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Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities [article]

Matteo Beccari, Martin Hutzenthaler, Arnulf Jentzen, Ryan Kurniawan, Felix Lindner, Diyora Salimova
2019 arXiv   pre-print
In this work we solve this problem by proving that full-discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic  ...  This article also contains a short literature overview on existing numerical approximation results for stochastic differential equations with superlinearly growing nonlinearities.  ...  We also refer, e.g., to [42, 54, 59, 69, 71, 76, 126, 130, 159, 160] for lower bounds for strong and weak approximation errors for numerical approximation schemes for SDEs with non-globally Lipschitz  ... 
arXiv:1903.06066v1 fatcat:5qwiayif3jaw5jbgukwktsw6l4

Approximate implicitization of triangular Bézier surfaces

Oliver J. D. Barrowclough, Tor Dokken
2010 Proceedings of the 26th Spring Conference on Computer Graphics - SCCG '10  
We discuss how Dokken's methods of approximate implicitization can be applied to triangular Bézier surfaces in both the original and weak forms.  ...  A numerical approach to weak approximate implicitization is also considered and we show that symmetries within this algorithm can be exploited to reduce the computation time of M.  ...  A similar approach known as weak approximate implicitization was developed in [Dokken and Thomassen 2006 ].  ... 
doi:10.1145/1925059.1925084 dblp:conf/sccg/BarrowcloughD10 fatcat:r5zo6qxqmzc2fngdhc6wxu42tm

Implicit Second Order Weak Taylor Tau-Leaping Methods for the Stochastic Simulation of Chemical Kinetics

T.-H. Ahn, A. Sandu
2011 Procedia Computer Science  
In this paper we use stochastic Taylor expansions to propose implicit second order weak Taylor tau-leaping methods for the stochastic simulation of chemical kinetics.  ...  Three different schemes of the second order weak Taylor tau-leaping methods are numerically tested to demonstrate the performance with accuracy.  ...  Implicit Second Order Weak Taylor Tau-Leaping Method In this section new implicit tau-leaping method is introduced by analogy with the implicit order 2.0 weak Taylor scheme.  ... 
doi:10.1016/j.procs.2011.04.250 fatcat:uzrpfktsjfcnflz7hbrmpfia6y

Implicit Simulation Methods for Stochastic Chemical Kinetics [article]

Tae-Hyuk Ahn and Adrian Sandu and Xiaoying Han
2013 arXiv   pre-print
This paper develops fully implicit tau-leaping-like algorithms that treat implicitly both the mean and the variance of the Poisson variables.  ...  Discrete Time Approximations for SDEs This section discusses the numerical solution of stochastic differential equations (SDEs), with an emphasis on weak approximations [14] .  ...  Figure 9 : 9 The relationship between accuracy and CPU time for X 5 of the ELF system took 178,364 seconds (approximately 50 hours), while 100,000 simulations of the implicit order two weak Taylor method  ... 
arXiv:1303.3614v1 fatcat:z5gn74e2kfcwfgsdjlor67nrqm

On the Efficiency of Simplified Weak Taylor Schemes for Monte Carlo Simulation in Finance [chapter]

Nicola Bruti Liberati, Eckhard Platen
2004 Lecture Notes in Computer Science  
In the simplified schemes discrete random variables, instead of Gaussian ones, are generated to approximate multiple stochastic integrals.  ...  We present a numerical comparison between weak Taylor schemes and their simplified versions.  ...  The CPU times needed to compute 4 million approximate paths with 64 time steps with the Euler, fully implicit Euler and order 2.0 weak Taylor scheme amount to 107, 114 and 110 seconds, respectively.  ... 
doi:10.1007/978-3-540-25944-2_100 fatcat:qghw22t5mjd3zo27jcly4ukhky

Implicit Particle Methods and Their Connection with Variational Data Assimilation

Ethan Atkins, Matthias Morzfeld, Alexandre J. Chorin
2013 Monthly Weather Review  
Minimizations required by implicit particle methods are shown to be similar to those that one encounters in variational data assimilation, and the connection of implicit particle methods with variational  ...  The implicit particle filter is a sequential Monte Carlo method for data assimilation that guides the particles to the high-probability regions via a sequence of steps that includes minimizations.  ...  The output of the two filters is an approximation of the conditional mean, and the output of the weak constraint 4DVAR code is an approximation of the conditional mode.  ... 
doi:10.1175/mwr-d-12-00145.1 fatcat:qeafyh6ea5cbph764sfqg4qoem

Approximation of jump diffusions in finance and economics

Nicola Bruti-Liberati, Eckhard Platen
2007 Computational Economics  
Consequently, discrete time approximations are required. In this paper we give a survey of strong and weak numerical schemes for SDEs with jumps.  ...  Numerical results on the pricing of options on an index are presented using weak approximation methods. 200 Mathematics Subject Classification: primary 60H10; secondary 65C05.  ...  Finally, Bruti-Liberati & Platen (2005b) present convergence theorems for weak approximations of any weak order β ∈ {1, 2, . . .}, including derivative free, implicit, predictor-corrector and jump-adapted  ... 
doi:10.1007/s10614-006-9066-y fatcat:nvqsxhn4szbflj6xxvgyxc4kye

