High-dimensional Stochastic Problems Research Articles

Structural stochastic responses determination via a sample-based stochastic finite element method

This paper presents a new stochastic finite element method for computing structural stochastic responses of linear problems. The method provides a new expansion of stochastic response and decouples the stochastic response into a combination of a series of deterministic responses with random variable coefficients. A dedicated iterative algorithm is proposed to compute the deterministic responses and corresponding random variable coefficients one by one. The algorithm computes the deterministic responses and corresponding random variable coefficients in their individual spaces and is insensitive to stochastic dimensions, thus it can be applied to high-dimensional stochastic problems without extra difficulties. More importantly, the deterministic responses can be computed efficiently by use of existing finite element method (FEM) solvers, thus the proposed method can be easy to embed into existing FEM structural analysis software. Three practical examples, including low-dimensional and high-dimensional stochastic problems, are given to demonstrate the accuracy and effectiveness of the proposed method.

Technical Note—Preservation of Additive Convexity and Its Applications in Stochastic Optimization Problems

Preservation Results for Proving Additively Convex Value Functions for High-Dimensional Stochastic Optimization Problems

Deep Neural Networks Algorithms for Stochastic Control Problems on Finite Horizon: Convergence Analysis

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value function by Monte Carlo regression. This is achieved in the dynamic programming recursion by performance or hybrid iteration, and regress now methods from numerical probabilities. We provide a theoretical justification of these algorithms. Consistency and rate of convergence for the control and value function estimates are analyzed and expressed in terms of the universal approximation error of the neural networks, and of the statistical error when estimating network function, leaving aside the optimization error. Numerical results on various applications are presented in a companion paper (arxiv.org/abs/1812.05916) and illustrate the performance of the proposed algorithms.

Flow-driven spectral chaos (FSC) method for simulating long-time dynamics of arbitrary-order non-linear stochastic dynamical systems

Uncertainty quantification techniques such as the time-dependent generalized polynomial chaos (TD-gPC) use an adaptive orthogonal basis to better represent the stochastic part of the solution space (aka random function space) in time. However, because the random function space is constructed using tensor products, TD-gPC-based methods are known to suffer from the curse of dimensionality. In this paper, we introduce a new numerical method called the flow-driven spectral chaos (FSC) which overcomes this curse of dimensionality at the random-function-space level. The proposed method is not only computationally more efficient than existing TD-gPC-based methods but is also far more accurate. The FSC method uses the concept of enriched stochastic flow maps to track the evolution of a finite-dimensional random function space efficiently in time. To transfer the probability information from one random function space to another, two approaches are developed and studied herein. In the first approach, the probability information is transferred in the mean-square sense, whereas in the second approach the transfer is done exactly using a new theorem that was developed for this purpose. The FSC method can quantify uncertainties with high fidelity, especially for the long-time response of stochastic dynamical systems governed by ODEs of arbitrary order. Six representative numerical examples, including a nonlinear problem (the Van-der-Pol oscillator), are presented to demonstrate the performance of the FSC method and corroborate the claims of its superior numerical properties. Finally, a parametric, high-dimensional stochastic problem is used to demonstrate that when the FSC method is used in conjunction with Monte Carlo integration, the curse of dimensionality can be overcome altogether.

Deep Neural Network Approximation of Nonlinear Model Predictive Control

This paper focuses on developing effective computational methods to enable the real-time application of model predictive control (MPC) for nonlinear systems. To achieve this goal, we follow the idea of approximating the MPC control law with a Deep Neural Network (DNN). To train the deep neural network offline, we propose a new “optimize and train” method that combines the steps of data generation and neural network training into a single high-dimensional stochastic optimization problem. This approach directly optimizes the closed loop performance of the DNN controller over a finite horizon for a number of initial states. The large-scale optimization problem can be solved efficiently using parallel computing techniques. The benefits of this approach over the conventional “optimize then train” protocol is illustrated through numerical results.

