A new approach for predicting the extreme response of structures

Simulation of a non-Gaussian stochastic process based on a combined distribution of the UHPM and the GBD

Previous article Next article Second-Order Markov and Reciprocal Stationary Gaussian ProcessesR. N. MiroshinR. N. Miroshinhttps://doi.org/10.1137/1124099PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. N. Bernshtei˘n, On the dependencies among random variablesCollected Works, Vol. IV, Nauka, Moscow, 1964, 235–254, (In Russian.) Google Scholar[2] J. L. Doob, Stochastic processes, John Wiley & Sons Inc., New York, 1953viii+654 MR0058896 0053.26802 Google Scholar[3] Benton Jamison, Reciprocal processes: The stationary Gaussian case, Ann. Math. Statist., 41 (1970), 1624–1630 MR0267637 0248.60030 CrossrefGoogle Scholar[4] I. I. Gikhman and , A. V. Skorokhod, The Theory of Stochastic Processes, Vol. I, Springer-Verlag, New York, 1974 CrossrefGoogle Scholar[5] E. Kamke, Differentialgleichungen. Lösungsmethoden and Lösungen, Vol. 1, Chelsea, New York, 1974 Google Scholar[6] E. Wong, Some results concerning the zero-crossings of Gaussian noise, SIAM J. Appl. Math., 14 (1966), 1246–1254 10.1137/0114099 MR0207059 0147.15905 LinkGoogle Scholar[7] E. Wong, The distribution of intervals between zeros for a stationary Gaussian process, SIAM J. Appl. Math., 18 (1970), 67–73 10.1137/0118008 MR0256456 0221.60024 LinkGoogle Scholar[8] R. N. Miroshin, Convergence of Rice and Longuet-Higgins series for a Wong process, Theory Prob. Applications, 21 (1976), 863–866 10.1137/1121103 0374.60049 LinkGoogle Scholar[9] R. N. Miroshin, Asymptotics of the second factorial moment of the number of upward crossings of the line $kt+a$ by a Gaussian stationary process, Vestnik Leningrad. Univ., 23 (1968), 109–120 MR0235612 0165.52602 Google Scholar[10] M. Krein, Sur le problème du prolongement des fonctions hermitiennes positives et continues, C. R. (Doklady) Acad. Sci. URSS (N.S.), 26 (1940), 17–22 MR0004333 0022.35302 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Characterization of Multivariate Stationary Gaussian Reciprocal DiffusionsJournal of Multivariate Analysis, Vol. 62, No. 1 Cross Ref Gaussian reciprocal diffusions and positive definite sturm-liouville operators4 April 2007 | Stochastics and Stochastic Reports, Vol. 55, No. 3-4 Cross Ref Gaussian reciprocal processes revisitedStatistics & Probability Letters, Vol. 12, No. 4 Cross Ref Reciprocal covariance solutions of some matrix differential equationsStochastic Processes and their Applications, Vol. 37, No. 1 Cross Ref Characterization and Classification of Gaussian Second Order Reciprocal Processes Cross Ref Gaussian reciprocal processes and self-adjoint stochastic differential equations of second order2 May 2007 | Stochastics and Stochastic Reports, Vol. 34, No. 1-2 Cross Ref Reciprocal diffusions and stochastic differential equations of second order ∗Stochastics, Vol. 24, No. 4 Cross Ref Multivariate reciprocal stationary Gaussian processesJournal of Multivariate Analysis, Vol. 23, No. 1 Cross Ref On Miroshin’s Second-Order Reciprocal ProcessesJulia Abrahams3 August 2006 | SIAM Journal on Applied Mathematics, Vol. 44, No. 1AbstractPDF (316 KB)The Use of Rice SeriesR. N. Miroshin28 July 2006 | Theory of Probability & Its Applications, Vol. 28, No. 4AbstractPDF (1011 KB)The zero-crossing problem for some nonstationary Gaussian processes (Corresp.)IEEE Transactions on Information Theory, Vol. 28, No. 4 Cross Ref Convergence of Longuet-Higgins Series for Stationary Gaussian Markov Processes of First OrderR. N. Miroshin17 July 2006 | Theory of Probability & Its Applications, Vol. 26, No. 1AbstractPDF (1546 KB) Volume 24, Issue 4| 1980Theory of Probability & Its Applications History Submitted:11 July 1977Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1124099Article page range:pp. 845-852ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Gaussian Processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP [Formula: see text] that is the image of another GP [Formula: see text] under a linear transformation [Formula: see text] acting on the sample paths of [Formula: see text] are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when [Formula: see text] is an unbounded operator such as a differential operator, which is common in many modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator [Formula: see text] acting on the sample paths of a square-integrable (not necessarily Gaussian) stochastic process. Our proof technique relies upon Hille’s theorem for the Bochner integral of a Banach-valued random variable.

