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Journal articles on the topic 'Probabilistic representation of PDEs'

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Published: 1 June 2024

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1

BERNAL, FRANCISCO, GONÇALO DOS REIS, and GREIG SMITH. "Hybrid PDE solver for data-driven problems and modern branching." European Journal of Applied Mathematics 28, no. 6 (May 22, 2017): 949–72. http://dx.doi.org/10.1017/s0956792517000109.

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The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for non-linear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully non-linear case and open research questions.

2

BLOMKER, D., M. ROMITO, and R. TRIBE. "A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees." Annales de l'Institut Henri Poincare (B) Probability and Statistics 43, no. 2 (March 2007): 175–92. http://dx.doi.org/10.1016/j.anihpb.2006.02.001.

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3

Gevorkyan, Ashot S., Aleksander V. Bogdanov, Vladimir V. Mareev, and Koryun A. Movsesyan. "Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment." Mathematics 10, no. 20 (October 18, 2022): 3868. http://dx.doi.org/10.3390/math10203868.

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A self-organizing joint system classical oscillator–random environment is considered within the framework of a complex probabilistic process that satisfies a Langevin-type stochastic differential equation. Various types of randomness generated by the environment are considered. In the limit of statistical equilibrium (SEq), second-order partial differential equations (PDE) are derived that describe the distribution of classical environmental fields. The mathematical expectation of the oscillator trajectory is constructed in the form of a functional-integral representation, which, in the SEq limit, is compactified into a two-dimensional integral representation with an integrand: the solution of the second-order complex PDE. It is proved that the complex PDE in the general case is reduced to two independent PDEs of the second order with spatially deviating arguments. The geometric and topological features of the two-dimensional subspace on which these equations arise are studied in detail. An algorithm for parallel modeling of the problem has been developed.

4

Yan, Long, Bohang Xu, and Zhangjun Liu. "Dimension Reduction Method-Based Stochastic Wind Field Simulations for Dynamic Reliability Analysis of Communication Towers." Buildings 13, no. 10 (October 16, 2023): 2608. http://dx.doi.org/10.3390/buildings13102608.

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The communication tower is a lifeline engineering that ensures the normal operation of wireless communication systems. Extreme wind disasters are inevitable while it is in service. Two dimension-reduction (DR) probabilistic representations based on proper orthogonal decomposition (POD) and wavenumber spectral representation (WSR), say DR-POD and DR-WSR, were thus proposed in this study. In order to determine the least representative sample size that satisfies the engineering accuracy requirements, the simulation error and simulation duration of 10 simulation points distributed along the height direction of the communication tower under different representative sample numbers were compared. Furthermore, for the fluctuating wind field with different numbers of simulation points distributed along the height of the communication tower, the simulation accuracy as well as efficiency of the DR-POD and the DR-WSR were compared. Finally, a high-rise communication tower structure’s wind-induced dynamic response study and wind-resistance reliability analysis were performed utilizing an alliance of the probability density evolution method (PDEM) and two DR probabilistic models, taking 10 load points into account. The structural dynamic analysis illustrates that the reliability of the communication tower structure and the wind-induced dynamic response allying the two DR probabilistic models with the PDEM have outstanding consistency.

5

Ren, Panpan, and Feng-Yu Wang. "Space-distribution PDEs for path independent additive functionals of McKean–Vlasov SDEs." Infinite Dimensional Analysis, Quantum Probability and Related Topics 23, no. 03 (September 2020): 2050018. http://dx.doi.org/10.1142/s0219025720500186.

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Let [Formula: see text] be the space of probability measures on [Formula: see text] with finite second moment. The path independence of additive functionals of McKean–Vlasov SDEs is characterized by PDEs on the product space [Formula: see text] equipped with the usual derivative in space variable and Lions’ derivative in distribution. These PDEs are solved by using probabilistic arguments developed from Ref. 2. As a consequence, the path independence of Girsanov transformations is identified with nonlinear PDEs on [Formula: see text] whose solutions are given by probabilistic arguments as well. In particular, the corresponding results on the Girsanov transformation killing the drift term derived earlier for the classical SDEs are recovered as special situations.

