-Metrics satisfy the standard metric axioms. Novel networks for classification and regression tasks are defined and constructed using Q-Metrics. These new networks are shown to outperform conventional feed forward back propagation networks with the same size when tested on real data sets.
This document discusses various methods for approximating marginal likelihoods and Bayes factors, including:
1. Geyer's 1994 logistic regression approach for approximating marginal likelihoods using importance sampling.
2. Bridge sampling and its connection to Geyer's approach. Optimal bridge sampling requires knowledge of unknown normalizing constants.
3. Using mixtures of importance distributions and the target distribution as proposals to estimate marginal likelihoods through Rao-Blackwellization. This connects to bridge sampling estimates.
4. The document discusses various methods for approximating marginal likelihoods and comparing hypotheses using Bayes factors. It outlines the historical development and connections between different approximation techniques.
We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.
International journal of engineering and mathematical modelling vol2 no1_2015_1IJEMM
Our efforts are mostly concentrated on improving the convergence rate of the numerical procedures both from the viewpoint of cost-efficiency and accuracy by handling the parametrization of the shape to be optimized. We employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The algorithms mimick classical multigrid strategies and are found very effective in terms of convergence acceleration. In this paper, we analyse and demonstrate the efficiency of the two-level correction algorithm which is the basic block of a more general miltilevel strategy.
Efficient Solution of Two-Stage Stochastic Linear Programs Using Interior Poi...SSA KPI
The document describes efficient solution methods for two-stage stochastic linear programs (SLPs) using interior point methods. Interior point methods require solving large, dense systems of linear equations at each iteration, which can be computationally difficult for SLPs due to their structure leading to dense matrices. The paper reviews methods for improving computational efficiency, including reformulating the problem, exploiting special structures like transpose products, and explicitly factorizing the matrices to solve smaller independent systems in parallel. Computational results show explicit factorizations generally require the least effort.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
Immersive Learning That Works: Research Grounding and Paths ForwardLeonel Morgado
We will metaverse into the essence of immersive learning, into its three dimensions and conceptual models. This approach encompasses elements from teaching methodologies to social involvement, through organizational concerns and technologies. Challenging the perception of learning as knowledge transfer, we introduce a 'Uses, Practices & Strategies' model operationalized by the 'Immersive Learning Brain' and ‘Immersion Cube’ frameworks. This approach offers a comprehensive guide through the intricacies of immersive educational experiences and spotlighting research frontiers, along the immersion dimensions of system, narrative, and agency. Our discourse extends to stakeholders beyond the academic sphere, addressing the interests of technologists, instructional designers, and policymakers. We span various contexts, from formal education to organizational transformation to the new horizon of an AI-pervasive society. This keynote aims to unite the iLRN community in a collaborative journey towards a future where immersive learning research and practice coalesce, paving the way for innovative educational research and practice landscapes.
Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...AbdullaAlAsif1
The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
Phenomics assisted breeding in crop improvementIshaGoswami9
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The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
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hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
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Authoring a personal GPT for your research and practice: How we created the Q...
talk_NASPDE.pdf
1. Analysis of Numerical Smoothing with Hierarchical
Approximations: Applications in
Probabilities/Densities Computation and Option Pricing
Chiheb Ben Hammouda
Christian Bayer Raúl Tempone
Center for Uncertainty
Quantification
Center for Uncertainty
Quantification
Center for Uncertainty Quantification Logo Lock-up
NASPDE Workshop, Eindhoven, Netherlands
May 16, 2023
2. Related Manuscripts to the Talk
Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Numerical smoothing with hierarchical adaptive sparse grids and
quasi-Monte Carlo methods for efficient option pricing”. In:
Quantitative Finance 23.2 (2023), pp. 209–227.
Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Multilevel Monte Carlo with Numerical Smoothing for Robust
and Efficient Computation of Probabilities and Densities”. In:
arXiv preprint arXiv:2003.05708 (2022).
1
3. Outline
1 Framework and Motivation
2 The Numerical Smoothing
3 Analysis of Multilevel Monte Carlo with Numerical Smoothing
4 Analysis of Adaptive Sparse Grids Quadrature with Numerical
Smoothing
5 Conclusions
1
4. 1 Framework and Motivation
2 The Numerical Smoothing
3 Analysis of Multilevel Monte Carlo with Numerical Smoothing
4 Analysis of Adaptive Sparse Grids Quadrature with Numerical
Smoothing
5 Conclusions
5. Framework
Goal: Approximate efficiently E[g(X(T))]
Setting:
▸ Given a (smooth) φ ∶ Rd
→ R, the function g ∶ Rd
→ R:
☀ Indicator functions: g(x) = 1(φ(x)≥0) (probabilities, digital/barrier
options, sensitivities, . . . )
☀ Dirac Delta functions: g(x) = δ(φ(x)=0) (density estimation,
sensitivities, . . . )
▸ X: solution process of a d-dimensional system of SDEs,
approximated by X (via a discretization scheme with N time steps),
E.g., stochastic volatility model: E.g., the Heston model
dXt = µXtdt +
√
vtXtdWX
t
dvt = κ(θ − vt)dt + ξ
√
vtdWv
t ,
(WX
t ,Wv
t ): correlated Wiener processes with correlation ρ.
Challenge: High-dimensional, non-smooth integration problem
E[g(X
∆t
(T))] = ∫
Rd×N
G(z)ρd×N (z)dz
(1)
1 ...dz
(1)
N ...dz
(d)
1 ...dz
(d)
N ,
with G(⋅) maps N × d random inputs to g(X
∆t
(T)); and ρd×N (z):
joint density function of z. 2
6. Multilevel Monte Carlo (MLMC)
(Heinrich 2001; Kebaier et al. 2005; Giles 2008)
A hierarchy of nested meshes of [0,T] (sequence of finer discretizations).
h` ∶= K−`
∆t0: the time steps size for levels ` ≥ 0; K>1, K ∈ N. (h0 > ... > hL)
X` ∶= Xh`
: The approximate process generated using a step size of h`.
h0 h1 hL
.....
