Finite Number Of Random Variables Research Articles

Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach

In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.

Quantifying uncertainty of a mathematical model of drug transport in tumors

This paper presents a numerical simulation in the two-dimensional for a system of PDE governing drug transport in tumors with random coefficients, which is described as a random field. The continuous stochastic field is approximated by a finite number of random variables via the Karhunen–Loève expansion and transform the stochastic problem into a determinate one with a parameter in high dimension. Then we apply a finite difference scheme and the Euler–Maruyama Integrator in time. The Monte Carlo method is used to compute corresponding simple averages. We compute the error estimate using the Central Limits Theorem (CLT) and the error estimate for the finite difference method. Some numerical results are simulated to illustrate the theoretical analysis. We also propose a comparison between the stochastic and determinate solving processes of our system where we show the efficiency of our adopted method.

Bayesian inference with subset simulation in varying dimensions applied to the Karhunen–Loève expansion

AbstractUncertainties associated with spatially varying parameters are modeled through random fields discretized into a finite number of random variables. Standard discretization methods, such as the Karhunen–Loève expansion, use series representations for which the truncation order is specified a priori. However, when data is used to update random fields through Bayesian inference, a different truncation order might be necessary to adequately represent the posterior random field. This is an inference problem that not only requires the determination of the often high‐dimensional set of coefficients, but also their dimension. In this article, we develop a sequential algorithm to handle such inference settings and propose a penalizing prior distribution for the dimension parameter. The method is a variable‐dimensional extension of BUS (Bayesian Updating with Structural reliability methods), combined with subset simulation (SuS). The key idea is to replace the standard Markov Chain Monte Carlo (MCMC) algorithm within SuS by a trans‐dimensional MCMC sampler that is able to populate the discrete‐continuous parameter space. To address this task, we consider two types of MCMC algorithms that operate in a fixed‐dimensional saturated space. The performance of the proposed method with both MCMC variants is assessed numerically for two examples: a 1D cantilever beam with spatially varying flexibility and a 2D groundwater flow problem with uncertain permeability field.

Probabilistic analysis of consolidation problems using subset simulation

In geotechnical structures, the failure probability (P f ) is usually calculated using the Monte Carlo simulation. This method is very time-consuming, particularly when dealing with small failure probabilities. To overcome this problem and as an alternative to MCS, the Subset simulation approach was applied to study the coupled two-dimensional consolidation. This method aims at performing a probabilistic analysis of the consolidation coupled with a heterogeneous soil with spatially varying Young's modulus (E). The probabilistic numerical results have shown that the probability of exceeding an admissible vertical displacement calculated by Subset simulation is very close to that calculated by MCS, but with a very substantial reduction in the number of simulations. In this study, the random field has been discretized into a finite number of random variables using the Karhunen-Loeve expansion. A parametric study to investigate the effect of the soil variability on P f was presented and discussed. The effect of autocorrelation horizontal and vertical distances L x and L y of E on P f has shown that increasing L x and L y increases P f for both isotropic and anisotropic soils. In addition, this study has shown that P f is more sensitive to L y than L x . Finally, the increase in COV(E) increases P f .

Bayesian inference of random fields represented with the Karhunen–Loève expansion

The integration of data into engineering models involving uncertain and spatially varying parameters is oftentimes key to obtaining accurate predictions. Bayesian inference is effective in achieving such an integration. Uncertainties related to spatially varying parameters are typically represented through random fields discretized into a finite number of random variables. The prior correlation length and variance of the field, as well as the number of terms in the random field discretization, have a considerable impact on the outcome of the Bayesian inference, which has received little attention in the literature. Here, we investigate the implications of different choices in the prior random field model on the solution of Bayesian inverse problems. We employ the Karhunen–Loève expansion for the representation of random fields. We show that a higher-order Karhunen–Loève discretization is required in Bayesian inverse problems as compared to standard prior random field representations, since the updated fields are non-homogeneous. Furthermore, the smoothing effect of the forward operator has a large influence on the posterior solution, particularly if we are analyzing quantities of interest that are sensitive to local random fluctuations of the inverse quantity. This is also reflected in posterior predictions, such as the estimation of rare event probabilities. We illustrate these effects analytically through a 1D cantilever beam with spatially varying flexibility, and numerically using a 2D linear elasticity example where the Young’s modulus is spatially variable.