6

Xiao, Lishun, Shengjun Fan, and Dejian Tian. "A probabilistic approach to quasilinear parabolic PDEs with obstacle and Neumann problems." ESAIM: Probability and Statistics 24 (2020): 207–26. http://dx.doi.org/10.1051/ps/2019023.

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In this paper, by a probabilistic approach we prove that there exists a unique viscosity solution to obstacle problems of quasilinear parabolic PDEs combined with Neumann boundary conditions and algebra equations. The existence and uniqueness for adapted solutions of fully coupled forward-backward stochastic differential equations with reflections play a crucial role. Compared with existing works, in our result the spatial variable of solutions of PDEs lives in a region without convexity constraints, the second order coefficient of PDEs depends on the gradient of the solution, and the required conditions for the coefficients are weaker.

7

Haneche, M., K. Djaballah, and K. Khaldi. "An algorithm for probabilistic solution of parabolic PDEs." Sequential Analysis 40, no. 4 (October 2, 2021): 441–65. http://dx.doi.org/10.1080/07474946.2021.2010403.

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8

Denis, Laurent, Anis Matoussi, and Jing Zhang. "Quasilinear Stochastic PDEs with two obstacles: Probabilistic approach." Stochastic Processes and their Applications 133 (March 2021): 1–40. http://dx.doi.org/10.1016/j.spa.2020.11.002.

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9

Matoussi, Anis, Dylan Possamaï, and Wissal Sabbagh. "Probabilistic interpretation for solutions of fully nonlinear stochastic PDEs." Probability Theory and Related Fields 174, no. 1-2 (July 10, 2018): 177–233. http://dx.doi.org/10.1007/s00440-018-0859-4.

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10

Sow *, A. B., and E. Pardoux. "Probabilistic interpretation of a system of quasilinear parabolic PDEs." Stochastics and Stochastic Reports 76, no. 5 (October 2004): 429–77. http://dx.doi.org/10.1080/10451120412331303150.

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11

Fahim, Arash, Nizar Touzi, and Xavier Warin. "A probabilistic numerical method for fully nonlinear parabolic PDEs." Annals of Applied Probability 21, no. 4 (August 2011): 1322–64. http://dx.doi.org/10.1214/10-aap723.

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12

VAN GENNIP, YVES, and CAROLA-BIBIANE SCHÖNLIEB. "Introduction: Big data and partial differential equations." European Journal of Applied Mathematics 28, no. 6 (November 7, 2017): 877–85. http://dx.doi.org/10.1017/s0956792517000304.

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Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

13

Al-Najjar, Nabil I., Ramon Casadesus-Masanell, and Emre Ozdenoren. "Probabilistic representation of complexity." Journal of Economic Theory 111, no. 1 (July 2003): 49–87. http://dx.doi.org/10.1016/s0022-0531(03)00075-9.

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14

Kong, Tao, Weidong Zhao, and Tao Zhou. "Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs." Communications in Computational Physics 18, no. 5 (November 2015): 1482–503. http://dx.doi.org/10.4208/cicp.240515.280815a.

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AbstractIn this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our numerical schemes, one has the flexibility to choose the associated forward SDE, and a suitable choice can significantly reduce the computational complexity. Various numerical examples including the HJB equations are presented to show the effectiveness and accuracy of the proposed numerical schemes.

15

Lindstr�m, Sten, and Wlodzimierz Rabinowicz. "On probabilistic representation of non-probabilistic belief revision." Journal of Philosophical Logic 18, no. 1 (February 1989): 69–101. http://dx.doi.org/10.1007/bf00296175.

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16

Buckdahn, Rainer, Ying Hu, and Shige Peng. "Probabilistic approach to homogenization of viscosity solutions of parabolic PDEs." NoDEA : Nonlinear Differential Equations and Applications 6, no. 4 (December 1, 1999): 395–411. http://dx.doi.org/10.1007/s000300050010.

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17

Treanţă, Savin. "Gradient Structures Associated with a Polynomial Differential Equation." Mathematics 8, no. 4 (April 4, 2020): 535. http://dx.doi.org/10.3390/math8040535.