MLMC idea
E[g(X(T))] ≈ E[g(XL(T))] = E[g(X0(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
+
L
∑
`=1
E[g(X`(T)) − g(X`−1(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
(1)
Var[g(X0(T))] ≫ Var[g(X`(T)) − g(X`−1(T))] ↘ as ` ↗
M0 ≫ M` ↘ as ` ↗
MLMC estimator: ̂
QMLMC
∶=
L
∑
`=0
̂
Q`, (sample independently each term of (1) with MC)
̂
Q0 ∶=
1
M0
M0
∑
m0=1
g(X0(T;ωm0 )); ̂
Q` ∶=
1
M`
M`
∑
m`=1
(g(X`(T;ωm`
)) − g(X`−1(T;ωm`
))), 1 ≤ ` ≤ L
Compared to MC: MLMC reduces the variance of the deepest level using samples on
coarser (less expensive) levels. 3
7. Quadrature Methods
Naive Quadrature operator based on a Cartesian quadrature grid
E[g] = ∫
Rd
g(x)ρ(x)dx ≈
d
⊗
k=1
QNk
k [g] ∶=
N1
∑
i1=1
⋯
Nd
∑
id=1
wi1 ⋯wid
g (xi1 ,...,xid
)
" Caveat: Curse of dimension: i.e., total number of quadrature
points N = ∏d
k=1 Nk.
Solution:
1 Sparsification of the grid points to reduce computational work.
2 Dimension-adaptivity to detect important dimensions of the
integrand.
Notation:
{xik
,wik
}Nk
ik=1 are respectively the sets of quadrature points and
corresponding quadrature weights for the kth dimension, 1 ≤ k ≤ d.
QNk
k [.]: the univariate quadrature operator for the kth dimension.
4
8. Hierarchical Sparse grids: Construction
(Bungartz and Griebel 2004)
Let β = [β1,...,βd] ∈ Nd
+, m(β) ∶ N+ → N+ an increasing function,
1 1D quadrature operators: Q
m(βk)
k on m(βk) points, 1 ≤ k ≤ d.
2 Detail operator: ∆
m(βk)
k = Q
m(βk)
k − Q
m(βk−1)
k , Q
m(0)
k = 0.
3 Hierarchical surplus: ∆m(β)
= ⊗d
k=1 ∆
m(βk)
k .
4 Hierarchical sparse grid approximation: on an index set I ⊂ Nd
QI
d [g] = ∑
β∈I
∆m(β)
[g] (2)
5
9. Grids Construction
Tensor Product (TP) approach: I ∶= I` = {∣∣ β ∣∣∞≤ `; β ∈ Nd
+}.
Regular sparse grids (SG): I ∶ I` = {∣∣ β ∣∣1≤ ` + d − 1; β ∈ Nd
+}
Adaptive sparse grids (ASG) (Gerstner and Griebel 2003):
Adaptive and a posteriori construction of I = IASGQ
by profit rule
IASGQ
= {β ∈ Nd
+ ∶ Pβ ≥ T}, with Pβ =
∣∆Eβ∣
∆Wβ
:
▸ ∆Eβ = ∣Q
I∪{β}
d [g] − QI
d [g]∣;
▸ ∆Wβ = Work[Q
I∪{β}
d [g]] − Work[QI
d [g]].
Figure 1.1: 2-D Illustration (Chen 2018): Admissible index sets I (top) and
corresponding quadrature points (bottom). Left: TP; middle: SG; right: ASG .
6
10. Table 1: Complexity comparison of the different methods for approximating
E[g(X(T))] within a pre-selected error tolerance, TOL. Given the same initial
problem, and using a weak order one scheme, E.g, the Euler-Maruyama scheme.
Method General Complexity Optimal Complexity
MC O (TOL−3
) O (TOL−3
)
MLMC O (TOL−3+β
), 1
2
≤ β ≤ 1
O (TOL−2
)
Quasi-MC (QMC) O (TOL−1− 2
1+2δ ), 0 ≤ δ ≤ 1
2
O (TOL−2
)
ASGQ O (TOL−1− 2
p ), p > 0
O (TOL−1
)
Sufficient Regularity Conditions for Optimal Complexity:
▸ MLMC (Cliffe, Giles, Scheichl, and Teckentrup 2011; Giles 2015):
g is Lipschtiz ⇒ canonical complexity: O (TOL−2
).
▸ QMC (Dick, Kuo, and Sloan 2013):
1 g belongs to the d-dimensional weighted Sobolev space of functions
with square-integrable mixed (partial) first derivatives.
2 High anisotropy between the different dimensions.
▸ ASGQ (Chen 2018; Ernst, Sprungk, and Tamellini 2018):
p is related to the order of bounded weighted mixed (partial)
derivatives of g and the anisotropy between the different dimensions.
⇒ ASGQ Complexity: O (TOL−1− 2
p ) (O (TOL−1
) when p ≫ 1).
11. Our Proposed Solutions/Strategies
to Recover Optimal Complexities
1 Extracting the available hidden regularity in the problem:
▸ Bias-free mollification (Bayer, Ben Hammouda, and Tempone 2020)
by taking conditional expectations over subset of integration
variables in the context of rough stochastic volatility models.
/ Good choice of subset of integration variables not always trivial.
▸ Mapping the problem to frequency/Fourier space + parametric
smoothing (Bayer, Ben Hammouda, Papapantoleon, Samet, and
Tempone 2022).
" Fourier transform of the density function available/cheap to
compute.
▸ Numerical smoothing (Bayer, Ben Hammouda, and Tempone 2023;
Bayer, Hammouda, and Tempone 2022): Topic of this talk.
2 Dimension reduction techniques based on Richardson
extrapolation on the weak error.
3 Reducing the discontinuity dimension and creating
anisotropy between the different dimensions using Brownian
bridge/Haar wavelet construction.
