Wiener Noise Research Articles

Orders of convergence in the averaging principle for SPDEs: The case of a stochastically forced slow component

This article is devoted to the analysis of semilinear, parabolic, Stochastic Partial Differential Equations, with slow and fast time scales. Asymptotically, an averaging principle holds: the slow component converges to the solution of another semilinear, parabolic, SPDE, where the nonlinearity is averaged with respect to the invariant distribution of the fast process.We exhibit orders of convergence, in both strong and weak senses, in two relevant situations, depending on the spatial regularity of the fast process and on the covariance of the Wiener noise in the slow equation. In a very regular case, strong and weak orders are equal to 12 and 1. In a less regular case, the weak order is also twice the strong order.This study extends previous results concerning weak rates of convergence, where either no stochastic forcing term was included in the slow equation, or the covariance of the noise was extremely regular.An efficient numerical scheme, based on Heterogeneous Multiscale Methods, is briefly discussed.

PERFORMANCE ANALYSIS OF ADAPTIVE FILTER AND FIR WIENER FILTER FOR NOISE CANCELLATION IN AUDIO SIGNALS

Speech has always been one of the most important carriers of information for people and has become a challenge to maintain its high quality. When the speech signal and noise both change continuously, then arises the need for algorithm that will form best estimation of noise signal. In Adaptive Noise Cancellation and Wiener Noise Cancellation two inputs - primary and reference signals are used. The primary input receives signal from the signal source which has been corrupted with a noise uncorrelated to the signal. The reference input receives noise signal uncorrelated with the signal but correlated in some way to the noise signal in primary input. The reference input is filtered to obtain a close estimate of primary input noise which is then subtracted from the corrupted signal at the primary input to produce an estimate of a clean uncorrupted signal. The audio signal corrupted with noise is used as a primary input and a noise signal is used as reference input. Computer simulations are carried out using MATLAB and illustrated.

Infinite-dimensional bilinear and stochastic balanced truncation with explicit error bounds

Along the ideas of Curtain and Glover (in: Bart, Gohberg, Kaashoek (eds) Operator theory and systems, Birkhäuser, Boston, 1986), we extend the balanced truncation method for (infinite-dimensional) linear systems to arbitrary-dimensional bilinear and stochastic systems. In particular, we apply Hilbert space techniques used in many-body quantum mechanics to establish new fully explicit error bounds for the truncated system and prove convergence results. The functional analytic setting allows us to obtain mixed Hardy space error bounds for both finite-and infinite-dimensional systems, and it is then applied to the model reduction of stochastic evolution equations driven by Wiener noise.

A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations

In this paper the numerical solution of non-autonomous semilinear stochastic evolution equations driven by an additive Wiener noise is investigated. We introduce a novel fully discrete numerical approximation that combines a standard Galerkin finite element method with a randomized Runge-Kutta scheme. Convergence of the method to the mild solution is proven with respect to the $L^p$-norm, $p \in [2,\infty)$. We obtain the same temporal order of convergence as for Milstein-Galerkin finite element methods but without imposing any differentiability condition on the nonlinearity. The results are extended to also incorporate a spectral approximation of the driving Wiener process. An application to a stochastic partial differential equation is discussed and illustrated through a numerical experiment.

A note on doubly nonlinear SPDEs with singular drift in divergence form

We prove well-posedness for a class of second-order SPDEs with multiplicative Wiener noise and doubly nonlinear drift of the form –div \gamma(\nabla \cdot) + \beta(\cdot) , where \gamma is the subdifferential of a convex function on \mathbb R^d and \beta is a maximal monotone graph everywhere defined on \mathbb R , on which neither growth nor continuity assumptions are imposed.

Random attractor for the 3D viscous primitive equations driven by fractional noises

We develop a new and general method to prove the existence of the random attractor (strong attractor) for the primitive equations (PEs) of large-scale ocean and atmosphere dynamics under non-periodic boundary conditions and driven by infinite-dimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the time-uniform a priori estimates in some Sobolev space whose regularity is higher than the solution space. But this method can not be applied to the 3D stochastic PEs with the non-periodic boundary conditions. Therefore, the existence of universal attractor (weak attractor) was established in previous works (see [15,16]). The main idea of our method is that we first derive that P-almost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., using our method we only need to obtain time-uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system, while the common method need to establish time-uniform a priori estimates in a more regular functional space than the solution space. Take the stochastic PEs for example, as the unique strong solution to the stochastic PEs belongs to C([0,T];(H1(℧))3), in view of our method, we only need to obtain the time-uniform a priori estimates in the solution space (H1(℧))3 to prove the existence of random attractor for this stochastic system, while the common method need to establish time-uniform a priori estimates for the solution in the functional space (H2(℧))3. However, time-uniform a priori estimates in (H2(℧))3 for the solution to stochastic PEs are too difficult to be established. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by Wiener noises, fractional noises and even jump noises. In a forth coming paper, using this new method we [46] prove the existence of random attractor for the stochastic nematic liquid crystals equations. This is the first result about the long-time behavior of stochastic nematic liquid crystals equations.

