Stochastic Discretization Research Articles

Randomized complexity of parametric integration and the role of adaption II. Sobolev spaces

We study the complexity of randomized computation of integrals depending on a parameter, with integrands from Sobolev spaces. That is, for r,d1,d2∈N, 1≤p,q≤∞, D1=[0,1]d1, and D2=[0,1]d2 we are given f∈Wpr(D1×D2) and we seek to approximate(Sf)(s)=∫D2f(s,t)dt(s∈D1), with error measured in the Lq(D1)-norm. Information is standard, that is, function values of f. Our results extend previous work of Heinrich and Sindambiwe (1999) [10] for p=q=∞ and Wiegand (2006) [15] for 1≤p=q<∞. Wiegand's analysis was carried out under the assumption that Wpr(D1×D2) is continuously embedded in C(D1×D2) (embedding condition). We also study the case that the embedding condition does not hold. For this purpose a new ingredient is developed – a stochastic discretization technique. In Part I a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting was solved. Here a further aspect of this problem is settled.

Comments on “Fuzzy Simulations for Expected Values of Functions of Fuzzy Numbers and Intervals”

It must be pointed out that the conclusion about the perfect performance of a recently proposed fuzzy simulation technique for expected values of fuzzy functions by Liu <i>et al.</i> (IEEE Trans. Fuzzy Syst., 29(6), 2021), the <i>improved stochastic discretization algorithm</i> (iSDA), is absolute and dangerous, which requires an urgent correction. Therefore, in this note, first and foremost, several theorems are initiated to address and specify the usage scope of the iSDA, which is related to the type of fuzzy variables involved in the functions. Further, examples and counter-examples with schematic diagrams based on the proposed theorems are provided to illustrate the performance of the iSDA. It is seen that the simulation results of the iSDA would be uncontrollable and lead to unbearable error for counter-examples, which immediately shows the necessity of this note. Second and incidentally, the respective convergence of the iSDA and another algorithm, the <i>special numerical integration algorithm</i> (NIA-S), are proved as supplementary contents for the original paper. Finally, the conclusion about the comparison between the iSDA and NIA-S is also restated.

Numerical investigation and ANN-based prediction on compressive strength and size effect using the concrete mesoscale concretization model

The concrete mesoscale concretization model (CMCM) is proposed using the statistical distribution and local spatial correlation theory to describe the randomization, continuum, and correlation of the material properties of mesoscale components. The distribution of material properties was obtained by stochastic discretization, spatial correlation correction, and remapping operations. The hypotheses of elastic modulus control (EMC) and characteristic deformation control (CDC) are used to describe the variation of mechanical parameters in the concrete damage plasticity model. The uniaxial compression behavior and size effect of concrete with different side lengths, aspect ratios, aggregate contents, air contents, and contact frictions are simulated using the above model. Without the contact frictions, the size effect of the sample’s side length is apparent, while the size effect of aspect ratio is not observed. The increase of size effect degree is contributed by the increase of air content rather than aggregate content. When the loading boundaries are constrained by friction, both the side length and aspect ratio of samples show the size effect. The increase of aggregate content leads to the increase of size effect degree, while the increase of air content leads to a minor increase of size effect degree. The developed CMCM can capture the mechanical responses, and the ANN-based prediction model is applicable to explore the relationship between compressive strength and sample parameters.

Arbitrary multi-resolution multi-wavelet-based polynomial chaos expansion for data-driven uncertainty quantification

Various real world problems deal with data-driven uncertainty. In particular, in geophysical applications the amount of available data is often limited, posing a challenge in the construction of an appropriate stochastic discretization. Arbitrary polynomial chaos is an alternative to tackle this challenge. Approximating the dependence of model output on the uncertain model parameters by expansion in an orthogonal polynomial basis using data-driven principles. This type of global polynomial representation suffers often from Gibbs’ phenomena, especially if applied in non-linear convection dominated problems that require to deal with discontinuities. The multi-resolution or multi-element framework has been successfully used for reducing Gibbs’ phenomena in intrusive stochastic discretizations. In the present work, we introduce a multi-resolution extension of the arbitrary polynomial chaos expansion which is based on the construction of piecewise polynomial. Gaussian quadrature nodes and weights that are computed using only stochastic (localized) moments provided by the underlying raw data. We enhance our approach by a multi-wavelet based stochastic adaptivity that assures a significant reduction of the computational costs. Numerical experiments of increasing complexity demonstrate the performance of the non-intrusive implementation of the introduced methods in relevant scenarios. The use of a carbon dioxide storage benchmark scenario allows one to compare the presented methodology with other stochastic discretization techniques applied to this benchmark.