8
12. 1 Framework and Motivation
2 The Numerical Smoothing
3 Analysis of Multilevel Monte Carlo with Numerical Smoothing
4 Analysis of Adaptive Sparse Grids Quadrature with Numerical
Smoothing
5 Conclusions
13. Numerical Smoothing Steps
Motivating Example:
E[g(X
∆t
T )] =?
g ∶ Rd
→ R nonsmooth function: (E.g., g(x) = 1(φ(x)≥0) or δ(φ(x)=0))
X
∆t
T (∆t = T
N ) Euler discretization of d-dimensional SDE , E.g.,
dX
(i)
t = ai(Xt)dt + ∑d
j=1 bij(Xt)dW
(j)
t ,
where {W(j)
}d
j=1 are standard Brownian motions.
X
∆t
T = X
∆t
T (∆W
(1)
1 ,...,∆W
(1)
N ,...,∆W
(d)
1 ,...,∆W
(d)
N )
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
∶=∆W
.
The discontinuity is in (N × d)-dimensional space characterised by
φ(X
∆t
(T)) = 0.
9
14. Numerical Smoothing Steps
1 Identify hierarchical representation of integration variables ⇒
locate the discontinuity in a smaller dimensional space
(a) X
∆t
T (∆W) ≡ X
∆t
T (Z), Z = (Zi)dN
i=1 ∼ N(0,IdN ):
s.t. “Z1 ∶= (Z
(1)
1 ,...,Z
(d)
1 ) (coarse rdvs) substantially contribute
even for ∆t → 0”, through hierarchical path generation (Brownian
bridges / Haar wavelet construction)
⇒ Discontinuity in d-dimensional space instead of
(N × d)-dimensions.
Haar wavelet construction in one dimension
For i.i.d. standard normal rdvs Z1, Zn,k, n ∈ N0, k = 0,...,2n
− 1, we define
the (truncated) standard Brownian motion
WN
t ∶= Z1Ψ−1(t) +
N
∑
n=0
2n
−1
∑
k=0
Zn,kΨn,k(t).
with Ψ−1(⋅) and Ψn,k(⋅) are the antiderivatives of the Haar basis functions.
15. Numerical Smoothing Steps
1 Identify hierarchical representation of integration variables ⇒
locate the discontinuity in a smaller dimensional space
(b) Linear mapping using A: rotation matrix whose structure depends
on the function g.
Y = AZ1.
E.g., for an arithmetic basket option with payoff
g(x) = max(∑
d
i=1 cixi(T) − K,0), a suitable A is a rotation matrix,
with the first row leading to Y1 = ∑
d
i=1 Z
(i)
1 up to rescaling without
any constraint for the remaining rows (Gram-Schmidt procedure).
⇒ Discontinuity in 1-dimensional space instead of d-dimensions.
⇒ y∗
1 (y−1,z
(1)
−1 ,...,z
(d)
−1 ): the exact discontinuity location s.t
φ(X
∆t
(T)) = φ(X
∆t
(T;y∗
1 ;y−1,z
(1)
−1 ,...,z
(d)
−1 )) = 0. (3)
Notation
▸ x−j: vector of length d − 1 denoting all the variables other than xj
in x ∈ Rd
. 11
16. Numerical Smoothing Steps
2
E[g(X(T))] ≈ E[g(X
∆t
(T))]
= ∫
Rd×N
G(z)ρd×N (z)dz
(1)
1 ...dz
(1)
N ...dz
(d)
1 ...dz
(d)
N
= ∫
RdN−1
I(y−1,z
(1)
−1 ,...,z
(d)
−1 )ρd−1(y−1)dy−1ρdN−d(z
(1)
−1 ,...,z
(d)
−1 )dz
(1)
−1 ...dz
(d)
−1
= E[I(Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] ≈ E[I(Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )], (4)
3
I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) = ∫
R
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
= ∫
y∗
1
−∞
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1 + ∫
+∞
y∗
1
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
≈ I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) ∶=
Mlag
∑
k=0
ηkG(ζk (y∗
1),y−1,z
(1)
−1 ,...,z
(d)
−1 ), (5)
4 Compute the remaining (dN − 1)-integral in (4) by MLMC or QMC or ASGQ.
Notation
G maps N × d Gaussian random inputs to g(X
∆t
(T));
y∗
1 (y−1,z
(1)
−1 ,...,z
(d)
−1 ): the exact discontinuity location (see (3))
y∗
1(y−1,z
(1)
−1 ,...,z
(d)
−1 ): the approximated discontinuity location via root finding.
MLag: number of Laguerre quadrature points ζk ∈ R, and weights ηk;
ρd×N (z) = 1
(2π)d×N/2 e−1
2
zT z
.
12
17. Extending Numerical Smoothing for
Density Estimation
Goal: Approximate the density ρX at u, for a stochastic process X
ρX(u) = E[δ(X − u)], δ is the Dirac delta function.
" Without any smoothing techniques (regularization, kernel
density,. . . ) MC and MLMC fail due to the infinite variance caused
by the Dirac distribution function, δ(⋅).
Strategy in (Bayer, Hammouda, and Tempone 2022): Conditioning
with respect to Z−1
ρX(u) =
1
√
2π
E[exp(−(Y ∗
1 (u))
2
/2)∣
dY ∗
1
dx
(u)∣]
≈
1
√
2π
E
⎡
⎢
⎢
⎢
⎣
exp(−(Y
∗
1(u))
2
/2)
R
R
R
R
R
R
R
R
R
R
R
dY
∗
1
dx
(u)
R
R
R
R
R
R
R
R
R
R
R
⎤
⎥
⎥
⎥
⎦
, (6)
Y ∗
1 (x;Z−1): the exact singularity; Y
∗
1(x;Z−1): the approximated
singularity obtained by solving X
∆t
(T;Y
∗
(x),Z−1) = x.
18. Why not Kernel Density Techniques
in Multiple Dimensions?