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In this paper, by using the characteristic system method, the kernel of a polynomial differential equation involving a derivation in R n is described by solving the Cauchy Problem for the corresponding first order system of PDEs. Moreover, the kernel representation has a special significance on the space of solutions to the corresponding system of PDEs. As very important applications, it has been established that the mathematical framework developed in this work can be used for the study of some second-order PDEs involving a finite set of derivations.

18

Hongler, Max-Olivier. "Brownian Swarm Dynamics and Burgers’ Equation with Higher Order Dispersion." Symmetry 13, no. 1 (December 31, 2020): 57. http://dx.doi.org/10.3390/sym13010057.

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The concept of ranked order probability distribution unveils natural probabilistic interpretations for the kink waves (and hence the solitons) solving higher order dispersive Burgers’ type PDEs. Thanks to this underlying structure, it is possible to propose a systematic derivation of exact solutions for PDEs with a quadratic nonlinearity of the Burgers’ type but with arbitrary dispersive orders. As illustrations, we revisit the dissipative Kotrweg de Vries, Kuramoto-Sivashinski, and Kawahara equations (involving third, fourth, and fifth order dispersion dynamics), which in this context appear to be nothing but the simplest special cases of this infinitely rich class of nonlinear evolutions.

19

Halpern, J. Y., and D. Koller. "Representation Dependence in Probabilistic Inference." Journal of Artificial Intelligence Research 21 (March 1, 2004): 319–56. http://dx.doi.org/10.1613/jair.1292.

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Non-deductive reasoning systems are often representation dependent: representing the same situation in two different ways may cause such a system to return two different answers. Some have viewed this as a significant problem. For example, the principle of maximum entropyhas been subjected to much criticism due to its representation dependence. There has, however, been almost no work investigating representation dependence. In this paper, we formalize this notion and show that it is not a problem specific to maximum entropy. In fact, we show that any representation-independent probabilistic inference procedure that ignores irrelevant information is essentially entailment, in a precise sense. Moreover, we show that representation independence is incompatible with even a weak default assumption of independence. We then show that invariance under a restricted class of representation changes can form a reasonable compromise between representation independence and other desiderata, and provide a construction of a family of inference procedures that provides such restricted representation independence, using relative entropy.

20

Karpati, A., P. Adam, and J. Janszky. "Quantum operations in probabilistic representation." Physica Scripta T135 (July 2009): 014054. http://dx.doi.org/10.1088/0031-8949/2009/t135/014054.

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21

Barber, M. J., J. W. Clark, and C. H. Anderson. "Neural Representation of Probabilistic Information." Neural Computation 15, no. 8 (August 1, 2003): 1843–64. http://dx.doi.org/10.1162/08997660360675062.

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It has been proposed that populations of neurons process information in terms of probability density functions (PDFs) of analog variables. Such analog variables range, for example, from target luminance and depth on the sensory interface to eye position and joint angles on the motor output side. The requirement that analog variables must be processed leads inevitably to a probabilistic description, while the limited precision and lifetime of the neuronal processing units lead naturally to a population representation of information. We show how a time-dependent probability densityρ(x; t) over variable x, residing in a specified function space of dimension D, may be decoded from the neuronal activities in a population as a linear combination of certain decoding functions φi(x), with coefficients given by the N firing rates ai(t) (generally with D ≪ N). We show how the neuronal encoding process may be described by projecting a set of complementary encoding functions [Formula: see text]i(x) on the probability density ρ(x; t), and passing the result through a rectifying nonlinear activation function. We show how both encoders [Formula: see text]i (x) and decoders φi(x) may be determined by minimizing cost functions that quantify the inaccuracy of the representation. Expressing a given computation in terms of manipulation and transformation of probabilities, we show how this representation leads to a neural circuit that can carry out the required computation within a consistent Bayesian framework, with the synaptic weights being explicitly generated in terms of encoders, decoders, conditional probabilities, and priors.

