Professor Stig Larsson is a distinguished mathematician and academic affiliated with Chalmers University of Technology in Gothenburg, Sweden. His expertise is in the field of applied mathematics, particularly focusing on numerical methods for partial differential equations and finite element methods. Prof. Larsson has made significant contributions to the study of mathematical models and their computational solutions. His research interests extend to areas such as semilinear parabolic problems, dynamical systems, and stochastic partial differential equations. In addition to his academic work, he has authored a widely-used textbook titled "Partial Differential Equations with Numerical Methods".
Prof. Larsson will present a survey of methods for proving strong convergence and weak convergence of numerical methods for the stochastic heat and wave equations.
Strong convergence refers to convergence with respect to a norm, for example, mean square convergence. Proofs typically involve representation of the error using the semigroup theory or energy estimates and using the Ito isometry or the Burkholder--Davies--Gundy inequality.
Weak convergence involves the error in some functional of the solution. Proofs may involve representation of the weak error in terms of the Kolmogorov equation and may use integration by parts from the Malliavin calculus.
The main part of the lectures will be concerned with the linear stochastic heat and wave equations. As an example of a more difficult problem, Prof. Larsson will briefly discuss the stochastic Cahn--Hilliard equation (if time permits). A rough outline of topics is:
- Stochastic integration in Hilbert space. Stochastic evolution problem in Hilbert space. Semigroup, mild solution. Stochastic heat equation. Stochastic wave equation.
- Numerical approximation by finite elements and Euler's method. Strong convergence.
- Weak convergence. Malliavin calculus.
- Stochastic Cahn--Hilliard equation. (Time permitting.)