Self-Tuned Mirror Descent Schemes for Smooth and Nonsmooth High-Dimensional Stochastic Optimization

We consider randomized block coordinate stochastic mirror descent (RBSMD) methods for solving high-dimensional stochastic optimization problems with strongly convex objective functions. Our goal is to develop RBSMD schemes that achieve a rate of convergence with a minimum constant factor with respect to the choice of the stepsize sequence. To this end, we consider both subgradient and gradient RBSMD methods addressing nonsmooth and smooth problems, respectively. For each scheme, (i) we develop self-tuned stepsize rules characterized in terms of problem parameters and algorithm settings; (ii) we show that the non-averaging iterate generated by the underlying RBSMD method converges to the optimal solution both in an almost sure and a mean sense; (iii) we show that the mean squared error is minimized. When problem parameters are unknown, we develop a unifying self-tuned update rule that can be applied in both subgradient and gradient SMD methods, and show that for any arbitrary and small enough initial stepsize, a suitably defined error bound is minimized. We provide constant factor comparisons with standard SMD and RBSMD methods. Our numerical experiments performed on an SVM model display that the self-tuned schemes are significantly robust with respect to the choice of problem parameters, and the initial stepsize.

Dimensionwise multivariate orthogonal polynomials in general probability spaces

This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, encompassing hierarchically organized, measure-consistent multivariate orthogonal polynomials in dependent random variables. In contrast to the existing PDD, which is valid strictly for independent random variables, no tensor-product structure or product-type probability measure is imposed or necessary. Fundamental mathematical properties of GPDD are examined by creating dimensionwise decomposition of polynomial spaces, deriving statistical properties of random orthogonal polynomials, demonstrating completeness of orthogonal polynomials for requisite assumptions, and upholding mean-square convergence to the correct limit. The GPDD approximation proposed should be effective in solving high-dimensional stochastic problems subject to dependent variables with a large class of non-product-type probability measures.

Probabilistic stability analysis of an embankment dam considering soil spatial variability

A probabilistic stability analysis of an embankment dam taking into account soil spatial variabilities is presented in this paper. The effective cohesion and friction angle of the backfill are modelled as two anisotropic cross-correlated lognormal random fields by using the Karhunen–Loève Expansion. The probabilistic analyses are performed by a meta-modelling technique SPCE/GSA which combines the Sparse Polynomial Chaos Expansion (SPCE) and the Global Sensitivity Analysis (GSA). This technique is targeted to deal with high dimensional stochastic problems and can provide a variety of interesting results in terms of probability density function (PDF) and statistical moments of the model response, failure probability and sensitivity index of each input variable. Concerning the deterministic calculations, two models (limit equilibrium and finite difference) are created and compared in a probabilistic framework. The influences of the soil spatial variability and the cross-correlation between the shear strength parameters on the reliability analysis results are investigated.

Stochastic dynamic response analysis and reliability assessment of non-linear structures under fully non-stationary ground motions

In this paper, stochastic dynamic response analysis and reliability assessment of nonlinear structures under fully non-stationary ground motions is investigated based on the probability density evolution method (PDEM). To handle this problem, a new method is proposed for points selection in high-dimensional random-variate space for PDEM, which is of paramount importance to the tradeoff of accuracy and efficiency. First, the original high-dimensional space is decomposed into several two-dimensional orthogonal subspaces. Then, each subspace is partitioned via the Voronoi cells and the representative points and assigned probabilities in each subspace are specified accordingly. Finally, the representative points and assigned probabilities in each subspace are randomly paired to formulate the representative points and assigned probabilities in high-dimension. A numerical example is studied to validate the proposed method, which indicates that the proposed method is of efficiency and accuracy for high-dimensional stochastic dynamic problem of structures. The problem, which needs to be further investigated, is also pointed out.

