Heat1D: 1D heat Bayesian inverse problem using CUQIpy¶

Overview¶

This benchmark is built using the library CUQIpy. It defines a posterior distribution for a 1D heat inverse problem, with a Gaussian likelihood and Karhunen–Loève (KL) parameterization of the uncertain parameters. Posteriors for two cases are available. In the first case, the data is available everywhere in the domain and with a noise level of \(0.1\%\); and in the other case, the data is available only on the left half of the domain and with a noise level of \(5\%\). For more details see Alghamdi et al. (2023).

Below are plots for the small noise case (top plot) and the large noise case (bottom plot). In both plots, the panels show the following: (a) Noisy data, exact data, and exact solution (b) Samples \(95\%\) credible interval computed after mapping the KL coefficients \(\mathbb{x}\) samples to the corresponding function \(\mathbb{g}\) samples. (c) KL coefficients samples \(95\%\) credible interval.

Small noise case Large noise case

Authors¶

Run¶

docker run -it -p 4242:4242 linusseelinger/benchmark-heat-1d

Properties¶

Model	Description
Heat1DSmallNoise	Posterior distribution for the small noise case
Heat1DLargeNoise	Posterior distribution for the large noise and incomplete data case
Heat1DExactSolution	Exact solution of the 1D heat inverse problem
KLExpansionCoefficient2Function	Map from the KL coefficients to the corresponding function
KLExpansionFunction2Coefficient	Projection of functions onto the KL coefficients space

Heat1DSmallNoise¶

Mapping	Dimensions	Description
input	[20]	KL Coefficients \(\mathbf{x}\)
output	[1]	Posterior log PDF \(\pi(\mathbf{x}\mid\mathbf{b})\) of the small noise case

Feature	Supported
Evaluate	True
Gradient	False
ApplyJacobian	False
ApplyHessian	False

Config	Type	Default	Description
None

Heat1DLargeNoise¶

Mapping	Dimensions	Description
input	[20]	KL coefficients \(\mathbf{x}\)
output	[1]	Posterior log PDF \(\pi(\mathbf{x}\mid\mathbf{b})\) of the large noise and incomplete data case

Feature	Supported
Evaluate	True
Gradient	False
ApplyJacobian	False
ApplyHessian	False

Config	Type	Default	Description
None

Heat1DExactSolution¶

Mapping	Dimensions	Description
input	[0]	No input to be provided.
output	[100]	Returns the exact solution \(\mathbf{g}^\text{exact}\) for the heat 1D inverse problem (the discretized initial heat profile).

Feature	Supported
Evaluate	True
Gradient	False
ApplyJacobian	False
ApplyHessian	False

Config	Type	Default	Description
None

KLExpansionCoefficient2Function¶

Mapping	Dimensions	Description
input	[20]	KL coefficients \(\mathbf{x}\)
output	[100]	The function values on grid points: the result of the mapping from the KL coefficients space to the function space

Feature	Supported
Evaluate	True
Gradient	False
ApplyJacobian	False
ApplyHessian	False

Config	Type	Default	Description
None

KLExpansionFunction2Coefficient¶

Mapping	Dimensions	Description
input	[100]	Function values on grid points
output	[20]	The KL coefficients: the result of projecting the function values onto the KL coefficients space

Feature	Supported
Evaluate	True
Gradient	False
ApplyJacobian	False
ApplyHessian	False

Config	Type	Default	Description
None

Mount directories¶

Mount directory	Purpose
None

Source code¶

Sources here.

Description¶

This benchmark defines the posterior distribution of a Bayesian inverse problem governed by a 1D heat equation. The underlying inverse problem is to infer an initial temperature profile \(g(\xi)\) at time \(\tau=0\) on the unit interval \(\xi \in [0,1]\) form measurements of temperature \(u(\xi, \tau)\) at time \(\tau=\tau^\text{max}\).