Similar to approaches based on MLMC with parametric regularization (Giles,
Nagapetyan, and Ritter 2015) or QMC with kernel density techniques
(Ben Abdellah, L’Ecuyer, Owen, and Puchhammer 2021; L’Ecuyer, Puchhammer,
and Ben Abdellah 2022).
This class of approaches has a pointwise error that increases exponentially with
respect to the dimension of the state vector X.
For a d-dimensional problem, a kernel density estimator with a bandwidth matrix,
H = diag(h,...,h)
MSE ≈ c1M−1
h−d
+ c2h4
. (7)
M is the number of samples, and c1 and c2 are constants.
Our approach in high dimension: For u ∈ Rd
ρX(u) = E[δ(X − u)] = E[ρd (Y∗
(u))∣det(J(u))∣]
≈ E[ρd (Y
∗
(u))∣det(J(u))∣], (8)
where
▸ Y∗
(u;⋅): the exact discontinuity; Y
∗
(u;⋅): the approximated discontinuity.
▸ J is the Jacobian matrix, with Jij =
∂y∗
i
∂uj
; ρd(⋅) is the multivariate Gaussian density.
Exact conditioning with respect to the remaining Brownian bridge noise ⇒ the
smoothing error in our approach is insensitive to the dimension of the problem.
14
19. 1 Framework and Motivation
2 The Numerical Smoothing
3 Analysis of Multilevel Monte Carlo with Numerical Smoothing
4 Analysis of Adaptive Sparse Grids Quadrature with Numerical
Smoothing
5 Conclusions
20. Multilevel Monte Carlo with Numerical Smoothing:
Estimator and Notation
Recall
▸
E[g(X(T))] ≈ E[g(X
∆t
(T))] = E[I(Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] ≈ E[I(Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )],
▸
I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) = ∫
R
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
= ∫
y∗
1
−∞
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1 + ∫
+∞
y∗
1
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
≈ I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) ∶=
Mlag
∑
k=0
ηkG(ζk (y∗
1),y−1,z
(1)
−1 ,...,z
(d)
−1 ),
where y∗
1(y−1,z
(1)
−1 ,...,z
(d)
−1 ): the approximated discontinuity location via root finding.
I`:= I`(y`
−1,z
(1),`
−1 ,...,z
(d),`
−1 ): the level ` approximation of I in ̂
QMLMC
, computed
with step size ∆t`; MLag,` Laguerre points; TOLNewton,` as the tolerance of Newton
at level `.
̂
QMLMC
∶=
L
∑
`=L0
̂
Q`,
with
̂
QL0 ∶=
1
ML0
ML0
∑
mL0
=1
IL0,[mL0
]; ̂
Q` ∶=
1
M`
M`
∑
m`=1
(I`,[m`] − I`−1,[m`]), L0 + 1 ≤ ` ≤ L, (9)
21. Notation
X`(T) ∶= X`(T;(Z`
1,Z`
−1)).
For convenience, we denote X`(T) by X
N`
T and X
N`
k are the
Euler–Maruyama increments of X
N`
T for 0 ≤ k ≤ N` with
X
N`
T = X
N`
N`
.
Assumption 3.1
There are positive rdvs Cp with finite moments of all orders such that
∀N` ∈ N, ∀k1,...,kp ∈ {0,...,N` − 1} ∶
R
R
R
R
R
R
R
R
R
R
R
R
R
∂p
X
N`
T
∂X
N`
k1
⋯∂X
N`
kp
R
R
R
R
R
R
R
R
R
R
R
R
R
≤ Cp a.s.
Assumption 3.1 is fulfilled if the drift and diffusion coefficients are
smooth.
16
22. Assumption 3.2
For p ∈ N, there are positive rdvs Dp with finite moments of all orders
such that a
⎛
⎝
∂X
N`
T
∂y
(Z`
1,Z`
−1)
⎞
⎠
−p
≤ Cp a.s.
a
y ∶= z−1
In (Bayer, Ben Hammouda, and Tempone 2023), we show
sufficient conditions where this assumption is valid.
For instance, Assumption 3.2 is valid for
▸ one-dimensional SDEs with a linear or constant diffusion.
▸ multivariate SDEs with a linear drift and constant diffusion,
including the multivariate lognormal model (see (Bayer,
Siebenmorgen, and Tempone 2018)).
23. Multilevel Monte Carlo with Numerical Smoothing:
Variance Decay, Complexity and Robustness
Let g(x) = 1(φ(x)≥0) or δ (φ(x) = 0)
Theorem 3.3 ((Bayer, Hammouda, and Tempone 2022))
Under Assmuptions 3.1 and 3.2 and further regularity assumptions for the drift and
diffusion, using Euler–Maruyama, V` ∶= Var[I` − I`−1] = O (∆t1
` ), compared with
O (∆t
1/2
` ) for MLMC without smoothing.
" General MLMC Complexity: O (TOL
−2−max(0,γ−β
α
)
log (TOL)2×1{β=γ}
),
where α: weak rate; β: variance decay rate; γ : work growth rate.
Corollary 3.4 ((Bayer, Hammouda, and Tempone 2022))
Under Assmuptions 3.1 and 3.2 and further regularity assumptions for the drift and
diffusion, the complexity of MLMC combined with numerical smoothing is O (TOL−2
) up
to log terms, compared with O (TOL−2.5
) for MLMC without smoothing.
" Milstein scheme: we show that we obtain the canonical complexity (O (TOL−2
)).
Corollary 3.5 ((Bayer, Hammouda, and Tempone 2022))
Let κ` be the kurtosis of the random variable I` − I`−1, then under assmuptions 3.1 and
3.2 and further regularity assumptions for the drift and diffusion, we obtain κ` = O (1)
compared with O (∆t
−1/2
` ) for MLMC without smoothing.
24. Sketch of the Proof of Theorem 3.3: Notation and Goal
Notation
X`,X`−1: the coupled paths of the approximate process X, simulated with time
step sizes ∆t` and ∆t`−1, respectively.