A comparative study of different reliability methods for high dimensional stochastic problems related to earth dam stability analyses

This article presents a probabilistic stability analysis of an existing earth dam including uncertainty quantification of soil properties and a reliability analysis of the dam sliding stability. The analyses are conducted by exploiting the available field measurements, and then by performing the Monte Carlo Simulation (MCS). Random fields and random variables approaches are both used to model the soil variabilities. Two left-and-right-bounded distributions, beta and truncated normal, are considered for the input random variables in the reliability analysis, and the influence of the horizontal autocorrelation distance on the failure probability is investigated.A comparative study of different reliability methods is also carried out by comparing with the results of the MCS. The considered reliability methods are: the Subset Simulation (SS), the Moment Method (MM), the Sparse Polynomial Chaos Expansion in combination with the Global Sensitivity Analysis (SPCE/GSA) and the Sparse Polynomial Chaos Expansion in combination with the Sliced Inverse Regression (SPCE/SIR). The comparative study shows that all these methods can give accurate results in term of the dam failure probability with small errors. It is also found that the most accurate method is the SPCE/GSA and the most efficient method is the SS.

Deep Learning-Based BSDE Solver for Libor Market Model with Applications to Bermudan Swaption Pricing and Hedging

The Libor market model, also known as the BGM Model, is a term structure model of interest rates. It is widely used for pricing interest rate derivatives, especially Bermudan swaptions, and other exotic Libor callable derivatives. For numerical implementation the pricing of derivatives with Libor market models is mainly carried out with Monte Carlo simulation. The PDE grid approach is not particularly feasible due to the “Curse of Dimensionality”. The standard Monte Carlo method for American swaption pricing more or less uses regression to estimate the expected value as a linear combination of basis functions as demonstrated in the classical paper by Longstaff and Schwartz [Longstaff01]. However, [Longstaff01] only provides the lower bound for American option price. Another complexity arises from applying Monte Carlo simulation is the computation of the sensitivities of the option, the so-called “Greeks”, which are fundamental for a trader's hedging activity. Recently, an alternative numerical method based on deep learning and backward stochastic differential equations (BSDEs) appeared in quite a few researches [Weinan17, Jiequn17]. For European style options the feedforward deep neural networks (DNN) show not only feasibility but also efficiency to obtain both prices and numerical Greeks. The standard LMM implementation requires dimension five or higher in factor space even after PCA, which cannot be solved by traditional PDE solvers, such as finite differences or finite elements methods. In this paper, a new backward DNN solver is proposed for Bermudan swaptions. Our approach is representing financial pricing problems in the form of high dimensional stochastic optimal control problems, FBSDEs, or equivalent PDEs. We demonstrate that using backward DNN the high-dimension Bermudan swaption pricing and hedging can be solved effectively and efficiently. A comparison between Monte Carlo simulation and the new method for pricing vanilla interest rate options manifests the superior performance of the new method. We then use this method to calculate prices and Greeks of Bermudan swaptions as a prelude for other Libor callable derivatives.

Spectral representation of stochastic field data using sparse polynomial chaos expansions

Uncertainty quantification is an emerging research area aiming at quantifying the variation in engineering system outputs due to uncertain inputs. One approach to study problems in uncertainty quantification is using polynomial chaos expansions. Though, a well-known limitation of polynomial chaos approaches is that their computational cost becomes prohibitive when the dimension of the stochastic space is large. In this paper, we propose a procedure to solve high dimensional stochastic problems with a limited computational budget. The methodology is based on an existing non-intrusive model reduction scheme for polynomial chaos representation, introduced by Raisee et al. [1], that is further extended by introducing sparse polynomial chaos expansions. Specifically, an optimal stochastic basis is calculated from a coarse scale analysis, using proper orthogonal decomposition and sparse polynomial chaos and is then utilized in the fine scale analysis. This way, the computational expense on both the coarse and fine discretization levels is drastically reduced. Two application examples are considered to validate the proposed method and demonstrate its potential in solving high dimensional uncertainty quantification problems. One analytical stochastic problems is first studied, where up to 20 uncertainties were introduced in order to challenge the proposed method. A more realistic CFD type application is then discussed. It consists of a two dimensional NACA 0012 symmetric profile operating at subsonic flight conditions. It is shown that the proposed reduced order method based on sparse polynomial chaos expansions is able to predict statistical quantities with little loss of information, at a cheaper cost than other state-of-the-art techniques.