Level‐set topology optimization for robust design of structures under hybrid uncertainties

SummaryThis paper will develop a new robust topology optimization (RTO) method based on level sets for structures subject to hybrid uncertainties, with a more efficient Karhunen‐Loève hyperbolic Polynomial Chaos–Chebyshev Interval method to conduct the hybrid uncertain analysis. The loadings and material properties are considered hybrid uncertainties in structures. The parameters with sufficient information are regarded as random fields, while the parameters without sufficient information are treated as intervals. The Karhunen‐Loève expansion is applied to discretize random fields into a finite number of random variables, and then, the original hybrid uncertainty analysis is transformed into a new process with random and interval parameters, to which the hyperbolic Polynomial Chaos–Chebyshev Interval is employed for the uncertainty analysis. RTO is formulated to minimize a weighted sum of the mean and standard variance of the structural objective function under the worst‐case scenario. Several numerical examples are employed to demonstrate the effectiveness of the proposed RTO, and Monte Carlo simulation is used to validate the numerical accuracy of our proposed method.

Natural frequency analysis of functionally graded material beams with axially varying stochastic properties

• Functionally graded material properties of beams are modelled as random fields. • A generalized eigenvalue equation is derived for stochastic free vibration analysis. • Effective surrogate models are developed based on the polynomial chaos expansion. • Several examples are presented to demonstrate the engineering applications. The functionally graded material (FGM) has a potential to replace ordinary ones in engineering reality due to its superior thermal and dynamical characteristics. In this regard, the paper presents an effective approach for uncertain natural frequency analysis of composite beams with axially varying material properties. Rather than simply assuming the material model as a deterministic function, we further extend the FGM property as a random field, which is able to account for spatial variability in laboratory observations and in-field data. Due to the axially varying input uncertainty, natural frequencies of the stochastically FGM (S-FGM) beam become random variables . To this end, the Karhunen–Loève expansion is first introduced to represent the composite material random field as the summation of a finite number of random variables. Then, a generalized eigenvalue function is derived for stochastic natural frequency analysis of the composite beam. Once the mechanistic model is available, the brutal Monte-Carlo simulation (MCS) similar to the design of experiment can be used to estimate statistical characteristics of the uncertain natural frequency response. To alleviate the computational cost of the MCS method, a generalized polynomial chaos expansion model developed based on a rather small number of training samples is used to mimic the true natural frequency function. Case studies have demonstrated the effectiveness of the proposed approach for uncertain natural frequency analysis of functionally graded material beams with axially varying stochastic properties .

A multilevel sparse kernel-based stochastic collocation finite element method for elliptic problems with random coefficients

A new stochastic collocation finite element method is proposed for the numerical solution of elliptic boundary value problems (BVP) with random coefficients, assuming that the randomness is well-approximated by a finite number of random variables with given probability distributions. The proposed method consists of a finite element approximation in physical space, along with a stochastic collocation quadrature approach utilizing the recent Multilevel Sparse Kernel-Based Interpolation (MuSIK) technique (Georgoulis et al., 2013). MuSIK is based on a multilevel sparse grid-type algorithm with the basis functions consisting of directionally anisotropic Gaussian radial basis functions (kernels) placed at directionally-uniform grid-points. We prove that MuSIK is interpolatory at these nodes, and, therefore, can be naturally used to define a quadrature scheme. Numerical examples are also presented, assessing the performance of the new algorithm in the context of high-dimensional stochastic collocation finite element methods.

An IMEX finite element method for a linearized Cahn–Hilliard–Cook equation driven by the space derivative of a space–time white noise

We consider a model initial- and Dirichlet boundary- value problem for a linearized Cahn–Hilliard–Cook equation, in one space dimension, forced by the space derivative of a space–time white noise. First, we introduce a canvas problem, the solution to which is a regular approximation of the mild solution to the problem and depends on a finite number of random variables. Then, fully discrete approximations of the solution to the canvas problem are constructed using, for discretization in space, a Galerkin finite element method based on $$H^2$$ piecewise polynomials, and, for time-stepping, an implicit/explicit method. Finally, we derive a strong a priori estimate of the error approximating the mild solution to the problem by the canvas problem solution, and of the numerical approximation error of the solution to the canvas problem.