Robust Fuzzy Filter Design for Nonlinear Parabolic Partial Differential Systems with Continuous Wiener Noise and Discontinuous Poisson Noise

This paper addresses a robust \(H_{\infty }\) fuzzy filter design problem for nonlinear stochastic partial differential systems (NSPDSs) with continuous random fluctuation, discontinuous Poisson jumping noise, random external disturbance and measurement noise in the spatiotemporal domain. For NSPDSs, the robust \(H_{\infty }\) filter design problem through a measurement output needs to solve a complex second-order Hamilton Jacobi integral inequality. In order to simplify the design procedure, a fuzzy stochastic partial differential system based on a fuzzy interpolation approach is proposed to approximate the NSPDS. Then, the robust \(H_{\infty }\) fuzzy filter design problem can be reformulated as a diffusion matrix inequality (DMI) problem. Since the DMI problem is difficult to be solved via traditional algebraic techniques, we utilize the divergence theorem and Poincare inequality to transform the DMIs to a set of linear matrix inequalities (LMIs) which could be easily solved with the help of MATLAB LMI Toolbox. Finally, a robust state estimation problem of an ecology system with intrinsic spatiotemporal continuous Wiener noise and discontinuous Poisson jump fluctuation is provided as an example to illustrate the design procedure and to confirm the \(H_{\infty }\) filtering performance of the proposed \(H_{\infty }\) fuzzy filter design method.

Wong–Zakai approximation and support theorem for SPDEs with locally monotone coefficients

In this paper we present the Wong–Zakai approximation results for a class of nonlinear SPDEs with locally monotone coefficients and driven by multiplicative Wiener noise. This model extends the classical monotone one and includes examples like stochastic 2d Navier–Stokes equations, stochastic porous medium equations, stochastic p-Laplace equations and stochastic reaction–diffusion equations. As a corollary, our approximation results also describe the support of the distribution of solutions.

Pattern formation in time series systems due to viscoelastic behavior: Case studies in uniform distribution, normal distribution, stock market index, and music

A new methodology was introduced to investigate the pattern formation in time series systems due to their viscoelastic behavior. Four stochastic processes, uniform distribution, normal distribution, Nasdaq-100 stock market index, and a melody were studied within this context. The time series data were converted into vectorial forms in a scattering diagram. The sequential vectors can be split into its in-line (or conservative) and out-of-line (or dissipative) components. Thus, one can define the storage and loss modulus for conservative, and dissipative components, respectively. Instead of using the geometric Brownian equation which involves Wiener noise term, the changes were taken into consideration at every step by introducing “lethargy” concept and the deviation from it. Thus, the mathematics is somehow simplified, and the dynamical behavior of time series systems can be elucidated at every step of change. The viscoelastic behavior of time series systems reveals patterns of the viscoelastic parameters such as storage and loss modulus, and also of thermodynamic work-like and heat-like properties. Besides, there occur some minima and maxima in the distribution of the angles between the sequential vectors in the scattering diagram. The same is true for the change of entropy of the system.

A variational approach to dissipative SPDEs with singular drift

We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which no continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques with a priori estimates, both pathwise and in expectation, on solutions to regularized equations.

The Heston stochastic volatility model in Hilbert space

ABSTRACTWe extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein–Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modeling of forward curves in energy and commodity markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics.

The Gisin-Percival stochastic Schrödinger equation from standard quantum filtering theory

We show that the quantum state diffusion equation of Gisin and Percival, driven by complex Wiener noise, is equivalent up to a global stochastic phase to quantum trajectory models. With an appropriate feedback scheme, we set up an analog continuous measurement model which exactly simulates the Gisin-Percival quantum state diffusion.

On the stochastic Cahn–Hilliard equation with a singular double-well potential

We prove well-posedness and regularity for the stochastic pure Cahn–Hilliard equation under homogeneous Neumann boundary conditions, with both additive and multiplicative Wiener noise. In contrast with great part of the literature, the double-well potential is treated as generally as possible, its convex part being associated to a multivalued maximal monotone graph everywhere defined on the real line on which no growth nor smoothness assumptions are assumed. The regularity result allows to give appropriate sense to the chemical potential and to write a natural variational formulation of the problem. The proofs are based on suitable monotonicity and compactness arguments in a generalized variational framework.

Weak Error Estimates for Trajectories of SPDEs Under Spectral Galerkin Discretization

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection , and we study the approximation of the probability distribution of the tra-jectories: test functions are regular, but depend on the values of the process on the interval [0, T ]. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Ito map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Ito map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

Wonham Filtering by Observations with Multiplicative Noises

We solve the optimal filtering problem for states of a homogeneous finite-state Markov jump process by indirect observations in the presence of Wiener noise. The key feature of this problem is that the noise intensities in observations depend on the unobserved state. The filtering estimate is represented as a solution to some stochastic system with continuous and purely discontinuous martingales in the right-hand side. We discuss the theoretical results and present a numerical example that illustrates the properties of the obtained estimates.