An efficient adaptive sparse grid collocation method through derivative estimation

For uncertainty propagation of highly complex and/or nonlinear problems, one must resort to sample-based non-intrusive approaches (Le Maître and Knio, 2010). In such cases, minimizing the number of function evaluations required to evaluate the response surface is of paramount importance. Sparse grid approaches have proven effective in reducing the number of sample evaluations. For example, the discrete projection collocation method has the notable feature of exhibiting fast convergence rates when approximating smooth functions; however, it lacks the ability to accurately and efficiently track response functions that exhibit fluctuations, abrupt changes or discontinuities in very localized regions of the input domain. On the other hand, the piecewise linear collocation interpolation approach can track these localized variations in the response surface efficiently, but it converges slowly in the smooth regions. The proposed methodology, building on an existing work on adaptive hierarchical sparse grid collocation algorithm (Ma and Zabaras, 2009), is able to track localized behavior while also avoiding unnecessary function evaluations in smoother regions of the stochastic space by using a finite difference based one-dimensional derivative evaluation technique in all the dimensions. This derivative evaluation technique leads to faster convergence in the smoother regions than what is achieved in the existing collocation interpolation approaches. Illustrative examples show that this method is well suited to high-dimensional stochastic problems, and that stochastic elliptic problems with stochastic dimension as high as 100 can be dealt with effectively.

An efficient non-intrusive reduced basis model for high dimensional stochastic problems in CFD

The major challenge industrial applications of uncertainty quantification (UQ) are facing, is the curse of dimensionality as a result of a large number of uncertainties. In this work, an efficient non-intrusive model reduction scheme is presented for UQ using proper orthogonal decomposition and a non-intrusive polynomial chaos approach based on regression. The main idea is to extract the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use this basis for the fine scale analysis. To validate the developed non-intrusive, reduced-order model, it is applied to two CFD type applications: (1) flow over a 2D RAE2822 airfoil and (2) the NASA Rotor 37, a transonic axial flow compressor. For the RAE2822, the surface of the airfoil is assumed to be stochastic and is defined as a random field with a given covariance and modeled using Karhunen–Loe`ve expansion. For the NASA Rotor 37, the geometry of the rotor blade is parameterized into 2D sections of airfoils. A total of 21 uncertain parameters is considered containing both operational and geometrical uncertainties. The statistical results are compared with those of the standard polynomial chaos method. The results show that the developed model produces statistical results with good accuracy. It is found that the memory requirements and the computational cost of the reduced-order model is 5–10 times lower than that of the standard polynomial chaos.

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10 different input distributions or histograms.

A stochastic model reduction method for nonlinear unconfined flow with multiple random input fields

In this paper we present a stochastic model reduction method for efficiently solving nonlinear unconfined flow problems in heterogeneous random porous media. The input random fields of flow model are parameterized in a stochastic space for simulation. This often results in high stochastic dimensionality due to small correlation length of the covariance functions of the input fields. To efficiently treat the high-dimensional stochastic problem, we extend a recently proposed hybrid high-dimensional model representation (HDMR) technique to high-dimensional problems with multiple random input fields and integrate it with a sparse grid stochastic collocation method (SGSCM). Hybrid HDMR can decompose the high-dimensional model into a moderate M-dimensional model and a few one-dimensional models. The moderate dimensional model only depends on the most M important random dimensions, which are identified from the full stochastic space by sensitivity analysis. To extend the hybrid HDMR, we consider two different criteria for sensitivity test. Each of the derived low-dimensional stochastic models is solved by the SGSCM. This leads to a set of uncoupled deterministic problems at the collocation points, which can be solved by a deterministic solver. To demonstrate the efficiency and accuracy of the proposed method, a few numerical experiments are carried out for the unconfined flow problems in heterogeneous porous media with different correlation lengths. The results show that a good trade-off between computational complexity and approximation accuracy can be achieved for stochastic unconfined flow problems by selecting a suitable number of the most important dimensions in the M-dimensional model of hybrid HDMR.