Contents Preface Introduction 0.1. Introduction to Precise integration method 1: Introduction to analytical dynamics 1.1. Holonomic and nonholonomic constraints 1.2. Generalized displacement, degrees of freedom and virtual displacement 1.3. Principle of virtual displacement and the D'Alembert principle 1.4. Lagrange equation 1.5. Hamilton variational principle 1.6. Hamiltonian canonical equations 1.7. Canonical transformation 1.8. Symplectic description of the canonical transformation 1.9. Poisson bracket 1.10. Action 1.11. Hamilton-Jacobi equation 2: Vibration theory 2.1. Single degree of freedom vibration 2.2. Vibration of multi-degrees of freedom system 2.3. Small vibration of gyroscopic systems 2.4. Non-linear vibration of multi-degrees of freedom system 2.5. Discussion on the stability of gyroscopic system 3: Probability and stochastic process 3.1. Preliminary of probability theory 3.2. Preliminary of stochastic process 3.3. Quadratic moment stochastic process (regular process) 3.4. Normal stochastic process 3.5. Markoff process 3.6. Spectral density of stationary stochastic process 4: Random vibration of structures 4.1. Models of random excitation 4.2. Response of structures under stationary excitations 4.3. Response under excitation of non-stationary stochastic process 5: Elastic system with single continuous coordinate 5.1. Fundamental equations of Timoshenco beam theory 5.2. Potential energy density and mixed energy density 5.3. Separation of variables, Adjoint symplectic ortho-normality 5.4. Multiple eigenvalues and the Jordan normal form 5.5. Expansion solution of the inhomogeneous equation 5.6. Two end boundary conditions 5.7. Interval mixed energy and precise integration method 5.8. Eigenvector based solution of Riccati equations 5.9. Stepwise integrationmethod by means of sub-structural combination 5.10. Influence function of single continuous coordinate system 5.11. Power flow 5.12. Wave scattering analysis 5.13. Wave induced resonance 5.14. Wave propagation along periodical structures 6: Linear optimal control, theory and computation 6.1. State space of linear system 6.2. Theory of stability 6.3. Prediction, filtering and smoothing 6.4. Prediction and its computation 6.5. Kalman filter 6.6. Optimal smoothing and computations 6.7. Optimal control 6.8. Robust control Concluding remarks

Mass correction of heavy particles in metals

We consider the distribution of the free coordinates of a time-homogeneous Markov process at the time of its first passage into a prescribed stopping set. This calculation (for an uncontrolled process) is of interest because under some circumstances it enables one to calculate the optimal control for a related controlled process. Scaling assumptions are made which allow the application of large deviation techniques. However, the first-order evaluation obtained by these techniques is often too crude to be useful, and the second-order correction term must be calculated. An expression for this correction term as an integral over time is obtained in Equation (20). The integration can be performed in some cases to yield the conclusions of Theorems 1 and 2, expressed in Equations (7) and (9). Theorem 1 gives the probability density of the state vector (to the required degree of approximation) at a prescribed time for a class of processes we may reasonably termlinear.Theorem 2 evaluates (without any assumption of linearity) the ratio of this density to the probability density of the coordinates under general stopping rules.

Beam training and tracking (BTT) are key technologies for millimeter wave communications. However, since the effectiveness of BTT methods heavily depends on wireless environments, complexity and randomness of practical environments severely limit the application scope of many BTT algorithms and even invalidate them. To tackle this issue, from the perspective of stochastic process (SP), in this paper we propose to model beam directions as a SP and address the problem of BTT via process inference. The benefit of the SP design methodology is that environment priors and uncertainties can be naturally taken into account (e.g., to encode them into SP distribution) to improve prediction efficiencies (e.g., accuracy and robustness). We take the Gaussian process (GP) as an example to elaborate on the design methodology and propose novel learning methods to optimize the prediction models. In particular, beam training subset is optimized based on derived posterior distribution. The GP-based SP methodology enjoys two advantages. First, good performance can be achieved even for small data, which is very appealing in dynamic communication scenarios. Second, in contrast to most BTT algorithms that only predict a single beam, our algorithms output an optimizable beam subset, which enables a flexible tradeoff between training overhead and desired performance. Simulation results show the superiority of our approach.

