# A Bayesian inverse problem benchmark based on the Laplace equation ## Overview This model implements the benchmark described in [this preprint](https://doi.org/10.48550/arXiv.2102.07263), 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](mailto:David.Aristoff@colostate.edu) (Colorado State University) - [Wolfgang Bangerth](mailto:bangerth@colostate.edu) (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.](https://github.com/UM-Bridge/benchmarks/tree/main/models/dealii-laplace) ## Description See [this preprint](https://doi.org/10.48550/arXiv.2102.07263).