A Particle Method for Solving Fredholm Equations of the First Kind

Fredholm integral equations of the first kind are the prototypical example of ill-posed linear inverse problems. They model, among other things, reconstruction of distorted noisy observations and indirect density estimation and also appear in instrumental variable regression. However, their numerical solution remains a challenging problem. Many techniques currently available require a preliminary discretization of the domain of the solution and make strong assumptions about its regularity. For example, the popular expectation maximization smoothing (EMS) scheme requires the assumption of piecewise constant solutions which is inappropriate for most applications. We propose here a novel particle method that circumvents these two issues. This algorithm can be thought of as a Monte Carlo approximation of the EMS scheme which not only performs an adaptive stochastic discretization of the domain but also results in smooth approximate solutions. We analyze the theoretical properties of the EMS iteration and of the corresponding particle algorithm. Compared to standard EMS, we show experimentally that our novel particle method provides state-of-the-art performance for realistic systems, including motion deblurring and reconstruction of cross-section images of the brain from positron emission tomography.

On Fuzzy Simulations for Expected Values of Functions of Fuzzy Numbers and Intervals

Based on existing fuzzy simulation algorithms, this article presents two innovative techniques for approximating the expected values of fuzzy numbers' monotone functions, which is of utmost importance in fuzzy optimization literature. In this regard, the stochastic discretization algorithm of Liu and Liu (2002) is enhanced by updating the discretization procedure for the simulation of the membership function and the calculation formula for the expected values. This is achieved through initiating a novel uniform sampling process and employing a formula for discrete fuzzy numbers, respectively, as the generated membership function in the stochastic discretization algorithm would adversely affect its accuracy to some extent. What is more, considering that the bisection procedure involved in the numerical integration algorithm of Li (2015) is time-consuming and also, not necessary for the specified types of fuzzy numbers, a special numerical integration algorithm is proposed, which can simplify the simulation procedure by adopting the analytical expressions of α-optimistic values. Subsequently, concerning the extensive applications of regular fuzzy intervals, several theorems are introduced and proved as an extended effort to apply the improved stochastic discretization algorithm and the special numerical integration algorithm to the issues of fuzzy intervals. Throughout this article, a series of numerical experiments are conducted from which the superiority of both the two novel techniques over others are conspicuously displayed in aspects of accuracy, stability, and efficiency.

Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel

We introduce a construction of multiscale tight frames on general domains. The frame elements are obtained by spectral filtering of the integral operator associated with a reproducing kernel. Our construction extends classical wavelets as well as generalized wavelets on both continuous and discrete non-Euclidean structures such as Riemannian manifolds and weighted graphs. Moreover, it allows to study the relation between continuous and discrete frames in a random sampling regime, where discrete frames can be seen as Monte Carlo estimates of the continuous ones. Pairing spectral regularization with learning theory, we show that a sample frame tends to its population counterpart, and derive explicit finite-sample rates on spaces of Sobolev and Besov regularity. Our results prove the stability of frames constructed on empirical data, in the sense that all stochastic discretizations have the same underlying limit regardless of the set of initial training samples.

Discrete approximation of the viscous HJ equation

We consider a stochastic discretization of the stationary viscous Hamilton-Jacobi equation on the flat d–dimensional torus \(\mathbb {T}^d\) associated with a Hamiltonian, convex and superlinear in the momentum variable. We show that each discrete problem admits a unique continuous solution on \(\mathbb {T}^d\), up to additive constants. By additionally assuming a technical condition on the associated Lagrangian, we show that each solution of the viscous Hamilton–Jacobi equation is the limit of solutions of the discrete problems, as the discretization step goes to zero.

Spatio-stochastic adaptive discontinuous Galerkin methods

We introduce a goal-oriented, adaptive framework for the uncertainty quantification of systems modeled by stochastically parametrized nonlinear hyperbolic and convection-dominated partial differential equations. Of particular interest are conservation laws in aerodynamics that may have a small number of stochastic parameters but exhibit strong nonlinearity and a wide range of scales. Our framework exploits localized structure in the spatio-parameter space to enable rapid, reliable uncertainty quantification for output quantities of interest. Our formulation comprises the following technical components: (i) a discontinuous Galerkin finite element method, which provides stability for convection-dominated problems; (ii) element-wise polynomial chaos expansions, which capture the parametric dependence of the solution in a way amenable to adaptation; (iii) the dual-weighted residual method, which provides global and element-wise error estimates for quantities of interest; and (iv) a projection-based anisotropic error indicator along with the associated adaptation mechanics that can detect and refine strongly directional features in the physical and/or parameter spaces simultaneously in an efficient manner. Both the spatial and stochastic discretization errors are controlled through the adaptive refinement of the spatial mesh or polynomial chaos expansion degree based on these anisotropic error indicators. We analyze stability, approximation properties, and a priori error bounds of the spatio-stochastic adaptive method. We finally demonstrate the effectiveness of our formulation for engineering-relevant transonic turbulent aerodynamics problems with uncertainties in flow conditions and turbulence parameters.

