New parallel stochastic global integral and local differential equation modeling and inversion

Optimal Control of Stochastic Systems

A Study on Uncertainty of Discharge in River Channel Using Stochastic Partial Differential Equation

Equations in Hirota's bilinear form appear naturally whenever one needs to write integrable differential equations in terms of "nice" functions. In the case of soliton equations nice typically means polynomials of exponentials with lin- ear exponents, whereas in the context of Painleve equations nice often means entire functions. The natural or original dependent variables of the equation are usually not the best in this respect and a change of the dependent vari- ables is necessary. This dependent variable transformation can be somewhat involved. Since the solutions of Painleve equations are meromorphic by definition (the movable singularities can be at worst poles) it is natural to express them as ratios of entire functions, and this leads to homogeneous equations and often to equations in Hirota's bilinear form. Indeed, bilinear forms were already derived by Painleve himself (1). It should be noted, however, that the converse of the above is not necessarily true: a bilinear form does not by itself imply that the independent functions are regular in any sense, in fact bilinear forms exist even for non integrable equations. In the field of integrable partial differential equations, the so-called τ - functions play a major role. These functions actually provide a huge pool of such "nice" functions, in both of the aforementioned interpretations, and they also possess an extremely rich algebraic and algebro-geometric structure. This is the main theme of Sato-theory, one of the great unifying theories for the description of integrable systems. The relevance of τ -functions for the study of the Painleve equations has been recognized ever since the seminal work of Jimbo and Miwa on isomonodromy deformations (2) and in particular since the results obtained by Okamoto, regarding the algebro-geometric properties of the Painleve equations (3).

Stochastic differential equation (SDE) models are the foundation for pricing and hedging financial derivatives. The drift and volatility functions in SDE models are typically chosen to be algebraic functions with a small number ( < 5 ) of parameters which can be calibrated to market data. A more flexible approach is to use neural networks to model the drift and volatility functions, which provides more degrees-of-freedom to match observed market data. Training of models requires optimizing over an SDE, which is computationally challenging. For European options, we develop a fast stochastic gradient descent (SGD) algorithm for training the neural network-SDE model. Our SGD algorithm uses two independent SDE paths to obtain an unbiased estimate of the direction of steepest descent. For American options, we optimize over the corresponding Kolmogorov partial differential equation (PDE). The neural network appears as coefficient functions in the PDE. Models are trained on large datasets (many contracts), requiring either large simulations (many Monte Carlo samples for the stock price paths) or large numbers of PDEs (a PDE must be solved for each contract). Numerical results are presented for real market data including S&P 500 index options, S&P 100 index options, and single-stock American options. The neural-network-based SDE models are compared against the Black-Scholes model, the Dupire's local volatility model, and the Heston model. Models are evaluated in terms of how accurate they are at pricing out-of-sample financial derivatives, which is a core task in derivative pricing at financial institutions. Specifically, we calibrate a neural network-SDE model to market data for a financial derivative on an asset with price S t with a payoff function g ( s ) , and we then evaluate its generalization accuracy for a financial derivative on the same asset S t but with a different payoff function f ( s ) . In addition to comparing out-of-sample pricing accuracy, we evaluate the hedging performance of the neural network-SDE model.

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

In Rani et al. (Numerical inversion of Laplace transform based on Bernstein operational matrix, Mathematical Methods in the Applied Sciences (2018) pp. 1–13), a numerical method is developed to find the inverse Laplace transform of certain functions using Bernstein operational matrix. Here, we describe Bernstein operational matrix of integration and propose an algorithm to solve linear time-varying systems governing differential equations. Apart from discussing error estimate, the method is implemented to linear differential equations on Bessel equation of order zero, damped harmonic oscillator, some higher order differential equations, singular integral equation, Volterra integral and integro-differential equations and nonlinear Volterra integral equations of the first kind. A comparison with some existing methods like Haar operational matrix, block pulse operational matrix and others are discussed. The method is simple and easy to implement on a variety of problems. Relative errors estimate just for 5th or 6th approximation show high applicability of the method.

