The solver provides the solution of the linear system of equations with Gaussian noise using belief propagation (BP) algorithm applied over the factor graph.

We observe a noisy linear system of equations with real coefficients and variables:

where x is the vector of the state variables (i.e., unknowns), f(x) is the vector of linear functions, b is the vector of observation values and u is the vector of uncorrelated observation errors. Note that the linear system of equations represents an overdetermined system.

The solution can be obtained by solving linear weighted least-squares (WLS) problem:

where A is the Jacobian matrix of linear functions or the coefficient matrix for our system, and W is a diagonal matrix containing inverses of observation variances.

Further, the solution to the problem can be found via maximization of the likelihood function which is defined via likelihoods of independent observations, and that can be efficiently solved utilizing factor graphs and the Gaussian belief propagation (BP) algorithm.

data.mat file with variables:

• data.A - coefficient matrix m x n (m>n);
• data.b - observation values column vector dimension of m x 1;
• data.v - observation variances column vector dimension of m x 1;

1. Post-Processing Options:

   • user.save - write data to a text file;
   • user.radius - compute spectral radius for synchronous and randomized damping scheduling, if spectral radius is less than 1 the BP algorithm converges;
   • user.error - compute mean absolute error, root mean square error and weighted residual sum of squares for solution;

2. Design of Iteration Scheme:

   • user.stop - the BP algorithm in the iteration loop is running until the criterion is reached, where the criterion is applied on the vector of mean-value messages from factor nodes to variable nodes in two consecutive iterations;
   • user.maxi - the upper limit on BP iterations;

3. Convergence Parameters:

   • user.prob - a Bernoulli random variable with probability "prob" independently sampled for each mean value message from indirect factor node to a variable node, with values between 0 and 1;
   • user.alph - the damped message is evaluated as a linear combination of the message from the previous and the current iteration, with weights "alph" and 1 - "alph", where "alph" is between 0 and 1;

Note: We use an improved BP algorithm that applies synchronous scheduling with randomized damping. The randomized damping parameter pairs lead to a trade-off between the number of non-converging simulations and the rate of convergence. In general, for the selection of "prob" and "alph" for which only a small fraction of messages are combined with their values in a previous iteration, and that is a case for "prob" close to 0 or "alph" close to 1, we observe a large number of non-converging simulations.

4. Virtual Factor Nodes
   • user.mean - the mean value of virtual factor nodes;
   • user.vari - the variance value of the virtual factor nodes;

Note: The virtual factor node is a singly-connected factor node used if the variable node x is not directly observed. In a usual scenario, without prior knowledge, the variance of virtual factor nodes tend to infinity.

• M. Cosovic and D. Vukobratovic, "Distributed Gauss-Newton Method for State Estimation Using Belief Propagation," in IEEE Transactions on Power Systems, vol. 34, no. 1, pp. 648-658, Jan. 2019. arxiv.org
• M. Cosovic, "Design and Analysis of Distributed State Estimation Algorithms Based on Belief Propagation and Applications in Smart Grids." arXiv preprint arXiv:1811.08355 (2018). arxiv.org