Numerical Approximation of Solutions to Stochastic Partial Differential Equations and Their Moments
Doctoral thesis, 2018
The first part of this thesis focusses on the numerical approximation of the first two moments of solutions to parabolic stochastic partial differential equations (SPDEs) with additive or multiplicative noise. More precisely, in Paper I an earlier result (A. Lang, S. Larsson, and Ch. Schwab, Covariance structure of parabolic stochastic partial differential equations, Stoch. PDE: Anal. Comp., 1(2013), pp. 351–364), which shows that the second moment of the solution to a parabolic SPDE driven by additive Wiener noise solves a well-posed deterministic space-time variational problem, is extended to the class of SPDEs with multiplicative Lévy noise. In contrast to the additive case, this variational formulation is not posed on Hilbert tensor product spaces as trial–test spaces, but on projective–injective tensor product spaces, i.e., on non-reflexive Banach spaces. Well-posedness of this variational problem is derived for the case when the multiplicative noise term is sufficiently small. This result is improved in Paper II by disposing of the smallness assumption. Furthermore, the deterministic equations in variational form are used to derive numerical methods for approximating the first and the second moment of solutions to stochastic ordinary and partial differential equations without Monte Carlo sampling. Petrov–Galerkin discretizations are proposed and their stability and convergence are analyzed.
In the second part the numerical solution of fractional order elliptic SPDEs with spatial white noise is considered. Such equations are particularly interesting for applications in statistics, as they can be used to approximate Gaussian Matérn fields. Specifically, in Paper III a numerical scheme is proposed, which is based on a finite element discretization in space and a quadrature for an integral representation of the fractional inverse involving only non-fractional inverses. For the resulting approximation, an explicit rate of convergence to the true solution in the strong mean-square sense is derived. Subsequently, in Paper IV weak convergence of this approximation is established. Finally, in Paper V a similar method, which exploits a rational approximation of the fractional power operator instead of the quadrature, is introduced and its performance with respect to accuracy and computing time is compared to the quadrature approach from Paper III and to existing methods for inference in spatial statistics.
In the second part the numerical solution of fractional order elliptic SPDEs with spatial white noise is considered. Such equations are particularly interesting for applications in statistics, as they can be used to approximate Gaussian Matérn fields. Specifically, in Paper III a numerical scheme is proposed, which is based on a finite element discretization in space and a quadrature for an integral representation of the fractional inverse involving only non-fractional inverses. For the resulting approximation, an explicit rate of convergence to the true solution in the strong mean-square sense is derived. Subsequently, in Paper IV weak convergence of this approximation is established. Finally, in Paper V a similar method, which exploits a rational approximation of the fractional power operator instead of the quadrature, is introduced and its performance with respect to accuracy and computing time is compared to the quadrature approach from Paper III and to existing methods for inference in spatial statistics.
(Petrov–)Galerkin discretizations
Strong and weak convergence
Fractional operators
Finite element methods
Space-time variational problems
Tensor product spaces
Stochastic partial differential equations
White noise
Lecture hall Pascal, Mathematical Sciences, Hörsalsvägen 1
Opponent: Prof. Dr. Peter Kloeden, Institute of Mathematics, Goethe University, Frankfurt am Main, Germany
Author
Kristin Kirchner
Chalmers, Mathematical Sciences, Applied Mathematics and Statistics
Covariance structure of parabolic stochastic partial differential equations with multiplicative Lévy noise
Journal of Differential Equations,; Vol. 262(2017)p. 5896-5927
Journal article
Kristin Kirchner. Numerical methods for the deterministic second moment equation of parabolic stochastic PDEs
David Bolin, Kristin Kirchner, and Mihály Kovács. Numerical solution of fractional elliptic stochastic PDEs with spatial white noise
David Bolin, Kristin Kirchner, and Mihály Kovács. Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise
David Bolin and Kristin Kirchner. The rational SPDE approach for Gaussian random fields with general smoothness
Prognoser görs varje dag i tidningar, radio och på TV: Om väder, pollen, översvämningar, luftföroreningar och epidemier för att nämna några exempel. Alla dessa har gemensamt att de ofta är baserade på prediktioner med hjälp av statistiska modeller. För att beräkna dessa prediktioner antar man att storheten som är av intresse (till exempel temperatur, nederbörd eller ozon) vid varje position är slumpmässig, det vill säga att den är modellerad med ett stokastiskt fält.