W` and B`: coupling Wiener and related Brownian bridge processes at levels `
and ` − 1, respectively.
For t ∈ [0,T], e`(t;Y,B`) is defined as
(X` − X`−1)(t) = ∫
t
0
(a(X`(s)) − a(X`−1(s)))ds + ∫
t
0
(b(X`(s)) − b(X`−1(s)))dW`(s)
= ∫
t
0
(a(X`(s)) − a(X`−1(s)))ds + ∫
t
0
(b(X`(s)) − b(X`−1(s)))
Y
√
T
ds
+ ∫
t
0
(b(X`(s)) − b(X`−1(s)))dB`(s)
=∶ e`(t;Y,B`),
where a(X(s)) = a(X(tn)), b(X(s)) = b(X(tn)), for tn ≤ s < tn+1, on the time
grid 0 = t0 < t1 < ... < tN = T.
g̃δ: a C∞
mollified version of g, with δ > 0,
Goal: We want to show V` ∶= Var[I` − I`−1] ≤ E[(I` − I`−1)
2
] = O (∆t`).
19
25. Sketch of the Proof of Theorem 3.3: Step 1
For Euler–Maruyama scheme and p ≥ 1,
Under Global Lipschitzity of drift and diffusion coefficients
Assumption, we have (Kloeden and Platen 1992)
E[e2p
` (T)] = O (∆tp
` ). (10)
In (Bayer, Hammouda, and Tempone 2022), we prove (using the
Grönwall, Hölder, Burkholder-Davis-Gundy and Jensen
inequalities) that
E[(∂ye`)2p
(T)] = O (∆tp
` ). (11)
20
26. Sketch of the Proof of Theorem 3.3: Step 2
For δ > 0, we have
∆Iδ
` (B`) ∶= (I
δ
` − I
δ
`−1)(B`) ∶= ∫
R
(g̃δ(X`(T;y,B`)) − g̃δ(X`−1(T;y,B`)))ρ1(y)dy
= ∫
R
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
∫
1
0
g̃′
δ
⎛
⎜
⎜
⎜
⎜
⎝
X`−1(T;y,B`) + θe`(T;y,B`)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
∶=z(θ;y,B`)
⎞
⎟
⎟
⎟
⎟
⎠
dθ
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
e`(T;y,B`) ρ1(y)dy, θ ∈ (0,1) (using Mean value theorem)
= ∫
1
0
[∫
R
∂yg̃δ(z(θ;y,B`))(∂yz(θ;y,B`))
−1
e`(T;y,B`) ρ1(y)dy]dθ (using ∂yg̃δ = g̃′
δ∂yz and Fubini’s theorem)
= −∫
1
0
[∫
R
e`(T;y,B`)g̃δ(z(θ;y,B`))(∂y ((∂yz(θ;y,B`))
−1
) − y (∂yz(θ;y,B`))
−1
)ρ1(y)dy]dθ
− ∫
1
0
[∫
R
∂ye`(T;y,B`)g̃δ(z(θ;y,B`))(∂yz(θ;y,B`))
−1
ρ1(y)dy]dθ. (12)
(Integration by parts and we prove that the boundary terms vanish to get the last terms)
Taking δ → 0 and applying the dominated convergence theorem to
(12):
∆I`(B`) ∶= (I` − I`−1)(B`)
= −∫
1
0
[∫
R
e`(T;y,B`)g(z(θ;y,B`))(∂y ((∂yz(θ;y,B`))
−1
) − y (∂yz(θ;y,B`))
−1
)ρ1(y)dy]dθ
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
(I)
−∫
1
0
[∫
R
∂ye`(T;y,B`)g(z(θ;y,B`))(∂yz(θ;y,B`))
−1
ρ1(y)dy]dθ
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
(II)
.
(13)
21
27. Sketch of the Proof of Theorem 3.3: Step 3
For the term (I) in (13), taking expectation with respect to the Brownian
bridge and using Hölder’s inequality twice (p,q,p1,q1 ∈ (1,+∞), 1
p + 1
q = 1
and 1
p1
+ 1
q1
= 1), result in
E [(I)
2
] ≤ (E [∣∣g(z(⋅;⋅,B`))(∂y ((∂yz(⋅;⋅,B`))
−1
) − Y (∂yz(⋅;⋅,B`))
−1
)∣∣
2q1
Lq
ρ1
([0,1]×R)
])
1/q1
× (E [∣∣e`(T;⋅,B`)∣∣
2p1
Lp
ρ1
(R)])
1/p1
= O (∆t`). (14)
Choosing p and p1 such that 2p1
p ≤ 1, and applying Jensen’s inequality:
(E [∣∣e`(T;⋅,B`)∣∣
2p1
Lp
ρ1
(R)])
1/p1
=
⎛
⎝
E
⎡
⎢
⎢
⎢
⎢
⎣
(∫
R
∣ep
` (T;y,B`)∣ρ1dy)
2p1
p
⎤
⎥
⎥
⎥
⎥
⎦
⎞
⎠
1/p1
≤ (E [∫
R
∣ep
` (T;y,B`)∣ρ1dy])
2
p
= O (∆t`) (using Fubini’s theorem and (10)). (15)
Using Assumptions 3.2 and 3.1, we show that
(E [∣∣g(z(⋅;⋅,B`))(∂y ((∂yz(⋅;⋅,B`))−1
) − Y (∂yz(⋅;⋅,B`))−1
)∣∣
2q1
Lq
ρ1
([0,1]×R)
])
1/q1
< ∞,
22
28. Sketch of the Proof of Theorem 3.3: Step 4
For the term (II) in (13), we redo same steps as for term (I)
E [(II)
2
] ≤ (E [∣∣g(z(⋅;⋅,B`))(∂yz(⋅;⋅,B`))
−1
∣∣
2q1
Lq
ρ1
([0,1]×R)
])
1/q1
× (E [∣∣∂ye`(T;⋅,B`)∣∣
2p1
Lp
ρ1
(R)])
1/p1
= O (∆t`) (16)
Using (11), we show (EB`
[∣∣∂ye`(T;⋅,B`)∣∣2p1
Lp
ρ1
(R)
])
1/p1
= O (∆t`).