The ideas of the method of fictitious domains and homotopy are combined with an aim to reduce the solution of boundary-value problems for multidimensional partial differential equations (PDE) in domains of any shape to an exponentially convergent sequence of PDE in a parallelepiped (or, in the 2D case, in a rectangle). This enables us to decrease the required computer time due to the elimination of the necessity of triangulation of the domain by a grid with N inner nodes (thus, the Delaunay algorithm in the 2D case requires $$ \mathcal{O} $$ (N log N) operations).

Background: Growth trajectories are highly variable between children, making epidemiological analyses challenging both to the identification of malnutrition interventions at the population level and also risk assessment at individual level. We introduce stochastic differential equation (SDE) models into child growth research. SDEs describe flexible dynamic processes comprising: drift - gradual smooth changes – such as physiology or gut microbiome, and diffusion - sudden perturbations, such as illness or infection. Methods: We present a case study applying SDE models to child growth trajectory data from the Haydom, Tanzania and Venda, South Africa sites within the MAL-ED cohort. These data comprise n=460 children aged 0-24 months. A comparison with classical curve fitting (linear mixed models) is also presented. Results: The SDE models offered a wide range of new flexible shapes and parameterizations compared to classical additive models, with performance as good or better than standard approaches. The predictions from the SDE models suggest distinct longitudinal clusters that form distinct ‘streams’ hidden by the large between-child variability. Conclusions: Using SDE models to predict future growth trajectories revealed new insights in the observed data, where trajectories appear to cluster together in bands, which may have a future risk assessment application. SDEs offer an attractive approach for child growth modelling and potentially offer new insights.

The Electromagnetic Stirring (EMS) technology has wide applications in the metal continuous casting. Stirring consists of exciting variable magnetic field in metal liquid, the magnetic field generates induction current. The induction current in the metal flow interacts with magnetic filed to generate EM force, which makes perfect motion to improve casting. The EM force strength is proportional with square of the magnetic field strength. The key of the optimized control of EMS is to adjust several parameters for best EM force and deep magnetic penetration. In this paper, we present a new AGILD EMS modeling. We use Maxwell magnetic field differential equation to replace the reduced magnetic field equation. We present new volume magnetic differential integral equation and new boundary strip magnetic field differential integral equation which are proposed by XIe and Li in GLGEO. According to the new AGILD EM modeling in [1], we developed 3D AGILD EMS modeling. Our AGILD EMS modeling does not need any complex boundary condition. The matrix equations are decomposed by GILD parallel processes. Therefore, our AGILD EMS modeling is fast and accurate without any boundary error. Our AGILD EMS modeling can simulate multiple strands and EMSs together. The simulation shows that the AGILD EMS modeling is very powerful tool for the design of excellent EMS system in Billent/Bloom and metal continuous casting system. The simulation is performed by using GLGEO’s new GL[2-4] and AGILD EMS modeling software. The EMS is an established technology and installed in Billet, Bloom and metal casters. The stirring consists of exciting variable magnetic field which is penetrating in the steel, iron, and metal liquid. The magnetic field generates induction electric current in the metal liquid. The induction current in the metal liquid interacts with the magnetic field to generate EM force which drives the metal liquid flow motion such that the casting is perfectly performing. The casting quality is improved by reduced segregation and porosity through transformation to an equiaxed solidification structure. The steel and iron productions are increasing due to the increased casting speeds. The temperature and carbon component are key factors for high quality steel. The magnetic field penetrating distribution depends on the material properties and high temperature. Moreover, the electromagnetic properties depend on the temperature. The temperature difference between outer and insider of the caster mold is very large. We consider the magnetic conductivity be space variable. The reduced magnetic field equation is no longer to govern this case. In this paper, we use the magnetic field Maxwell equation to replace the reduced magnetic equation. Moreover, we present a new strip magnetic field differential integral equation which is proposed by Xie and Li in [1]. We use the new strip integral equation on the double layered boundary and the magnetic field Garlekin equation in the internal domain to construct the AGILD EMS modeling for the metal continuous casting. It has more advantages over FEM and FD modeling. The AGILD and GL EMS modeling, GL and AGILD nondestructive testing, GL and AGILD heating modeling, and GL and AGILD heavier viscosity flow modeling for the continuous casting are developed by Xie and Li in GLGEO. Many EM research works on motor need the condition that the magnetic resistivity monotone increasedly depends on the magnetic induction. It is very strong and unpractical condition. In practice, magnetic resistivity is not monotone increasing but has a minimum point at B = B0. Our nonlinear FEM EM modeling [10-12] are available for the above practical condition, i.e. the magnetic resistivity is not monotone increasing. We prove that if H(B) monotone increasedly depends on B, the nonlinear FEM is convergent. Because H(B) = υ(B)B, so, we only need that υ(B)B monotone increasedly depends on the B, which is widely generalized. Moreover, the H(B)’s monotone property is the practical physical property. Our nonlinear FEM magnetic DC modeling is briefly described in this paper. The organization of the paper is as follows. We present the new strip differential integral equation in the section 2 which is proposed in [1]. The double layered collocation FEM equation for the magnetic field is