Även om ett specifikt utfall av ett sådant stokastiskt fält är slumpmässigt så följer dess beteende vissa matematiska lagar. En specifik klass av stokastiska fält utgörs av normalfördelade fält. Dessa är attraktiva för tillämpningarna ovan eftersom de matematiska lagarna som karakteriserar dem är, i viss mening, enkla och lätta att använda. För statistisk prediktion är det speciellt viktigt att kunna simulera utfall av normalfördelade fält.
I denna avhandling fokuserar vi bland annat på problemet att beräkna lösningar till en specifik klass av stokastiska partiella differentialekvationer. Dessa ekvationer har egenskapen att deras lösningar är normalfördelade fält. Genom att formulera en metod för att lösa dessa ekvationer approximativt och snabbt med en dator får vi därför ett effektivt sätt att simulera utfall för normalfördelade fält. Som nämns ovan är dessa utfall viktiga för prediktion i spatial statistik och resultaten kan tillämpas inom många områden.
Även om ett specifikt utfall av ett sådant stokastiskt fält är slumpmässigt så följer dess beteende vissa matematiska lagar. En specifik klass av stokastiska fält utgörs av normalfördelade fält. Dessa är attraktiva för tillämpningarna ovan eftersom de matematiska lagarna som karakteriserar dem är, i viss mening, enkla och lätta att använda. För statistisk prediktion är det speciellt viktigt att kunna simulera utfall av normalfördelade fält.
I denna avhandling fokuserar vi bland annat på problemet att beräkna lösningar till en specifik klass av stokastiska partiella differentialekvationer. Dessa ekvationer har egenskapen att deras lösningar är normalfördelade fält. Genom att formulera en metod för att lösa dessa ekvationer approximativt och snabbt med en dator får vi därför ett effektivt sätt att simulera utfall för normalfördelade fält. Som nämns ovan är dessa utfall viktiga för prediktion i spatial statistik och resultaten kan tillämpas inom många områden.
Every day, forecasts are made in the newspapers, on the radio, and on television: about weather, pollen count, floods, air pollution, and epidemics, to name a few. All of them have in common that their prognosis typically is based on predictions using statistical models. To arrive at these predictions, one assumes that, at every considered location, the quantity of interest (e.g., temperature, precipitation, ozone) is random, i.e., it is modeled by a random field.
Even though a specific outcome of such a field is random, its behavior follows certain mathematical laws. A specific class of these random fields is formed by Gaussian random fields. They are highly attractive for the applications mentioned above, since the mathematical laws characterizing them are, in a sense, simple and practicable. For the statistical predictions, it is particularly important to be able to create realizations (i.e., possible outcomes) of these Gaussian random fields.
In this thesis, we consider (among other topics) the problem computing solutions to a specific class of stochastic partial differential equations. These equations have the property that their solutions are Gaussian random fields. Thus, by formulating a method for solving these stochastic partial differential equations approximately and quickly with a computer, we thereby describe an efficient way of creating realizations of Gaussian random fields. As outlined above, these realizations are of importance for predictions in spatial statistics and there are numerous applications for the outcomes of this work.
Even though a specific outcome of such a field is random, its behavior follows certain mathematical laws. A specific class of these random fields is formed by Gaussian random fields. They are highly attractive for the applications mentioned above, since the mathematical laws characterizing them are, in a sense, simple and practicable. For the statistical predictions, it is particularly important to be able to create realizations (i.e., possible outcomes) of these Gaussian random fields.
In this thesis, we consider (among other topics) the problem computing solutions to a specific class of stochastic partial differential equations. These equations have the property that their solutions are Gaussian random fields. Thus, by formulating a method for solving these stochastic partial differential equations approximately and quickly with a computer, we thereby describe an efficient way of creating realizations of Gaussian random fields. As outlined above, these realizations are of importance for predictions in spatial statistics and there are numerous applications for the outcomes of this work.
Subject Categories
Computational Mathematics
Probability Theory and Statistics
Mathematical Analysis
ISBN
978-91-7597-697-6
Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4378
Publisher
Chalmers
Lecture hall Pascal, Mathematical Sciences, Hörsalsvägen 1
Opponent: Prof. Dr. Peter Kloeden, Institute of Mathematics, Goethe University, Frankfurt am Main, Germany