New Fuzzy Simulation-Based Algorithms for the Credibility of Fuzzy Events

Accurate and computationally efficient methodologies for calculating the credibility of a fuzzy event not only constitute great challenges to researchers, but their importance to practitioners is steadily increasing ever since. In this regard, in this article, we develop two fuzzy simulation-based algorithms using respectively the uniform discretization method and the bisection approach. Facilitating the monotonicity property of shape functions of regular fuzzy numbers, we demonstrate the stability and reliability of our treatment using relevant theorems. Subsequently, a comparison analysis is conducted, which verifies higher levels of accuracy and efficiency for them when the standard stochastic discretization algorithm is used as a benchmark. Furthermore, these novel simulation-based algorithms are extended to consider the case of credibility of joint fuzzy events. A series of numerical experiments are developed for demonstrating clearly the accuracy and computational efficiency of the treatment.

Bayesian network classifiers using ensembles and smoothing

Bayesian network classifiers are, functionally, an interesting class of models, because they can be learnt out-of-core, i.e. without needing to hold the whole training data in main memory. The selective K-dependence Bayesian network classifier (SKDB) is state of the art in this class of models and has shown to rival random forest (RF) on problems with categorical data. In this paper, we introduce an ensembling technique for SKDB, called ensemble of SKDB (ESKDB). We show that ESKDB significantly outperforms RF on categorical and numerical data, as well as rivalling XGBoost. ESKDB combines three main components: (1) an effective strategy to vary the networks that is built by single classifiers (to make it an ensemble), (2) a stochastic discretization method which allows to both tackle numerical data as well as further increases the variance between different components of our ensemble and (3) a superior smoothing technique to ensure proper calibration of ESKDB’s probabilities. We conduct a large set of experiments with 72 datasets to study the properties of ESKDB (through a sensitivity analysis) and show its competitiveness with the state of the art.

A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

This article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented. For the stochastic discretization, a non-intrusive approach, namely the Stochastic Collocation method is used. The spatio-temporal discretization relies on the Runge–Kutta Discontinuous Galerkin method. The a posteriori estimator is derived using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. The scaling properties of the residuals are investigated and the efficiency of the proposed adaptive algorithms is illustrated in various numerical examples.

Preconditioned Krylov subspace and GMRHSS iteration methods for solving the nonsymmetric saddle point problems

In the present paper, we propose a separate approach as a new strategy to solve the saddle point problem arising from the stochastic Galerkin finite element discretization of Stokes problems. The preconditioner is obtained by replacing the (1,1) and (1,2) blocks in the RHSS preconditioner by others well chosen and the parameter α in (2,2) −block of the RHSS preconditioner by another parameter β. The proposed preconditioner can be used as a preconditioner corresponding to the stationary itearative method or to accelerate the convergence of the generalized minimal residual method (GMRES). The convergence properties of the GMRHSS iteration method are derived. Meanwhile, we analyzed the eigenvalue distribution and the eigenvectors of the preconditioned matrix. Finally, numerical results show the effectiveness of the proposed preconditioner as compared with other preconditioners.

Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance

We introduce a general framework for approximating parabolic Stochastic Partial Differential Equations (SPDEs) based on fluctuation-dissipation balance. Using this approach we formulate Stochastic Discontinuous Galerkin Methods (SDGM). We show how methods with linear-time computational complexity can be developed for handling domains with general geometry and generating stochastic terms, handling both Dirichlet and Neumann boundary conditions. We demonstrate our approach on example systems and contrast with alternative approaches using direct stochastic discretizations based on random fluxes. We show how our Fluctuation-Dissipation Discretizations (FDD) framework allows us to compensate for discrepancies in dissipative properties between the continuous operators and their discretized versions. This allows us to handle general heterogeneous discretizations, accurately capturing statistical relations. Our FDD framework provides a general approach for formulating SDGM discretizations and other numerical methods for robust approximation of stochastic differential equations.