Some singular integral evolution equations with wide class of closed operators are studied in Banach space. The considered integral equations are investigated without the existence of the resolvent of the closed operators. Also, some non-linear singular evolution equations are studied. An abstract parabolic transform is constructed to study the solutions of the considered ill-posed problems. Applications to fractional evolution equations and Hilfer fractional evolution equations are given. All the results can be applied to general singular integro-differential equations. The Fourier Transform plays an important role in constructing solutions of the Cauchy problems for parabolic and hyperbolic partial differential equations. This means that the Fourier transform is suitable but under conditions on the characteristic forms of the partial differential operators. Also, the Laplace transform plays an important role in studying the Cauchy problem for abstract differential equations in Banach space. But in this case, we need the existence of the resolvent of the considered abstract operators. This note is devoted to exploring the Cauchy problem for general singular integro-partial differential equations without conditions on the characteristic forms and also to study general singular integral evolution equations. Our approach is based on applying the new parabolic transform. This transform generalizes the methods developed within the regularization theory of ill-posed problems.

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

ABSTRACTThis article proposes two novel numerical techniques for solving two‐dimensional stochastic Itô–Volterra Fredholm integral equations. Many two‐dimensional stochastic integral equations present significant challenges for analytical solutions. Consequently, possessing an efficient method to get very precise numerical solutions for these stochastic integral equations is of paramount importance. In this article, two novel numerical techniques based on two‐dimensional wavelets have been successfully introduced to acquire the numerical solutions of the two‐dimensional stochastic Itô–Volterra Fredholm integral equations. This efficient and noteworthy method transforms the stochastic integral equation into a system of algebraic equations, which is then solved using an appropriate numerical technique. Moreover, convergence analysis and the error analysis have also been well established successfully. At the end, some test problems have been presented to illustrate the accuracy, efficiency, simplicity, and plausibility of the proposed techniques.

Semi-analytical Karhunen–Loeve representation of irregular waves based on the prolate spheroidal wave functions

In this study, we propose a new method that is useful for estimating unknown parameter values of stochastic differential equation (SDE) models, based on probability density function (PDF) data measured from random dynamical systems. As our method does not require explicit description of PDF, it can be applied to the SDE models even when their PDFs are hardly derived in explicit forms due to multiplicative-noise terms, nonlinear terms, and so on. Therefore, our method is expected to provide a versatile tool to dynamically parameterize measured PDF data. In our proposed method, it is assumed that a measured PDF is obtained from a random dynamical system whose structure is described by a known SDE model with unknown parameter values. With the help of It\^o calculus, the Fokker-Planck equation (FPE) is derived from the SDE model. The measured PDF and a candidate of parameter values are substituted into the FPE to calculate a FPE residual. Our method is applied to two random vibration systems. Their FPE residuals tend to zero as the parameter values tend to exact values, showing that our proposed FPE residual can be utilized for unknown parameter estimation of SDE models.