An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficient

AbstractIn this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions.Our ANOVA-DSM consists of offline and online stages. In the offline stage, the original high-dimensional stochastic problem is decomposed into a series of low-dimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using the Karhunen-Loève expansion (KLE) and a two-level preconditioning optimization approach. Multiple trial functions are used to enrich the stochastic basis and improve the accuracy. In the online stage, we solve each stochastic subproblem for any given forcing function by projecting the stochastic solution into the data-driven stochastic basis constructed offline. In our ANOVA-DSM framework, solving the original highdimensional stochastic problem is reduced to solving a series of ANOVA-decomposed stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided to further reduce the number of the stochastic subproblems and speed up our method. To demonstrate the accuracy and efficiency of our method, numerical examples are presented for one- and two-dimensional elliptic PDEs with random coefficients.

Partitioned treatment of uncertainty in coupled domain problems: A separated representation approach

This work is concerned with the propagation of uncertainty across coupled domain problems with high-dimensional random inputs. A stochastic model reduction approach based on low-rank separated representations is proposed for the partitioned treatment of the uncertainty space. The construction of the coupled domain solution is achieved though a sequence of approximations with respect to the dimensionality of the random inputs associated with each individual sub-domain and not the combined dimensionality, hence drastically reducing the overall computational cost. The coupling between the sub-domain solutions is done via the classical finite element tearing and interconnecting (FETI) method, thus providing a well suited framework for parallel computing. Two high-dimensional stochastic problems, a 2D elliptic PDE with random diffusion coefficient and a stochastic linear elasticity problem, have been considered to study the performance and accuracy of the proposed stochastic coupling approach.

ANALYSIS OF VARIANCE-BASED MIXED MULTISCALE FINITE ELEMENT METHOD AND APPLICATIONS IN STOCHASTIC TWO-PHASE FLOWS

The stochastic partial differential systems have been widely used to model physical processes, where the inputs involve large uncertainties. Flows in random and heterogeneous porous media is one of the cases where the random inputs (e.g., permeability) are often modeled as a stochastic field with high-dimensional random parameters. To treat the high dimensionality and heterogeneity efficiently, model reduction is employed in both stochastic space and physical space. An analysis of variance (ANOVA)-based mixed multiscale finite element method (MsFEM) is developed to decompose the high-dimensional stochastic problem into a set of lower-dimensional stochastic subproblems, which require much less computational complexity and significantly reduce the computational cost in stochastic space, and the mixed MsFEM can capture the heterogeneities on a coarse grid to greatly reduce the computational cost in the spatial domain. In addition, to enhance the efficiency of the traditional ANOVA method, an adaptive ANOVA method based on a new adaptive criterion is developed, where the most active dimensions can be selected to greatly reduce the computational cost before conducting ANOVA decomposition. This novel adaptive criterion is based on variance-decomposition method coupled with sparse-grid probabilistic collocation method or multilevel Monte Carlo method. The advantage of this adaptive criterion lies in its much lower computational overhead for identifying the active dimensions and interactions. A number of numerical examples in two-phase stochastic flows are presented and demonstrate the accuracy and performance of the adaptive ANOVA-based mixed MsFEM.

An Optimal Approximate Dynamic Programming Algorithm for Concave, Scalar Storage Problems With Vector-Valued Controls

We prove convergence of an approximate dynamic programming algorithm for a class of high-dimensional stochastic control problems linked by a scalar storage device, given a technical condition. Our problem is motivated by the problem of optimizing energy flows for a power grid supported by grid-level storage. The problem is formulated as a stochastic, dynamic program, where we estimate the value of resources in storage using a piecewise linear value function approximation. Given the technical condition, we provide a rigorous convergence proof for an approximate dynamic programming algorithm, which can capture the presence of both the amount of energy held in storage as well as other exogenous variables. Our algorithm exploits the natural concavity of the problem to avoid any need for explicit exploration policies.