Bayesian Probabilistic Numerical Methods (Part I) Chris. J. Oates Newcastle University Alan Turing Institute Joint work with: Jon Cockayne, Tim Sullivan and Mark Girolami June 2017 @ ICERM Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 1 / 25

Motivation: Computational Pipelines Numerical analysis for the “drag and drop” era of computational pipelines: [Fig: IBM High Performance Computation] The sophistication and scale of modern computer models creates an urgent need to better understand the propagation and accumulation of numerical error within arbitrary - often large - pipelines of computation, so that “numerical risk” to end-users can be controlled. Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 2 / 25

Motivation: Solution of Poisson’s Equation Consider numerical solution for x ∈ X of the Poisson equation − ∆ x = f in D x = g on ∂ D based on (noiseless) information of the form − ∆ x ( t 1 ) f ( t 1 ) . . . . . . − ∆ x ( t m ) f ( t m ) { t i } m { t i } d A ( x ) = = , i =1 ∈ D , i = m +1 ∈ ∂ D . x ( t m +1 ) g ( t m +1 ) . . . . . . x ( t n ) g ( t n ) This is an ill-posed inverse problem and must be regularised. The onus is on us to establish principled statistical foundations that are general. Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 3 / 25

Motivation: Solution of Poisson’s Equation Consider numerical solution for x ∈ X of the Poisson equation − ∆ x = f in D x = g on ∂ D based on (noiseless) information of the form − ∆ x ( t 1 ) f ( t 1 ) . . . . . . − ∆ x ( t m ) f ( t m ) { t i } m { t i } d A ( x ) = = , i =1 ∈ D , i = m +1 ∈ ∂ D . x ( t m +1 ) g ( t m +1 ) . . . . . . x ( t n ) g ( t n ) This is an ill-posed inverse problem and must be regularised. The onus is on us to establish principled statistical foundations that are general. Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 3 / 25

Motivation: Solution of Poisson’s Equation Consider numerical solution for x ∈ X of the Poisson equation − ∆ x = f in D x = g on ∂ D based on (noiseless) information of the form − ∆ x ( t 1 ) f ( t 1 ) . . . . . . − ∆ x ( t m ) f ( t m ) { t i } m { t i } d A ( x ) = = , i =1 ∈ D , i = m +1 ∈ ∂ D . x ( t m +1 ) g ( t m +1 ) . . . . . . x ( t n ) g ( t n ) This is an ill-posed inverse problem and must be regularised. The onus is on us to establish principled statistical foundations that are general. Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 3 / 25

Insight: Numerical Analysis as Bayesian Inversion The Bayesian approach, popularised in Stuart (2010), can be used: a prior measure P x is placed on X a posterior measure P x | a is defined as the “restriction of P x to those functions x ∈ X for which − ∆ x ( t 1 ) . . A ( x ) = a e.g. A ( x ) = = a . − ∆ x ( t n ) is satisfied” (to be formalised). = ⇒ Principled and general uncertainty quantification for numerical methods. Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 4 / 25

Insight: Numerical Analysis as Bayesian Inversion The Bayesian approach, popularised in Stuart (2010), can be used: a prior measure P x is placed on X a posterior measure P x | a is defined as the “restriction of P x to those functions x ∈ X for which − ∆ x ( t 1 ) . . A ( x ) = a e.g. A ( x ) = = a . − ∆ x ( t n ) is satisfied” (to be formalised). = ⇒ Principled and general uncertainty quantification for numerical methods. Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 4 / 25

The Research Agenda Part I First Job: Elicit the Abstract Structure 1 Second Job: Check Well-Defined, Existence and Uniqueness 2 Third Job: Characterise Optimal Information 3 Part II Fourth Job: Algorithms to Access P x | a 4 Fifth Job: Extend to Pipelines of Computation 5 Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 5 / 25

First Job: Elicit the Abstract Structure Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 6 / 25

Abstract Structure Abstractly, consider an unobserved state variable x ∈ X together with: A quantity of interest , denoted Q ( x ) ∈ Q An information operator , denoted x �→ A ( x ) ∈ A . Examples: Task Q ( x ) A ( x ) { x ( t i ) } n � Integration x ( t ) ν ( d t ) i =1 { x ( t i ) } n Optimisation arg max x ( t ) i =1 {− ∆ x ( t i ) } m i =1 ∪ { x ( t i ) } n Solution of Poisson Eqn x ( · ) i = m +1 Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 7 / 25

Abstract Structure Abstractly, consider an unobserved state variable x ∈ X together with: A quantity of interest , denoted Q ( x ) ∈ Q An information operator , denoted x �→ A ( x ) ∈ A . Examples: Task Q ( x ) A ( x ) { x ( t i ) } n � Integration x ( t ) ν ( d t ) i =1 { x ( t i ) } n Optimisation arg max x ( t ) i =1 {− ∆ x ( t i ) } m i =1 ∪ { x ( t i ) } n Solution of Poisson Eqn x ( · ) i = m +1 Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 7 / 25

Abstract Structure Let P • denote the set of distributions on • . Let M # µ denote the “pushforward” measure, st ( M # µ )( S ) = µ ( M − 1 ( S )). Classical Numerical Probabilistic Numerical Method Method Assumed e.g. smoothness P x ∈ P X Inputs Information a ∈ A a ∈ A Output b ( a ) ∈ Q B ( P x , a ) ∈ P Q A Probabilistic Numerical Method is Bayesian iff B ( P x , a ) = Q # P x | a . Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 8 / 25

Abstract Structure Let P • denote the set of distributions on • . Let M # µ denote the “pushforward” measure, st ( M # µ )( S ) = µ ( M − 1 ( S )). Classical Numerical Probabilistic Numerical Method Method Assumed e.g. smoothness P x ∈ P X Inputs Information a ∈ A a ∈ A Output b ( a ) ∈ Q B ( P x , a ) ∈ P Q A Probabilistic Numerical Method is Bayesian iff B ( P x , a ) = Q # P x | a . Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 8 / 25

Abstract Structure Let P • denote the set of distributions on • . Let M # µ denote the “pushforward” measure, st ( M # µ )( S ) = µ ( M − 1 ( S )). Classical Numerical Probabilistic Numerical Method Method Assumed e.g. smoothness P x ∈ P X Inputs Information a ∈ A a ∈ A Output b ( a ) ∈ Q B ( P x , a ) ∈ P Q A Probabilistic Numerical Method is Bayesian iff B ( P x , a ) = Q # P x | a . Chris. J. Oates Probabilistic Numerical Methods (I) June 2017 @ ICERM 8 / 25

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