АЛГОРИТМ ОПРЕДЕЛЕНИЯ СДВИГА ФАЗ ГАРМОНИЧЕСКИХ СИГНАЛОВ С ИСПОЛЬЗОВАНИЕМ СТОХАСТИЧЕСКОЙ ДИСКРЕТИЗАЦИИ

The article focuses on the problem of estimating basic parameters of harmonic signals. An algorithm for determining the phase shift of such signals by the probabilistic-statistical method is proposed. An important advantage of the algorithm is the fact that the circulation time is aliquant to the signal period or shorter. It is important when measuring parameters of infra-low-frequency radio signals. Mathematical modeling of the algorithm was carried out to study errors and assess the possibility of practical application of the algorithm for processing infra-low-frequency radio signals at stochastic discretization. The results of a numerical experiment show that the error of determining the phase shift of harmonic signals using the developed algorithm on the basis of integral samples with stochastic discretization of time intervals according to the uniform distribution law is 1/100 of a percent, and little depends on the accuracy of signal discretization by level. The error obtained corresponds to the accuracy of discretization when translating the 12-bit ADCs into analog-digital converters. The proposed algorithm can be applied for processing infra-low-frequency radio signals in acoustics and communication, as well as in geophysical and biomedical research.

A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method

Abstract In this article we present an a posteriori error estimator for the spatial–stochastic error of a Galerkin-type discretization of an initial value problem for a random hyperbolic conservation law. For the stochastic discretization we use the stochastic Galerkin method and for the spatial–temporal discretization of the stochastic Galerkin system a Runge–Kutta discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos (2016, Hyperbolic Conservation Laws in Continuum Physics, 4th edn., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Berlin, Springer, pp. xxxviii+826), this leads to computable error bounds for the space–stochastic discretization error. Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretization. We conclude with some numerical examples confirming the theoretical findings.

Теорема о стохастической дискретизации изображений в радиолокации и связи.

Теорема о стохастической дискретизации изображений в радиолокации и связи.

Stochastic hyperelastic modeling considering dependency of material parameters

This paper investigates the uncertainty of a hyperelastic model by treating random material parameters as stochastic variables. For its stochastic discretization a polynomial chaos expansion (PCE) is used. An important aspect in our work is the consideration of stochastic dependencies in the stochastic modeling of Ogden’s material model. To this end, artificial experiments are generated using the auto-regressive moving average process based on real experiments. The parameter identification for all data provides statistics of Ogden’s material parameters, which are subsequently used for stochastic modeling. Stochastic dependencies are incorporated into the PCE using a Nataf transformation from dependent distributed random variables to independent standard normal distributed ones. The representative numerical example shows that our proposed method adequately takes into account the stochastic dependencies of Ogden’s material parameters.

АЛГОРИТМ ОПРЕДЕЛЕНИЯ ЧАСТОТЫ ГАРМОНИЧЕСКОГО СИГНАЛА С ИСПОЛЬЗОВАНИЕМ СТОХАСТИЧЕСКОЙ ДИСКРЕТИЗАЦИИ

The paper deals with the development of an algorithm for determining frequency of harmonic signals using a probabilistic-statistical method. The main feature of this algorithm is a short time of addressing to the investigated signal, which much shorter than signal period, according to three integrated sample collections with digital processing. Instantaneous values of the investigated signal in each sampling are based on stochastic discretization over time, according to the uniform distribution law. The main advantages of the algorithm are the short time of access to the signal under study and high accuracy of frequency measurement, which is essential for the infralow frequency signals with a duration period measured in minutes, hours, days, etc. There has been performed a numerical experiment in order to evaluate an error in determining frequency of such signals, depending on the accuracy of their sampling by real analog-to-digital converters. The paper shows that the error of frequency determined by the developed algorithm makes a few hundredths of a percent and scarcely depends on accuracy of a signal discretization by a certain level. The error obtained corresponds to discretization accuracy under conversion into accepted values of analogue-to-digital converters from 6- to 16-bit analogue-to-digital converters. The present algorithm may find practical use in radio technical processing of infralow frequency signals in acoustics, hydro-acoustics, seismic acoustics, underwater and underground communication.

Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins

In this work we propose an Uncertainty Quantification methodology for sedimentary basins evolution under mechanical and geochemical compaction processes, which we model as a coupled, time-dependent, non-linear, monodimensional (depth-only) system of PDEs with uncertain parameters. While in previous works (Formaggia et al. 2013, Porta et al. 2014) we assumed a simplified depositional history with only one material, in this work we consider multi-layered basins, in which each layer is characterized by a different material, and hence by different properties. This setting requires several improvements with respect to our earlier works, both concerning the deterministic solver and the stochastic discretization. On the deterministic side, we replace the previous fixed-point iterative solver with a more efficient Newton solver at each step of the time-discretization. On the stochastic side, the multi-layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters, that need an appropriate treatment for surrogate modeling techniques, such as sparse grids, to be effective. We propose an innovative methodology to this end which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space. The reference coordinate system is built upon exploiting physical features of the problem at hand. We employ the fact that the locations of the material interfaces display a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations. We showcase the capabilities of our numerical methodologies through two synthetic test cases. In particular, we show that our methodology reproduces with high accuracy multi-modal probability density functions displayed by target state variables (e.g., porosity).