On the Bhattacharyya distance and the divergence between Gaussian processes

Environmental contours are often applied in probabilistic structural reliability analysis to identify extreme environ-mental conditions that may give rise to extreme loads and responses. They facilitate approximate long term analysis of critical structural responses in situations where computationally heavy and time-consuming response calculations makes full long-term analysis infeasible. The environmental contour method identifies extreme environmental conditions that are expected to give rise to extreme structural response of marine structures. The extreme responses can then be estimated by performing response calculations for environmental conditions along the contours. Response-based analysis is an alternative, where extreme value analysis is performed on the actual response rather than on the environmental conditions. For complex structures, this is often not practical due to computationally heavy response calculations. However, by establishing statistical emulators of the response, using machine learning techniques, one may obtain long time-series of the structural response and use this to estimate extreme responses. In this paper, the contour method will be compared to response-based estimation of extreme vertical bending moment for a tanker. A response emulator based on Gaussian processes regression with adaptive sampling has been established based on response calculations from a hydrodynamic model. Long time-series of sea-state parameters such as significant wave height and wave period are used to construct N-year environmental contours and the extreme N-year response is estimated from numerical calculations for identified sea states. At the same time, the response emulator is applied on the time series to provide long time-series of structural response, in this case vertical bending moment of a tanker. Extreme value analysis is then performed directly on the responses to estimate the N-year extreme response. The results from either method will then be compared, and it is possible to evaluate the accuracy of the environmental contour method in estimating the response.

The paper proposes an algorithm that makes numerical integrations with conditions for stability to be usable for computing inelastic seismic responses of structures. The numerical integrations with conditions cannot be applied to solving directly the equation of the motion, but the numerical integrations utilized for linear problems generally give exacter responses than that without any conditions. The method of modal analysis can clear the condition by applying the eigenvalue analysis to the equation of the motion expressed by the tangent stiffness in the inelasticity. The paper shows considerable difference between the result by the numerical integration with conditions and that without any condition, and the former result will be exacter than the latter from several proofs in the discussion

We investigate the bridge problem for stochastic processes, that is, we analyze the statistical properties of trajectories constrained to begin and terminate at a fixed position within a time interval τ. Our primary focus is the time-reversal symmetry of these trajectories: under which conditions do the statistical properties remain invariant under the transformation t→τ-t? To address this question, we compare the stochastic differential equation describing the bridge, derived equivalently via Doob's transform or stochastic optimal control, with the corresponding equation for the time-reversed bridge. We aim to provide a concise overview of these well-established derivation techniques and subsequently obtain a local condition for the time-reversal asymmetry that is specifically valid for the bridge. We are specifically interested in cases in which detailed balance is not satisfied and aim to eventually quantify the bridge asymmetry and understand how to use it to derive useful information about the underlying out-of-equilibrium dynamics. To this end, we derived a necessary condition for time-reversal symmetry, expressed in terms of the current velocity of the original stochastic process and a quantity linked to detailed balance. As expected, this formulation demonstrates that the bridge is symmetric when detailed balance holds, a sufficient condition that was already known. However, it also suggests that a bridge can exhibit symmetry even when the underlying process violates detailed balance. While we did not identify a specific instance of complete symmetry under broken detailed balance, we present an example of partial symmetry. In this case, some, but not all, components of the bridge display time-reversal symmetry. This example is drawn from a minimal non-equilibrium model, namely Brownian Gyrators, that are linear stochastic processes. We examined non-equilibrium systems driven by a "mechanical" force, specifically those in which the linear drift cannot be expressed as the gradient of a potential. While Gaussian processes like Brownian Gyrators offer valuable insights, it is known that they can be overly simplistic, even in their time-reversal properties. Therefore, we transformed the model into polar coordinates, obtaining a non-Gaussian process representing the squared modulus of the original process. Despite this increased complexity and the violation of detailed balance in the full process, we demonstrate through exact calculations that the bridge of the squared modulus in the isotropic case, constrained to start and end at the origin, exhibits perfect time-reversal symmetry.

: The 16th conference on Stochastic processes and their applications was held at Stanford University between August 17 and 21 1987. The conference was attended by 169 researchers. There were 19 invited papers and several sessions of contributed papers. Ample time was allowed for interactions. Keywords: Shape; Inequalities; Network simulation; Travelling waves; Finite particles; Boundary value problems; Random walks; Gaussian processes; Estimation; Statistical inference; Queueing networks; Deviation; Martingales; Economic models; Optimization; Reliability.