In this example, the governing partial differential equation that can be solved to find the temperature \(u(\xi, \tau)\) given the initial temperature profile \(g(\xi)\) is

\[\begin{split} \begin{align} \frac{\partial u(\xi,\tau)}{\partial t} - \frac{\partial^2 u(\xi,\tau)}{\partial \xi^2} & = 0, \quad \xi\in[0,1], \quad 0\le \tau \le \tau^\mathrm{max}, \\ u(0,\tau)= u(1,\tau)&= 0,\\ u(\xi,0)&= g(\xi), \end{align} \end{split}\]

assuming zero source term and a constant diffusion coefficient of value 1. We discretize the system using first order finite difference method in space on a regular grid of \(n=100\) nodes, and forward Euler in time. We denote by \(\mathbf{g}\) and \(\mathbf{u}\) the discretization of \(g\) and \(u\).

Furthermore, we parameterize \(\mathbf{g}\) using a truncated Karhunen–Loève (KL) expansion to impose some regularity and spatial correlation and reduce the dimension of the discretized unknown parameter from \(n\) to \(n_\text{KL}\), where \(n_\text{KL} \ll n\). For given expansion basis \(\mathbf{a}_i\) for \(i=1,...,n_\text{KL}\), parameterization \(\mathbf{g}\) in terms of the KL expansion coefficients \(\mathbf{x}=[x_1, ..., x_{n_\text{KL}}]\) can be written as

\[ \mathbf{g} =\sum_{i=1}^{n_\text{KL}} x_i \mathbf{a}_i. \]

In the Bayesian setting, we consider \(\mathbf{x}\) an \(n_\text{KL}\)-dimensional random vector representing the unknown KL coefficients. And thus the corresponding \(\mathbf{g}\) and \(\mathbf{u}\) are random vectors as well. We define the inverse problem as

\[ \mathbf{b} = \mathbf{A}(\mathbf{x}) + \mathbf{e}, \]

where \(\mathbf{b}\) is an \(m\)-dimensional random vector representing the observed data, in this case measured temperature profile at time \(\tau^\text{max}\), and \(\mathbf{e}\) is an \(m\)-dimensional random vector representing the noise. \(\mathbf{A}\) is the forward operator that maps \(\mathbf{x}\) to \(\mathbf{b}\). Applying \(\mathbf{A}\) involves solving the heat PDE above for given \(\mathbf{x}\) and extracting the observations \(\mathbf{b}\) from the space-time solution \(\mathbf{u}\).

This benchmark defines a posterior distribution over \(\mathbf{x}\) given \(\mathbf{b}\) as

\[ \pi(\mathbf{x}\mid \mathbf{b}) \propto \pi(\mathbf{b}\mid \mathbf{x})\pi(\mathbf{x}), \]

where \(\pi(\mathbf{b}|\mathbf{x})\) is a likelihood function and \(\pi(\mathbf{x})\) is a prior distribution.

The noise is assumed to be Gaussian with a known noise level, and so the likelihood is defined via

\[ \mathbf{b} \mid \mathbf{x} \sim \mathcal{N}(\mathbf{A}\mathbf{x}, \sigma^2\mathbf{I}_m), \]

where \(\mathbf{I}_m\) is the \(m\times m\) identity matrix and \(\sigma\) defines the noise level.

Two setups of this Bayesian problem are available

Heat1DLargeNoise: A posterior for which the data is available everywhere in the domain (on grid points) and at time \(\tau^\text{max}\), with a noise level of \(0.1\%\)
Heat1DSmallNoise A posterior for which the data is available only on the left half of the domain (on grid points) and at time \(\tau^\text{max}\), with a noise level of \(5\%\)

See um-bridge Clients for more details on how to use such UM-Bridge models.

In addition to the two HTTP models for the posterior, there is also an HTTP model for the exact solution of the problem. This model is called Heat1DExactSolution and returns exact initial heat profile used to generate the synthetic data when called. The map from the coefficients \(\mathbf{x}\) to the discretized function \(\mathbf{g}\) is provided via the HTTP model KLExpansionCoefficient2Function and the projection of \(\mathbf{g}\) on the coefficient space \(\mathbf{x}\) is provided by the HTTP model KLExpansionFunction2Coefficient.

Using CUQIpy, this benchmark is defined in the files heat1D_problem.py, data_script.py, and server.py provided here.