Using Assumptions 3.2 and 3.1, we show that
(E [∣∣g(z(⋅;⋅,B`))(∂yz(⋅;⋅,B`))−1
∣∣
2q1
Lq
ρ1
([0,1]×R)
])
1/q1
< ∞,
23
29. Error Discussion for MLMC
̂
QMLMC
: the MLMC estimator
E[g(X(T)] − ̂
QMLMC
= E[g(X(T))] − E[g(X
∆tL
(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error I: bias or weak error of O(∆tL)
+ E[IL (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − E[IL (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error II: numerical smoothing error of O(M
−s/2
Lag,L
)+O(TOLNewton,L)
+ E[IL (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − ̂
QMLMC
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error III: MLMC statistical error of O
⎛
⎝
√
∑L
`=L0
√
MLag,`+log(TOL−1
Newton,`
)
⎞
⎠
Notations
y∗
1: the approximated location of the non smoothness obtained by Newton
iteration ⇒ ∣y∗
1 − y∗
1∣ = TOLNewton
MLag is the number of points used by the Laguerre quadrature for the one
dimensional pre-integration step.
s > 0: For the parts of the domain separated by the discontinuity location,
derivatives of G with respect to y1 are bounded up to order s. 24
30. MLMC for Probability in the GBM model: Euler
0 2 4 6 8
-10
-8
-6
-4
-2
0 2 4 6 8
-15
-10
-5
0
0 2 4 6 8
2
4
6
8
10
0 2 4 6 8
102
103
kurtosis
(a) without numerical smoothing.
0 2 4 6 8
-20
-18
-16
-14
-12
-10
-8
0 2 4 6 8
-15
-10
-5
0
0 2 4 6 8
2
4
6
8
10
0 2 4 6 8
3
4
5
6
kurtosis
(b) With numerical smoothing.
Figure 3.1: MLMC with Euler–Maruyama scheme for probability computation
under the geometric Brownian motion (GBM): Variance, cost, L1
-distance
and kurtosis per level. P`: the numerical approximation of the QoI at level `.
25
31. MLMC for Probability in the GBM Model: Milstein
0 2 4 6 8
-15
-10
-5
0 2 4 6 8
-25
-20
-15
-10
-5
0
0 2 4 6 8
2
4
6
8
10
0 2 4 6 8
103
104
105
kurtosis
(a) without numerical smoothing.
0 2 4 6 8
-35
-30
-25
-20
0 2 4 6 8
-25
-20
-15
-10
-5
0
0 2 4 6 8
2
4
6
8
10
0 2 4 6 8
101
kurtosis
(b) With numerical smoothing.
Figure 3.2: MLMC with Milstein scheme for probability computation under
the geometric Brownian motion (GBM): Variance, cost, L1
-distance and
kurtosis per level. P`: the numerical approximation of the QoI at level `.
26
32. Probability Computation under the GBM Model:
Numerical Complexity Comparison
10-4
10-3
10-2
10-1
TOL
10-6
10-4
10-2
100
102
104
106
E[W]
MLMC without smoothing (Euler)
TOL-2.5
MLMC without smoothing (Milstein)
MLMC+ Numerical Smoothing (Euler)
TOL-2
log(TOL)2
MLMC+ Numerical Smoothing (Milstein)
TOL-2
Figure 3.3: Probability Computation under GBM: Comparison of the
numerical complexity of the different MLMC estimators.
27
33. 1 Framework and Motivation
2 The Numerical Smoothing
3 Analysis of Multilevel Monte Carlo with Numerical Smoothing
4 Analysis of Adaptive Sparse Grids Quadrature with Numerical
Smoothing
5 Conclusions
34. Notations
The Haar basis functions, ψn,k, of L2
([0,1]) with support
[2−n
k,2−n
(k + 1)]:
ψ−1(t) ∶= 1[0,1](t); ψn,k(t) ∶= 2n/2
ψ (2n
t − k), n ∈ N0, k = 0,...,2n
− 1,
where ψ(⋅) is the Haar mother wavelet:
ψ(t) ∶=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
1, 0 ≤ t < 1
2
,
−1, 1
2
≤ t < 1,
0, else,
Grid Dn ∶= {tn
` ∣ ` = 0,...,2n+1
} with tn
` ∶= `
2n+1 T.
For i.i.d. standard normal rdvs Z−1, Zn,k, n ∈ N0, k = 0,...,2n
− 1,
we define the (truncated) standard Brownian motion
WN
t ∶= Z−1Ψ−1(t) +
N
∑
n=0
2n
−1
∑
k=0
Zn,kΨn,k(t).
with Ψ−1(⋅) and Ψn,k(⋅) are the antiderivatives of the Haar basis
functions. 28
35. The Smoothness Theorem
We define the deterministic function HN
∶ R2N+1−1
→ R as
HN
(zN
) ∶= EZ−1
[g (XN
T (Z−1,zN
))], (17)
where ZN
∶= (Zn,k)n=0,...,N, k=0,...2n−1.
Theorem 4.1 ((Bayer, Ben Hammouda, and Tempone 2023))
Given Assumptions 3.1 and 3.2. Then, for any p ∈ N and indices
n1,...,np and k1,...,kp (satisfying 0 ≤ kj < 2nj ), the function HN
defined in (17) satisfies the following (with constants independent of
nj,kj)a
∂p
HN
∂zn1,k1 ⋯∂znp,kp
(zN
) = Ô
P (2− ∑
p
j=1 nj/2
).
In particular, HN
is of class C∞
.
a
The constants increase in p and N.
36. Sketch of the Proof
1 We consider a mollified version gδ of g and the corresponding function
HN
δ (defined by replacing g with gδ in (17)).