This chapter is focused on the estimation of wind farm power fluctuations from the behaviour of a single turbine during continuous operation (special events such as turbine tripping, grid transients, sudden voltages changes, etc. are not considered). The time scope ranges from seconds to some minutes and the geographic scope is bounded to one or a few nearby wind farms. One of the objectives of this chapter is to explain quantitatively the wind power variability in a farm from the behaviour of a single turbine. For short intervals and inside a wind farm, the model is based on the experience with a logger system designed and installed in four wind farms (Sanz et al., 2000a), the classic theory of Gaussian (normal) stochastic processes, the wind coherence model (Schlez & Infield, 1998), and the general coherence function derived by Riso Institute in Horns Rev wind farm (Martins et al., 2006; Sorensen et al., 2008a). For larger distances and slower variations, the model has been tested with meteorological data from the weather network. The complexities inherent to stochastic processes are partially circumvented presenting some case studies with meaningful graphs and using classical tools of signal processing and time series analysis when possible. The sum of the power from many turbines is a stochastic process that is the outcome of many interactions from different sources. The sum of the power variations from more than four turbines converges approximately to a Gaussian process despite of the process nature (deterministic, stochastic, broadband or narrowband), analogously to the martingale central limit theorem (Hall & Heyde, 1980). The only required condition is the negligible effect of synchronization forces among turbine oscillations. The data logged at some wind farms are smooth and they have good mathematical properties except during special events such as turbine breaker trips or severe weather. This chapter will show that, under some circumstances, the power output of a wind farm can be approximated to a Gaussian process and its auto spectrum density can be estimated from the spectrum of a turbine, wind farm dimensions and wind coherence. The wind farm power variability is fully characterized by its auto spectrum provided the Gaussian approximation is accurate enough. Many interesting properties such as the mean power fluctuation shape during a period, the distribution of power variation in a time period, the more extreme power variation expected during a short period, etc. can be estimated applying the outstanding properties of Gaussian processes according to (Bierbooms, 2008) and (Mur-Amada, 2009).

Offshore structures are subjected to stochastic loads from wave, current, and wind. A long-term response analysis is important for evaluating the structural safety; however, it is computationally demanding to perform the large number of nonlinear time-domain simulations. Metamodeling is an attractive prospect, but there are two existing obstacles to overcome. First, existing metamodels are mostly confined to a single sea state and are unable to predict the response for different wave directions. Second, for floating systems, the inputs are usually vessel loads/motions, which are a priori unknown. This paper develops a new metamodel framework for accurately predicting the time series response under short-term uncertainties from the stochastic process, and seven long-term variables, namely the significant wave height, spectral period, wave direction, and speeds and directions of current and wind. The exogenous input is based only on the environmental variables, which can be predefined. The proposed metamodel is implemented on a floating system and used to predict the time series of riser stress, the extreme stress response and fatigue damage. Four cases with increasing number of long-term variables are tested. Excellent agreement is found between the metamodel prediction and numerical simulation even for the most critical case with all variables included.

There is an increasing interest in the photoinduced dynamics in the low frequency, ω, regime. The multiphoton absorptions by molecules in strong laser fields depend on the polarization of the laser and on the molecular structure. The unique properties of the interaction of atoms and molecules with lasers in the low-frequency regime imply new concepts and directions in strong-field light-matter interactions. Here we represent a perturbational approach for the calculations of the quasi-energy spectrum in the low-frequency regime, which avoids the construction of the Floquet operator with extremely large number of Floquet channels. The zero-order Hamiltonian in our perturbational approach is the adiabatic Hamiltonian where the atoms/molecules are exposed to a dc electric field rather than to ac-field. This is in the spirit of the first step in the Corkum three-step model. The second-order perturbation correction terms are obtained when iℏω∂∂τ serves as a perturbation and τ is a dimensionless variable. The second-order adiabatic perturbation scheme is found to be an excellent approach for calculating the ac-field Floquet solutions in our test case studies of a simple one-dimensional time-periodic model Hamiltonian. It is straightforward to implement the perturbation approach presented here for calculating atomic and molecular energy shifts (positions) due to the interaction with low-frequency ac-fields using high-level electronic structure methods. This is enabled since standard quantum chemistry packages allow the calculations of atomic and molecular energy shifts due to the interaction with dc-fields. In addition to the shift of the energy positions, the energy widths (inverse lifetimes) can be obtained at the same level of theory. These energy shifts are functions of the laser parameters (low frequency, intensity, and polarization).