22

Soldatova, Larisa N., Andrey Rzhetsky, Kurt De Grave, and Ross D. King. "Representation of probabilistic scientific knowledge." Journal of Biomedical Semantics 4, Suppl 1 (2013): S7. http://dx.doi.org/10.1186/2041-1480-4-s1-s7.

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23

Haba, Z. "Probabilistic representation of quantum dynamics." Physics Letters A 175, no. 6 (April 1993): 371–76. http://dx.doi.org/10.1016/0375-9601(93)90984-8.

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24

Gikhman, Il I. "Probabilistic representation of quantum evolution." Ukrainian Mathematical Journal 44, no. 10 (October 1992): 1203–8. http://dx.doi.org/10.1007/bf01057675.

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25

Feldman, Jacob. "Symbolic representation of probabilistic worlds." Cognition 123, no. 1 (April 2012): 61–83. http://dx.doi.org/10.1016/j.cognition.2011.12.008.

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26

Zheng, Yu, and Yong Chen. "Ordered analytic representation of pdes by hamiltonian canonical system." Applied Mathematics-A Journal of Chinese Universities 17, no. 2 (June 2002): 177–82. http://dx.doi.org/10.1007/s11766-002-0042-6.

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27

Pham, Huyên. "Feynman-Kac Representation of Fully Nonlinear PDEs and Applications." Acta Mathematica Vietnamica 40, no. 2 (May 31, 2015): 255–69. http://dx.doi.org/10.1007/s40306-015-0128-x.

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28

OUKNINE, YOUSSEF, and DJIBRIL NDIAYE. "WEAK SOLUTIONS OF SEMILINEAR PDEs WITH OBSTACLE(S) IN SOBOLEV SPACES AND THEIR PROBABILISTIC INTERPRETATION VIA THE RFBSDEs AND DRFBSDEs." Stochastics and Dynamics 08, no. 02 (June 2008): 247–69. http://dx.doi.org/10.1142/s0219493708002305.

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We prove the existence and uniqueness of the solution of a semilinear PDEs with obstacle(s) under Lipschitz condition. We give a probabilistic interpretation of the solution in Sobolev spaces using reflected forward–backward stochastic differential equations, doubly reflected forward–backward stochastic differential equations and the penalization method.

29

Briand, Philippe, and Ying Hu. "Probabilistic approach to singular perturbations of semilinear and quasilinear parabolic PDEs." Nonlinear Analysis: Theory, Methods & Applications 35, no. 7 (March 1999): 815–31. http://dx.doi.org/10.1016/s0362-546x(97)00681-0.

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30

Belopolskaya, Ya. "Probabilistic approach to solution of nonlinear PDES arising in financial mathematics." Journal of Mathematical Sciences 167, no. 4 (May 25, 2010): 444–60. http://dx.doi.org/10.1007/s10958-010-9930-0.

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31

Wang, Jinxia, and Xicheng Zhang. "Probabilistic approach for systems of second order quasi-linear parabolic PDEs." Journal of Mathematical Analysis and Applications 388, no. 2 (April 2012): 676–94. http://dx.doi.org/10.1016/j.jmaa.2011.09.059.

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32

Block, Ned. "If perception is probabilistic, why does it not seem probabilistic?" Philosophical Transactions of the Royal Society B: Biological Sciences 373, no. 1755 (July 30, 2018): 20170341. http://dx.doi.org/10.1098/rstb.2017.0341.

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The success of the Bayesian perspective in explaining perceptual phenomena has motivated the view that perceptual representation is probabilistic. But if perceptual representation is probabilistic, why does normal conscious perception not reflect the full probability functions that the probabilistic point of view endorses? For example, neurons in cortical area MT that respond to the direction of motion are broadly tuned: a patch of cortex that is tuned to vertical motion also responds to horizontal motion, but when we see vertical motion, foveally, in good conditions, it does not look at all horizontal. The standard solution in terms of sampling runs into the problem that sampling is an account of perceptual decision rather than perception. This paper argues that the best Bayesian approach to this problem does not require probabilistic representation. This article is part of the theme issue ‘Perceptual consciousness and cognitive access'.

