A Bayesian inverse problem benchmark based on the Laplace equation¶

Overview¶

This model implements the benchmark described in this preprint, currently accepted for publication in SIAM Review. This benchmark implements an inverse problem where one tries to infer the stiffness properties of a membrane – that is, the coefficient a(x) in the equation -div [ a(x) grad u(x) ] = f(x) – from measurements of the displacement u(x).

In the concrete case considered by the benchmark, the 64 inputs correspond to an 8x8 grid of stiffness values of the membrane (i.e., the coefficient a(x) is discretized on an 8x8 grid of piecewise constant values). The membrane deforms under a (known) form, and the 169 outputs correspond to the values of the resulting deformation at a 13x13 grid of points.

The complete benchmark compares these 169 outputs with actual measurements (obtained through independent computations that solve the forward problem with a different method) and augments it with a prior probability distribution on the set of 64 parameters. The result is a posterior probability distribution whose properties (such as mean and covariance) the benchmark probes.

The complete benchmark, along with its solution, is described in great detail in the paper mentioned above.

Authors¶

David Aristoff (Colorado State University)
Wolfgang Bangerth (Colorado State University)

Run¶

docker run -it -p 4242:4242 linusseelinger/model-laplace:latest

Properties¶

Model	Description
forward	Forward evaluation of the Laplace equation
posterior	Posterior density

forward¶

Mapping	Dimensions	Description
input	[64]	A set of 64 parameters corresponding to an 8x8 grid of stiffness values of a membrane
output	[169]	A set of 169 displacement values corresponding to the displacement of the membrane at a grid of 13x13 points

Feature	Supported
Evaluate	True
Gradient	False
ApplyJacobian	False
ApplyHessian	False

Config	Type	Default	Description
None

posterior¶

Mapping	Dimensions	Description
input	[64]	A set of 64 parameters corresponding to an 8x8 grid of stiffness values of a membrane
output	[1]	The posterior of the input parameters

Feature	Supported
Evaluate	True
Gradient	False
ApplyJacobian	False
ApplyHessian	False

Config	Type	Default	Description
None

Mount directories¶

Mount directory	Purpose
None

Source code¶

Model sources here.

Description¶

See this preprint.