An efficient PDE-constrained stochastic inverse algorithm for probabilistic geotechnical site characterization using geophysical measurements

This paper develops an efficient, PDE-constrained stochastic inverse analysis methodology to probabilistically estimate site-specific elastic parameters of soil from sparse geophysical test measurements by accounting for the uncertain spatial variability of soil deposits and any measurement uncertainty associated with the geophysical experiment. Hypothesizing the soil parameters at any site to be three-dimensional, heterogeneous, anisotropic random fields, the methodology first probabilistically simulates the geophysical experiment using the finite element method in conjunction with a stochastic collocation approach to compute statistical measures of a quantity of interest such as the soil displacement or acceleration throughout the soil domain. To this end, the random fields are discretized into finite number of random variables by utilizing a Gaussian mixture model that allows for mimicking the soil formation process. The parameters of the random fields are initially assumed based on the generic data available in the literature for the geological soil type. The stochastic collocation approach utilizes a recently developed non-product quadrature method, conjugate unscented transformation, to accurately estimate the statistical moments corresponding to the model response variables in a computationally efficient manner. The methodology, then, employs a minimum variance framework to fuse the finite element model output with sparse real measurements to update the initially assumed soil statistical parameters. The methodology is illustrated through numerical geophysical experiments at a fictitious geotechnical site and is verified with three very different true profiles of the soil modulus. Moreover, a probabilistic sensitivity analysis is carried out by varying the number and locations of sensors. It is observed that by judiciously selecting the sensor locations, following a set of information maps, obtained by exploiting the equations of the minimum variance scheme, more information may be extracted from any geophysical experiments, leading to less uncertain estimates of the soil parameters.

A Chebyshev polynomial-based Galerkin method for the discretization of spatially varying random properties

The uncertainty related to the spatially varying random property requires to be described as a random field, which in definition consists of an infinite number of random variables that are attached to each point of a continuous domain of the random field. The utility of Karhunen–Loeve expansion allows to represent the random field as the summation of a series of deterministic functions and random variables. By this way, the random field is represented by using a finite number of random variables. However, the K–L expansion depends crucially upon the eigen-solutions of the Fredholm integral equation of the second kind. Therefore, the interest of practical random field simulations is mainly concerned with containing as few random variables as possible, but needs to guarantee small approximation errors for the reproduced results in the meantime. To this end, the Chebyshev polynomial is introduced in the paper for the solution of the integral eigenvalue problem. As an effective class of basis functions for the Galerkin projection, the convergence rate of the Chebyshev polynomial-based Galerkin method is assessed by means of the global variance and covariance errors. And its accuracy is examined by reproducing several stationary and non-stationary, Gaussian, and non-Gaussian random fields. Together with the practical simulations of the concrete compressive strength field, numerical results show that the Chebyshev polynomial-based Galerkin scheme has engineering applications.

Stochastic regularity of a quadratic observable of high-frequency waves

We consider high-frequency waves satisfying the scalar wave equation with highly oscillatory initial data. The wave speed, and the phase and amplitude of the initial data are assumed to be uncertain, described by a finite number of random variables with known probability distributions. We define quantities of interest (QoIs), or observables, as local averages of the squared modulus of the wave solution. We aim to quantify the regularity of these QoIs in terms of the input random parameters, and the wave length, i.e., to estimate the size of their derivatives. The regularity is important for uncertainty quantification methods based on interpolation in the stochastic space. In particular, the size of the derivatives should be bounded independently of the wave length. In this paper, we are able to show that when these QoIs are approximated by Gaussian beam superpositions, they indeed have this property, despite the highly oscillatory character of the waves.

A posteriori error estimation for elliptic partial differential equations with small uncertainties

In this article, a finite element error analysis is performed on a class of linear and nonlinear elliptic problems with small uncertain input. Using a perturbation approach, the exact (random) solution is expanded up to a certain order with respect to a parameter that controls the amount of randomness in the input and discretized by finite elements. We start by studying a diffusion (linear) model problem with a random coefficient characterized via a finite number of random variables. The main focus of the article is the derivation of a priori and a posteriori error estimates of the error between the exact and approximate solution in various norms, including goal‐oriented error estimation. The analysis is then extended to a class of nonlinear problems. We finally illustrate the theoretical results through numerical examples, along with a comparison with the Stochastic Collocation method in terms of computational costs. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 175–212, 2016

Sensitivity analysis of complex models: Coping with dynamic and static inputs

In this paper, we address the issue of conducting a sensitivity analysis of complex models with both static and dynamic uncertain inputs. While several approaches have been proposed to compute the sensitivity indices of the static inputs (i.e. parameters), the one of the dynamic inputs (i.e. stochastic fields) have been rarely addressed. For this purpose, we first treat each dynamic as a Gaussian process. Then, the truncated Karhunen–Loève expansion of each dynamic input is performed. Such an expansion allows to generate independent Gaussian processes from a finite number of independent random variables. Given that a dynamic input is represented by a finite number of random variables, its variance-based sensitivity index is defined by the sensitivity index of this group of variables. Besides, an efficient sampling-based strategy is described to estimate the first-order indices of all the input factors by only using two input samples. The approach is applied to a building energy model, in order to assess the impact of the uncertainties of the material properties (static inputs) and the weather data (dynamic inputs) on the energy performance of a real low energy consumption house.

Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs

AbstractIn this article, we propose and analyse a sparse grid collocation method to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive an optimality system. A Galerkin approximation in physical space and a sparse grid collocation in the probability space is used. Error estimates for a fully discrete solution using an appropriate norm are provided, and we analyse the computational efficiency. Computational evidence complements the present theory, to show the effectiveness of our stochastic collocation method.

A MULTI-FIDELITY STOCHASTIC COLLOCATION METHOD FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH RANDOM INPUT DATA

Over the last few years there have been dramatic advances in the area of uncertainty quantification. In particular, we have seen a surge of interest in developing efficient, scalable, stable, and convergent computational methods for solving differential equations with random inputs. Stochastic collocation (SC) methods, which inherit both the ease of implementation of sampling methods like Monte Carlo and the robustness of nonsampling ones like stochastic Galerkin to a great deal, have proved extremely useful in dealing with differential equations driven by random inputs. In this work we propose a novel enhancement to stochastic collocation methods using deterministic model reduction techniques. Linear parabolic partial differential equations with random forcing terms are analysed. The input data are assumed to be represented by a finite number of random variables. A rigorous convergence analysis, supported by numerical results, shows that the proposed technique is not only reliable and robust but also efficient.

Monte Carlo Finite Volume Element Methods for the Convection-Diffusion Equation with a Random Diffusion Coefficient

The paper presents a framework for the construction of Monte Carlo finite volume element method (MCFVEM) for the convection-diffusion equation with a random diffusion coefficient, which is described as a random field. We first approximate the continuous stochastic field by a finite number of random variables via the Karhunen-Loève expansion and transform the initial stochastic problem into a deterministic one with a parameter in high dimensions. Then we generate independent identically distributed approximations of the solution by sampling the coefficient of the equation and employing finite volume element variational formulation. Finally the Monte Carlo (MC) method is used to compute corresponding sample averages. Statistic error is estimated analytically and experimentally. A quasi-Monte Carlo (QMC) technique with Sobol sequences is also used to accelerate convergence, and experiments indicate that it can improve the efficiency of the Monte Carlo method.

Solving Continuous Models with Dependent Uncertainty: A Computational Approach

This paper presents a computational study on a quasi-Galerkin projection-based method to deal with a class of systems of random ordinary differential equations (r.o.d.e.’s) which is assumed to depend on a finite number of random variables (r.v.’s). This class of systems of r.o.d.e.’s appears in different areas, particularly in epidemiology modelling. In contrast with the other available Galerkin-based techniques, such as the generalized Polynomial Chaos, the proposed method expands the solution directly in terms of the random inputs rather than auxiliary r.v.’s. Theoretically, Galerkin projection-based methods take advantage of orthogonality with the aim of simplifying the involved computations when solving r.o.d.e.’s, which means to compute both the solution and its main statistical functions such as the expectation and the standard deviation. This approach requires the previous determination of an orthonormal basis which, in practice, could become computationally burden and, as a consequence, could ruin the method. Motivated by this fact, we present a technique to deal with r.o.d.e.’s that avoids constructing an orthogonal basis and keeps computationally competitive even assuming statistical dependence among the random input parameters. Through a wide range of examples, including a classical epidemiologic model, we show the ability of the method to solve r.o.d.e.’s.

A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties

We here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables. It is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales. An efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale, involving a deterministic operator, and local problems at a micro scale for which we have the possibility to use fine approximation spaces. Global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions. Convergence properties of these tensor based methods, which are closely related to spectral decompositions, benefit from the separation of scales. Different types of uncertainties are considered at the micro level. They may be associated with some variability in the operator or source terms, or even with some geometrical variability. In the latter case, specific reformulations of local problems using fictitious domain methods are introduced.

A stochastic collocation method for the second order wave equation with a discontinuous random speed

In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.