Core Concepts

This paper presents an abstract framework to obtain convergence rates for the approximation of random evolution equations corresponding to a random family of forms determined by finite-dimensional noise. The full discretisation error in space, time, and randomness is considered, where polynomial chaos expansion (PCE) is used for the semi-discretisation in randomness.

Abstract

The key highlights and insights of this content are:
The authors consider random evolution equations formulated as abstract Cauchy problems with random generators. They aim to provide approximation methods and convergence results for such equations.
The approximation involves three independent discretisations: in randomness using polynomial chaos expansion (PCE), in space using a Galerkin method, and in time using A-stable schemes.
The main result is a set of regularity conditions on the random forms under which convergence of polynomial order in randomness is obtained, depending on the smoothness of the coefficients and the Sobolev regularity of the initial value. In space and time, the same convergence rates as in the deterministic setting are achieved.
The authors derive error estimates for vector-valued PCE and a quantified version of the Trotter-Kato theorem for form-induced semigroups, which are key tools in the analysis.
The results are illustrated through a random heat equation example with random anisotropic diffusion coefficients.
The authors work mainly from a functional analytic point of view and focus on the Hilbert space setting.

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by Katharina Kl... at **arxiv.org** 04-12-2024

Deeper Inquiries

To extend the framework to handle more general types of random inputs beyond finite-dimensional noise, we can consider utilizing techniques from stochastic analysis and probability theory. One approach could involve incorporating random processes with more complex structures, such as continuous-time stochastic processes or random fields. This would require extending the analysis to include infinite-dimensional spaces and function spaces, allowing for a broader range of random inputs. Additionally, techniques from functional analysis and probability theory, such as stochastic integration and stochastic differential equations, could be employed to model and analyze the behavior of these more general random inputs. By incorporating these advanced mathematical tools, the framework can be adapted to handle a wider variety of random evolution equations with more complex random inputs.

Applying the presented approach to high-dimensional random evolution equations may pose several challenges. One major challenge is the curse of dimensionality, where the computational complexity increases exponentially with the number of random variables. This can lead to difficulties in numerical simulations and analysis, requiring advanced computational techniques and algorithms to handle high-dimensional spaces efficiently. Additionally, the analysis may need to be adapted to account for the increased complexity and dimensionality of the random inputs, potentially requiring modifications to the error estimates and convergence results. Techniques such as dimensionality reduction, sparse grid methods, and tensor approximations could be employed to address these challenges and improve the computational efficiency of the analysis for high-dimensional random evolution equations.

The requirement for additional spatial regularity in the case of non-symmetric forms can potentially be relaxed or optimized by considering alternative approaches or assumptions. One possible optimization could involve refining the error estimates and convergence results to account for the lack of symmetry in the forms. By carefully analyzing the impact of non-symmetry on the convergence rates and error bounds, it may be possible to identify conditions under which the spatial regularity requirement can be relaxed without compromising the accuracy of the results. Additionally, exploring alternative discretization methods or numerical techniques that are less sensitive to the lack of symmetry in the forms could help optimize the analysis and potentially reduce the spatial regularity constraints in the case of non-symmetric forms.

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