Higher-order semi-implicit Taylor schemes for Itô stochastic differential equations

R. Zeghdane, L. Abbaoui, A. Tocino
2011 Journal of Computational and Applied Mathematics  
one-dimensional noise, we give an infinite family of semi-implicit simplified schemes of weak order 4.0.  ...  The paper considers the derivation of families of semi-implicit schemes of weak order N = 3.0 (general case) and N = 4.0 (additive noise case) for the numerical solution of Itô stochastic differential  ...  is only interested in the moments of the solution, weak approximations are used.  ... 
doi:10.1016/ fatcat:gt4wuyykzvdo7jymt3qagkqt4m

On Weak Predictor–Corrector Schemes for Jump-Diffusion Processes in Finance [chapter]

Nicola Bruti-Liberati, Eckhard Platen
2012 Topics in Numerical Methods for Finance  
We consider in this paper weak discrete time approximations of jump-diffusion SDEs which are appropriate for problems such as derivative pricing and the evaluation of risk measures.  ...  We present regular and jump-adapted predictor-corrector schemes with first and second order of weak convergence.  ...  We refer to Bruti-Liberati & Platen (2006) for the weak convergence of explicit and implicit approximations for SDEs with jumps.  ... 
doi:10.1007/978-1-4614-3433-7_1 fatcat:haex7kidwvhwlh5ps5wpoexf4a

Envelope computation in the plane by approximate implicitization

Tino Schulz, Bert Jüttler
2011 Applicable Algebra in Engineering, Communication and Computing  
This connection enables us to use approximate implicitization for computing the (exact or approximate) implicit representation of the envelope.  ...  We analyze the connection of this function to the implicit equation of the envelope.  ...  Weak approximate implicitization In order to avoid these computational difficulties, the weak method for approximate implicitization (cf.  ... 
doi:10.1007/s00200-011-0149-1 fatcat:owm256xwdjbbvkzqk6ttfuqpzu

Some results for the weak convergence of semi-implicit split-step methods

Burhaneddin Izgi, Berivan Ari
2019 New Trends in Mathematical Sciences  
Finally, we present the weak convergence rate of SISS methods is approximately 1 with respect to the numerical results.  ...  In this paper, we obtain some results for the weak convergence of semi-implicit split-step (SISS) methods which are recently developed to solve a class of nonlinear stochastic differential equation with  ...  Therefore, we study the weak convergence of the semi-implicit split-step methods in this paper.  ... 
doi:10.20852/ntmsci.2019.336 fatcat:admavdeabvgl7dcofre2522eei

On the numerical stability of simulation methods for SDEs under multiplicative noise in finance

Eckhard Platen, Lei Shi
2013 Quantitative finance (Print)  
The stability criterion presented in this paper is designed to handle both scenario and Monte Carlo simulation, that is, both strong and weak approximations.  ...  The result being that schemes, which have implicitness in both the drift and the diffusion terms, exhibit the largest stability regions.  ...  As pointed out in Kloeden & Platen (1999) , in a simplified weak scheme, terms that approximate the diffusion coefficient can also be made implicit.  ... 
doi:10.1080/14697688.2012.713981 fatcat:nk4gowfpdnhv3h5cbbngdxsasq

Page 1498 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
The paper deals with the higher order approximation of weak solutions in hyperbolic systems of conservation law equations, in particular, in gas dynamics.  ...  The author defines weak transition conditions for non-stationary shock waves (so-called e-Hugoniot conditions), which describe weak solutions in an e-neighbourhood of a shock, using an inte- gral conservation  ... 

Runge-Kutta Lawson schemes for stochastic differential equations [article]

Kristian Debrabant and Anne Kværnø and Nicky Cordua Mattsson
2020 arXiv   pre-print
Moreover, assume that (a) for > 0 are skew-symmetric, (b) and +1 are the numerical approximations of ( ) and (8) obtained by the SRK method (13) , (c) this approximation is of weak order˜, i.e. for  ...  Weak convergence.  ... 
arXiv:1909.11629v3 fatcat:sqwqeft2fze7zli24zj4etqvxq

Modification of transition's factor in the compact surface-potential-based MOSFET model

Tijana Kevkic, Vladica Stojanovic, Dragan Petkovic
2016 The University Thought: Publication in Natural Sciences  
The results of implicit SPB model and weak inversion approximation, given by Eq. (3) , are also depicted in this figure.  ...  The results of the weak inversion approximation (3) (dotted line), and the classical implicit ψ s model (dashed line) are also shown.  ... 
doi:10.5937/univtho6-11360 fatcat:64ljn2up2jgfjktxirnbpdfaya
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