2 We prove that we can interchange the integration and differentiation,
which implies
∂HN
δ (zN
)
∂zn,k
= E [g′
δ (XN
T (Z−1,zN
))
∂XN
T (Z−1,zN
)
∂zn,k
].
3 Multiplying and dividing by
∂XN
T (Z−1,zN )
∂y (where y ∶= z−1) and
replacing the expectation by an integral w.r.t. the standard normal
density, we obtain
∂HN
δ (zN
)
∂zn,k
= ∫
R
∂gδ (XN
T (y,zN
))
∂y
(
∂XN
T
∂y
(y,zN
))
−1
∂XN
T
∂zn,k
(y,zN
)
1
√
2π
e− y2
2 dy.
(18)
4 We show that integration by parts is possible, and then we can discard
the mollified version to obtain the smoothness of HN
because
∂HN
(zN
)
∂zn,k
= −∫
R
g (XN
T (y,zN
))
∂
∂y
⎡
⎢
⎢
⎢
⎢
⎣
(
∂XN
T
∂y
(y,zN
))
−1
∂XN
T
∂zn,k
(y,zN
)
1
√
2π
e− y2
2
⎤
⎥
⎥
⎥
⎥
⎦
dy.
5 The proof relies on successively applying the technique above of
dividing by
∂XN
T
∂y and then integrating by parts.
30
37. Error Discussion for ASGQ
QASGQ
N : the ASGQ estimator
E[g(X(T)] − QASGQ
N = E[g(X(T))] − E[g(X
∆t
(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error I: bias or weak error of O(∆t)
+ E[I (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − E[I (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error II: numerical smoothing error of O(M
−s/2
Lag
)+O(TOLNewton)
+ E[I (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − QASGQ
N
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error III: ASGQ error of O(M
−p/2
ASGQ
)
, (19)
Notations
MASGQ: the number of quadrature points used by the ASGQ estimator, and p > 0.
y∗
1: the approximated location of the non smoothness obtained by Newton iteration
⇒ ∣y∗
1 − y∗
1∣ = TOLNewton
MLag is the number of points used by the Laguerre quadrature for the one
dimensional pre-integration step.
s > 0: For the parts of the domain separated by the discontinuity location,
derivatives of G with respect to y1 are bounded up to order s.
31
38. Work and Complexity Discussion for ASGQ
An optimal performance of ASGQ is given by
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
min
(MASGQ,MLag,TOLNewton)
WorkASGQ ∝ MASGQ × MLag × ∆t−1
s.t. Etotal,ASGQ = TOL.
(20)
Etotal, ASGQ ∶= E[g(X(T)] − QASGQ
N
= O (∆t) + O (M
−p/2
ASGQ) + O (M
−s/2
Lag ) + O (TOLNewton).
We show in (Bayer, Ben Hammouda, and Tempone 2023) that
under certain conditions for the regularity parameters s and p
(p,s ≫ 1)
▸ WorkASGQ = O (TOL−1
) (for the best case) compared to
WorkMC = O (TOL−3
) (for the best case of MC).
32
39. ASGQ Quadrature Error Convergence
101
102
103
104
MASGQ
10-3
10
-2
10
-1
100
101
Relative
Quadrature
Error
ASGQ without smoothing
ASGQ with numerical smoothing
(a)
101
102
103
104
MASGQ
10-3
10
-2
10-1
100
101
Relative
Quadrature
Error
ASGQ without smoothing
ASGQ with numerical smoothing
(b)
Figure 4.1: Digital option under the Heston model (dimension=16):
Comparison of the relative quadrature error convergence for the ASGQ
method with/out numerical smoothing. (a) Without Richardson
extrapolation (N = 8), (b) with Richardson extrapolation (Nfine level = 8).
33
40. Comparing ASGQ with MC
In (Bayer, Ben Hammouda, and Tempone 2023):
For the different considered examples under Heston or discretized GBM models:
ASGQ with numerical smoothing 10 - 100× faster in dim. around 20 than MC.
10
-3
10
-2
10
-1
Total Relative Error
10
-1
10
0
10
1
10
2
103
Computational
Work
MC (without smoothing, without Richardson extrapolation)
MC (without smoothing, with Richardson extrapolation)
ASGQ (with smoothing, without Richardson extrapolation)
ASGQ (with smoothing, with Richardson extrapolation)
Figure 4.2: Digital option under Heston: Computational work comparison for ASGQ with
numerical smoothing and MC with the different configurations in terms of the level of
Richardson extrapolation.
34
41. 1 Framework and Motivation
2 The Numerical Smoothing
3 Analysis of Multilevel Monte Carlo with Numerical Smoothing
4 Analysis of Adaptive Sparse Grids Quadrature with Numerical
Smoothing
5 Conclusions
42. Conclusions
Contributions in the context deterministic quadrature methods
1 A novel numerical smoothing technique, combined with ASGQ/QMC,
Hierarchical Brownian Bridge and Richardson extrapolation.
2 Our approach outperforms substantially the MC method for moderate/high
dimensions and for dynamics where discretization is needed.
3 We provide a regularity analysis for the smoothed integrand in the time stepping
setting, and a related error discussion of our approach.
Contributions in the context MLMC methods
1 The numerical smoothing approach is adapted to the MLMC context for efficient
probability computation, univariate/multivariate density estimation, and option
pricing.
2 Compared to the case without smoothing
☀ We significantly reduce the kurtosis at the deep levels of MLMC which improves
the robustness of the estimator.
☀ We improve the strong convergence rate (we recover the MLMC complexities
obtained for Lipschitz functionals) ⇒ improvement of MLMC complexity from
O (TOL−2.5
) to O (TOL−2
).
