# Registered Data

## [00059] Numerical solutions for differential equations: Probabilistic approaches and statistical perspectives

**Session Date & Time**:- 00059 (1/2) : 1C (Aug.21, 13:20-15:00)
- 00059 (2/2) : 1D (Aug.21, 15:30-17:10)

**Type**: Proposal of Minisymposium**Abstract**: Many applications involve predicting the dynamics of a system by solving differential equations. Due to the increased demand for predictive power of these models, numerically solving a differential equation is now often combined with parameter estimation or uncertainty quantification. This paradigm shift drives the need for probabilistic approaches that are compatible with statistical inference, or that improve the robustness of inference to possibly inaccurate mathematical models. The talks in this minisymposium will present recent work that addresses these challenges for deterministic ODEs and PDEs, by using ideas from numerical analysis, probability theory, and Bayesian statistical inference.**Organizer(s)**: Han Cheng Lie, Takeru Matsuda, Yuto Miyatake**Classification**:__62F15__,__62G99__,__65L05__,__65N75__**Speakers Info**:- Oksana Chkrebtii (The Ohio State University)
- Connor Duffin (The University of Cambridge)
- Toni Karvonen (The University of Helsinki)
- Chris Oates (Newcastle University)
- Yanni Papandreou (Imperial College London)
- Tim Sullivan (The University of Warwick)
- Onur Teymur (The University of Kent)
- Yue Wu (University of Strathclyde)

**Talks in Minisymposium**:**[00066] Probabilistic Numerical Methods****Author(s)**:**Chris Oates**(Newcastle University)

**Abstract**: The scale and complexity of modern scientific computer codes typically precludes a detailed analysis of how the code is numerically implemented. For example, multi-scale and multi-physics models of the human heart call on diverse numerical sub-routines to integrate differential equations, perform interpolation and optimise over some parameters of the model. As such, the computer output is acknowledged to be inexact and some alternative form of uncertainty quantification is needed for the output to be properly interpreted. This talk will provide an introduction to probabilistic numerical methods, which aim to provide probabilistic uncertainty quantification for computer code output. These methods are composed of "modules", such that a probabilistic description of numerical error can be automatically propagated, and some of the most useful modules will be discussed.

**[00148] Theoretical Guarantees for the Statistical Finite Element Method****Author(s)**:**Yanni Papandreou**(Imperial College London)- Jon Cockayne (University of Southampton)
- Mark Girolami (University of Cambridge)
- Andrew B. Duncan (Imperial College London)

**Abstract**: The statistical finite element method (StatFEM) is an emerging probabilistic method that allows observations of a physical system to be synthesized with the numerical solution of a PDE intended to describe it in a coherent statistical framework, to compensate for model error. This work presents a new theoretical analysis of the statistical finite element method demonstrating that it has similar convergence properties to the finite element method on which it is based. Our results constitute a bound on the Wasserstein-2 distance between the ideal prior and posterior and the StatFEM approximation thereof, and show that this distance converges at the same mesh-dependent rate as finite element solutions converge to the true solution. Several numerical examples are presented to demonstrate our theory, including an example which test the robustness of StatFEM when extended to nonlinear quantities of interest.

**[00186] Posterior error estimates for statistical finite element methods with Sobolev priors****Author(s)**:**Toni Karvonen**(University of Helsinki)- Fehmi Cirak (University of Cambridge)
- Mark Girolami (University of Cambridge)

**Abstract**: The statistical finite element, statFEM, approach synthesises measurement data with finite element models and allows for making predictions about the system response. Suppose that noisy measurement data are generated by a deterministic true system response function satisfying a second-order elliptic partial differential equation for an unknown true source term. In this setting, we provide probabilistic error analysis for a prototypical statFEM setup based on a Gaussian process prior whose covariance kernel induces a Sobolev space.

**[00714] Approximating the solutions of delay differential equations via the randomized Euler method****Author(s)**:**Yue Wu**(University of Strathclyde)- Fabio Difonzo (University of Bari Aldo Moro)
- Pawel Przybyl (AGH University of Science and Technology)

**Abstract**: In this talk, we consider Caratheodory delay ODEs with time-irregular coefficients, where a randomized Euler scheme is proposed to approximate the exact solution. This is the case when there is a lack of convergence for deterministic algorithms.

**[02954] Statistical finite elements for misspecified models****Author(s)**:**Connor Duffin**(University of Cambridge)- Edward Cripps (University of Western Australia)
- Thomas Stemler (University of Western Australia)
- Mark Girolami (University of Cambridge)

**Abstract**: I will present a statistical finite element method for nonlinear, time-dependent problems. This is a statistical augmentation of the finite element method which admits model misspecification inside of the governing equations, via Gaussian processes. The method is Bayesian, sequentially updates model mismatch upon receipt of observed data, and ensures scalability through low-rank approximations to the posterior. In this talk I will present statFEM and discuss various case studies with experimental and synthetic data.

**[04408] GParareal: Towards a time-parallel probabilistic ODE solver****Author(s)**:**T J Sullivan**(University of Warwick)- Kamran Pentland (University of Warwick)
- Massimiliano Tamborrino (University of Warwick)
- Lynton Appel (Culham Centre for Fusion Energy)
- James Buchanan (Culham Centre for Fusion Energy)

**Abstract**: Numerical solution of complex ODEs can be accelerated with time-parallel integration, predicting the solution serially using a cheap solver and correcting these values in parallel using an expensive solver. We propose a time-parallel ODE solver (GParareal) that models the prediction-correction term using a Gaussian process emulator. GParareal compares favourably with the classic parareal algorithm, locates solutions to certain ODEs where parareal fails, and can also use archives of legacy solutions to further accelerate convergence.

**[04863] The Bayesian approach to inverse Robin problems****Author(s)**:**Ieva Kazlauskaite**(University of Cambridge)

**Abstract**: In this talk, I will present the Bayesian approach to inverse Robin problems. The problem of interest is a certain elliptic boundary value problem of determining a Robin coefficient on a hidden part of the boundary from Cauchy data on the observable part. Such an inverse problem arises naturally in the initialisation of large-scale ice sheet models that are crucial in climate and sea-level predictions. We motivate the Bayesian approach for a prototypical Robin inverse problem by showing that the posterior mean converges in probability to the data-generating ground truth as the number of observations increases. Related to the stability theory for inverse Robin problems, we establish a logarithmic convergence rate for regular Robin coefficients, whereas for analytic coefficients we can attain an algebraic rate. Further, our numerical results demonstrate the effectiveness of the approach in recovering the Robin coefficient for an ice sheet model.

**[04930] Prior models for enforcing boundary constraints in state-space probabilistic PDE solvers****Author(s)**:**Oksana Chkrebtii**(The Ohio State University)- Yue Ma (The Ohio State Univeristy)

**Abstract**: Probabilistic numerics is an active field of research that seeks to construct stochastic analogues of numerical methods, including the solution of ordinary and partial differential equations. Probabilistic solvers for partial differential equations require the specification of flexible prior models that respect physical constraints while allowing for computational efficiencies of the sequential updates. We focus on state-space based probabilistic PDE solvers and describe advances in nonparametric modeling of system states with boundary constraints.