On Well-Posedness of Semilinear Stochastic Evolution Equations on $L_p$ Spaces

We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations on $L_p$ spaces, driven by multiplicative Wiener noise, with a drift term given by a superposi...

Robust Stochastic Stability Analysis Method of DFIG Integration on Power System Considering Virtual Inertia Control

Concerning the stochastic excitation caused by wind power fluctuation and the stochastic parameter of virtual inertia caused by wind speed uncertainty, a power system stability analysis method considering the Wiener noise is proposed in this paper. First, based on the reduced-order model of DFIG retaining virtual inertia control and phase-locked loop (phase-locked loop) dynamics, the analytic function relationship between stochastic parameters and elements in the state matrix is derived using the sensitivity analysis method; thus, the model of interconnected system with stochastic parameters considering the Wiener noise is established. And by constructing the Lyapunov functional containing the model of stochastic parameters, the linear matrix inequality (LMI) that satisfies the robust stochastic stability criterion is derived. Furthermore, by transforming the solving of LMI to a feasibility problem, the stability of system is identified. And then, by calculating online system instability probability under the current operating condition, the probability stability analysis method is proposed. On this basis, a system stochastic stability degree index is defined in this paper, which quantifies the effect of stochastic parameters on system small-signal stability, and could provide more effective information for the safe and stable operation of power system. Simulation tests on the IEEE four-generator two-area system and the New England 10-machine 39-bus system verify that the proposed method could effectively identify the small-signal stability of power system under the joint influence of stochastic parameter and stochastic external excitation, and is applicable to complex system with multiple generators.

MTF and DQE enhancement using an apodized-aperture x-ray detector design.

Acquisition of high-quality x-ray images using low patient exposures requires detectors with high detective quantum efficiency (DQE). We describe a novel apodized-aperture pixel (AAP) design that increases high-frequency modulation transfer function (MTF) and DQE values. The AAP design makes a separation of physical sensor elements from image pixels by using very small sensor elements (e.g., 0.010-0.025 mm) to synthesize desired larger image pixels (e.g., 0.1-0.2 mm). A cascaded systems model of signal and noise propagation is developed to describe the benefits of the AAP approach in terms of the MTF, Wiener noise power spectrum (NPS), and DQE. The theoretical model was validated experimentally using a CMOS/CsI detector with 0.05 mm sensor elements to synthesize 0.20 mm image pixels and a clinical Se detector with 0.07 mm sensor elements to synthesize 0.28 mm pixels. A Monte Carlo study and x-ray images of a star-pattern and rat leg are used to visually compare AAP images. When used with a high-resolution converter layer and sensor elements one quarter the size of image pixels, the MTF is increased by 53% and the DQE by a factor of 2.3× at the image sampling cut-off frequency. Both simulated and demonstration images show improved detectability of high-frequency content and removal of aliasing artifacts. Evidence of Gibbs ringing is sometimes seen near high-contrast edges. It is shown that the AAP approach preserves the MTF of the small sensor elements and attenuates frequencies above the image sampling cut-off frequency. This has the double benefit of improving the MTF while reducing both signal and noise aliasing, resulting in an increase of the DQE at high spatial frequencies. For optimal implementation, the converter layer must have very high spatial resolution and the detector must have low readout noise.

State Estimation by Continuous-Time Observations in Multiplicative Noise

The paper is devoted to the Bayesian estimation of finite-state random vector given the indirect non-stationary continuous-time observations corrupted by a Wiener noise. The key feature is that the noise intensity is a function of the estimated vector, hence the traditional optimal filtering framework fails in this case.The estimate is obtained both in the explicit integral form and as the solution to a stochastic Differential system with some jump processes in the right hand side. The presence of the multiplicative noise gives a possibility to raise the estimation quality up to the exact value restoration.The numerical procedure for the estimate calculation is accompanied with the accuracy analysis. An example illustrating the performance of the proposed estimate is also presented.

Classical and quantum stochastic models of resistive and memristive circuits

The purpose of this paper is to examine stochastic Markovian models for circuits in phase space for which the drift term is equivalent to the standard circuit equations. In particular, we include dissipative components corresponding to both a resistor and a memristor in series. We obtain a dilation of the problem which is canonical in the sense that the underlying Poisson bracket structure is preserved under the stochastic flow. We do this first of all for standard Wiener noise but also treat the problem using a new concept of symplectic noise, where the Poisson structure is extended to the noise as well as the circuit variables, and in particular where we have canonically conjugate noises. Finally, we construct a dilation which describes the quantum mechanical analogue.