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.
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¶
Description¶
See this preprint.