3 When estimating densities: Compared to the smoothing strategies based on
MLMC with parametric regularization as in (Giles, Nagapetyan, and Ritter 2015)
or QMC with kernel density techniques as in (Ben Abdellah, L’Ecuyer, Owen, and
Puchhammer 2021), the error of our approach does not increase exponentially
with respect to the dimension of state vector
35
43. Future Work and Extensions
1 Extend our techniques to efficiently compute
▸ Sensitivities (Financial Greeks): ∂
∂α
E[f(ω,α)].
▸ Risk quantities ⇒ nested expectations problems
E [g (E [f(X,Y )∣X])].
▸ Computing nonsmooth quantities (such as probabilities) of a
functional of a solution arising from a random PDE.
2 Combine the numerical smoothing technique with multilevel QMC
to profit from the good features of QMC and MLMC.
3 Combine the numerical smoothing technique with importance
sampling ideas in (Hammouda, Rached, and Tempone 2020;
Hammouda, Rached, Tempone, and Wiechert 2021) to improve
MLMC efficiency for rare events probabilities.
36
44. Related References
Thank you for your attention!
[1] C. Bayer, C. Ben Hammouda, R. Tempone. Numerical Smoothing with
Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for
Efficient Option Pricing. Quantitative Finance 23, no. 2 (2023): 209-227.
[2] C. Bayer, C. Ben Hammouda, R. Tempone. Multilevel Monte Carlo with
Numerical Smoothing for Robust and Efficient Computation of Probabilities
and Densities, arXiv:2003.05708 (2022).
[3] C. Bayer, C. Ben Hammouda, A. Papapantoleon, M. Samet, R. Tempone.
Optimal Damping with Hierarchical Adaptive Quadrature for Efficient
Fourier Pricing of Multi-Asset Options in Lévy Models, arXiv:2203.08196
(2022).
[4] C. Bayer, C. Ben Hammouda, R. Tempone. Hierarchical adaptive sparse
grids and quasi-Monte Carlo for option pricing under the rough Bergomi
model, Quantitative Finance, 2020.
37
45. References I
[1] Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Hierarchical adaptive sparse grids and quasi-Monte Carlo for
option pricing under the rough Bergomi model”. In: Quantitative
Finance 20.9 (2020), pp. 1457–1473.
[2] Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Numerical smoothing with hierarchical adaptive sparse grids
and quasi-Monte Carlo methods for efficient option pricing”. In:
Quantitative Finance 23.2 (2023), pp. 209–227.
[3] Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Multilevel Monte Carlo with Numerical Smoothing for Robust
and Efficient Computation of Probabilities and Densities”. In:
arXiv preprint arXiv:2003.05708 (2022).
[4] Christian Bayer, Markus Siebenmorgen, and Rául Tempone.
“Smoothing the payoff for efficient computation of basket option
pricing.”. In: Quantitative Finance 18.3 (2018), pp. 491–505.
38
46. References II
[5] Christian Bayer et al. “Optimal Damping with Hierarchical
Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset
Options in Lévy Models”. In: arXiv preprint arXiv:2203.08196
(2022).
[6] Amal Ben Abdellah et al. “Density estimation by randomized
quasi-Monte Carlo”. In: SIAM/ASA Journal on Uncertainty
Quantification 9.1 (2021), pp. 280–301.
[7] Hans-Joachim Bungartz and Michael Griebel. “Sparse grids”. In:
Acta numerica 13 (2004), pp. 147–269.
[8] Peng Chen. “Sparse quadrature for high-dimensional integration
with Gaussian measure”. In: ESAIM: Mathematical Modelling
and Numerical Analysis 52.2 (2018), pp. 631–657.
39
47. References III
[9] K Andrew Cliffe et al. “Multilevel Monte Carlo methods and
applications to elliptic PDEs with random coefficients”. In:
Computing and Visualization in Science 14.1 (2011), p. 3.
[10] Josef Dick, Frances Y Kuo, and Ian H Sloan. “High-dimensional
integration: the quasi-Monte Carlo way”. In: Acta Numerica 22
(2013), pp. 133–288.
[11] Oliver G Ernst, Bjorn Sprungk, and Lorenzo Tamellini.
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many Gaussian random variables (with application to elliptic
PDEs)”. In: SIAM Journal on Numerical Analysis 56.2 (2018),
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[12] Thomas Gerstner and Michael Griebel. “Dimension–adaptive
tensor–product quadrature”. In: Computing 71 (2003), pp. 65–87.
40
48. References IV
[13] Michael B Giles. “Multilevel Monte Carlo methods”. In: Acta
Numerica 24 (2015), pp. 259–328.
[14] Michael B Giles. “Multilevel Monte Carlo path simulation”. In:
Operations Research 56.3 (2008), pp. 607–617.
[15] Michael B Giles, Tigran Nagapetyan, and Klaus Ritter.
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and densities”. In: SIAM/ASA Journal on Uncertainty
Quantification 3.1 (2015), pp. 267–295.
[16] Chiheb Ben Hammouda, Nadhir Ben Rached, and Raúl Tempone.
“Importance sampling for a robust and efficient multilevel Monte
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and Computing 30.6 (2020), pp. 1665–1689.
41
49. References V
[17] Chiheb Ben Hammouda et al. “Optimal Importance Sampling
via Stochastic Optimal Control for Stochastic Reaction
Networks”. In: arXiv preprint arXiv:2110.14335 (2021).
[18] Stefan Heinrich. “Multilevel monte carlo methods”. In:
International Conference on Large-Scale Scientific Computing.
Springer. 2001, pp. 58–67.
[19] Ahmed Kebaier et al. “Statistical Romberg extrapolation: a new
variance reduction method and applications to option pricing”.
In: The Annals of Applied Probability 15.4 (2005), pp. 2681–2705.
[20] Peter E Kloeden and Eckhard Platen. “Stochastic differential
equations”. In: Numerical solution of stochastic differential
equations. Springer, 1992, pp. 103–160.
42
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[21] Pierre L’Ecuyer, Florian Puchhammer, and Amal Ben Abdellah.
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Conditioning”. In: INFORMS Journal on Computing (2022).
43