33

Zhu, Bo, and Baoyan Han. "Stochastic PDEs and Infinite Horizon Backward Doubly Stochastic Differential Equations." Journal of Applied Mathematics 2012 (2012): 1–17. http://dx.doi.org/10.1155/2012/582645.

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We give a sufficient condition on the coefficients of a class of infinite horizon BDSDEs, under which the infinite horizon BDSDEs have a unique solution for any given square integrable terminal values. We also show continuous dependence theorem and convergence theorem for this kind of equations. A probabilistic interpretation for solutions to a class of stochastic partial differential equations is given.

34

Heath, David, and Martin Schweizer. "Martingales versus PDEs in finance: an equivalence result with examples." Journal of Applied Probability 37, no. 4 (December 2000): 947–57. http://dx.doi.org/10.1239/jap/1014843075.

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We provide a set of verifiable sufficient conditions for proving in a number of practical examples the equivalence of the martingale and the PDE approaches to the valuation of derivatives. The key idea is to use a combination of analytic and probabilistic assumptions that covers typical models in finance falling outside the range of standard results from the literature. Applications include Heston's stochastic volatility model and the Black-Karasinski term structure model.

35

Heath, David, and Martin Schweizer. "Martingales versus PDEs in finance: an equivalence result with examples." Journal of Applied Probability 37, no. 04 (December 2000): 947–57. http://dx.doi.org/10.1017/s0021900200018143.

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We provide a set of verifiable sufficient conditions for proving in a number of practical examples the equivalence of the martingale and the PDE approaches to the valuation of derivatives. The key idea is to use a combination of analytic and probabilistic assumptions that covers typical models in finance falling outside the range of standard results from the literature. Applications include Heston's stochastic volatility model and the Black-Karasinski term structure model.

36

D'Ambrosio, Lorenzo, and Marius Ghergu. "Representation formulae for nonhomogeneous differential operators and applications to PDEs." Journal of Differential Equations 317 (April 2022): 706–53. http://dx.doi.org/10.1016/j.jde.2022.02.013.

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37

Henry-Labordère, Pierre, Nadia Oudjane, Xiaolu Tan, Nizar Touzi, and Xavier Warin. "Branching diffusion representation of semilinear PDEs and Monte Carlo approximation." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 55, no. 1 (February 2019): 184–210. http://dx.doi.org/10.1214/17-aihp880.

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38

Harraj, N., Y. Ouknine, and I. Turpin. "Double-barriers-reflected BSDEs with jumps and viscosity solutions of parabolic integrodifferential PDEs." Journal of Applied Mathematics and Stochastic Analysis 2005, no. 1 (January 1, 2005): 37–53. http://dx.doi.org/10.1155/jamsa.2005.37.

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We give a probabilistic interpretation of the viscosity solutions of parabolic integrodifferential partial equations with two obstacles via the solutions of forward-backward stochastic differential equations with jumps.

39

Moghaddam, B., and A. Pentland. "Probabilistic visual learning for object representation." IEEE Transactions on Pattern Analysis and Machine Intelligence 19, no. 7 (July 1997): 696–710. http://dx.doi.org/10.1109/34.598227.

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40

Höhna, Sebastian, Tracy A. Heath, Bastien Boussau, Michael J. Landis, Fredrik Ronquist, and John P. Huelsenbeck. "Probabilistic Graphical Model Representation in Phylogenetics." Systematic Biology 63, no. 5 (June 20, 2014): 753–71. http://dx.doi.org/10.1093/sysbio/syu039.

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41

Wang, Haijun, Shengyan Zhang, Yujie Du, Hongjuan Ge, and Bo Hu. "Visual tracking via probabilistic collaborative representation." Journal of Electronic Imaging 26, no. 1 (February 1, 2017): 013010. http://dx.doi.org/10.1117/1.jei.26.1.013010.

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42

Beccaria, M. "Probabilistic representation of fermionic lattice systems." Nuclear Physics B - Proceedings Supplements 83-84, no. 1-3 (March 2000): 911–13. http://dx.doi.org/10.1016/s0920-5632(00)00413-8.

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43

Beccaria, Matteo, Carlo Presilla, Gian Fabrizio De Angelis, and Giovanni Jona-Lasinio. "Probabilistic representation of fermionic lattice systems." Nuclear Physics B - Proceedings Supplements 83-84 (April 2000): 911–13. http://dx.doi.org/10.1016/s0920-5632(00)91842-5.

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Khrennikov, Andrei. "Probabilistic pathway representation of cognitive information." Journal of Theoretical Biology 231, no. 4 (December 2004): 597–613. http://dx.doi.org/10.1016/j.jtbi.2004.07.015.

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Mao, L., and H. Resat. "Probabilistic representation of gene regulatory networks." Bioinformatics 20, no. 14 (April 8, 2004): 2258–69. http://dx.doi.org/10.1093/bioinformatics/bth236.

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46

Chen, Zhen-Qing. "Time fractional equations and probabilistic representation." Chaos, Solitons & Fractals 102 (September 2017): 168–74. http://dx.doi.org/10.1016/j.chaos.2017.04.029.

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Hu, Mingshang, and Falei Wang. "Probabilistic approach to singular perturbations of viscosity solutions to nonlinear parabolic PDEs." Stochastic Processes and their Applications 141 (November 2021): 139–71. http://dx.doi.org/10.1016/j.spa.2021.07.006.

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48

Bahlali, K., A. Elouaflin, and M. N'zi. "Backward stochastic differential equations with stochastic monotone coefficients." Journal of Applied Mathematics and Stochastic Analysis 2004, no. 4 (January 1, 2004): 317–35. http://dx.doi.org/10.1155/s1048953304310038.

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We prove an existence and uniqueness result for backward stochastic differential equations whose coefficients satisfy a stochastic monotonicity condition. In this setting, we deal with both constant and random terminal times. In the random case, the terminal time is allowed to take infinite values. But in a Markovian framework, that is coupled with a forward SDE, our result provides a probabilistic interpretation of solutions to nonlinear PDEs.

49

Peng, Xiang, Qilong Gao, Jiquan Li, Zhenyu Liu, Bing Yi, and Shaofei Jiang. "Probabilistic Representation Approach for Multiple Types of Epistemic Uncertainties Based on Cubic Normal Transformation." Applied Sciences 10, no. 14 (July 8, 2020): 4698. http://dx.doi.org/10.3390/app10144698.

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Many non-probabilistic approaches have been widely regarded as mathematical tools for the representation of epistemic uncertainties. However, their heavy computational burden and low computational efficiency hinder their applications in practical engineering problems. In this article, a unified probabilistic representation approach for multiple types of epistemic uncertainties is proposed based on the cubic normal transformation method. The epistemic uncertainties can be represented using an interval approach, triangular fuzzy approach, or evidence theory. The uncertain intervals of four statistical moments, which contain mean, variance, skewness, and kurtosis, are calculated using the sampling analysis method. Subsequently, the probabilistic cubic normal distribution functions are conducted for sampling points of four statistical moments of epistemic uncertainties. Finally, a calculation procedure for the construction of probabilistic representation functions is proposed, and these epistemic uncertainties are represented with belief and plausibility continuous probabilistic measure functions. Two numerical examples and one engineering example demonstrate that the proposed approach can act as an accurate probabilistic representation function with high computational efficiency.

50

Li, Han Ling, Lei Hou, Jun Jie Zhao, De Zhi Lin, and Lin Qiu. "Dual Scaled Stochastic FEM Simulation of Porous Honeycomb Material." Applied Mechanics and Materials 443 (October 2013): 48–52. http://dx.doi.org/10.4028/www.scientific.net/amm.443.48.

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In this paper, a dual-scaled method is introduced to simulate the nonlinear property of porous honeycomb material. In microscopic scale, stochastic analysis upon a detailed representation of the hexagonal cells is applied. In macroscopic scale, coupled fluid-solid PDEs with a modified stochastic item are used to describe the rheology of non-Newtonian property of honeycomb. Semi-discrete finite element method (FEM) is applied to solve the PDEs. Comparison of stochastic dynamic system with definite dynamic system is introduced. Numerical Results of Euler explicit time scheme and Crank-Nicolson semi-implicit time scheme are presented.