# PDE as Nonlinear least squares problems with binary variables

 2 1 Suppose we have an edge-weighted i a heat network, then we want to identify which edges will be contribute in the heat convection. I am using heat equation on a network and then solve it via least-squares problem. I want to solve the heat equation T_t(x,t) = - L_x . T(x,t) + F(x,t) in an edge-weighted graph where L_x = \sum_i x_i e_{ij} is weighted Laplacian matrix of the graph, F() are sources on some nodes and T() is temperature of each node in each time step. Then using Theta method I conclude to the following least squares problem: \min_{x\in \{0,1\}} \|\hat{T}_j(t_i) - T_j(t_i, x)\|_2^2 I have done the part which variables(weights) are relaxed to be between [0,1]. I used L1 regularization and active set methods in Matlab's fmincon to approximate the edge weights and then simulate the solution graph using edge-weights. Next step would be to solve this problem using exact methods, for example:branch and bound, etc. My question: is there would be right way to go with B&B?(I know it would be expensive to apply exact methods, but will worth to check and compare the solutions). Any other suggestion would be really appreciate. I could not find some references on the discrete methods for nonlinear least squares. Would you lease let me know if you know? asked 10 Feb '15, 05:17 Saber 33●4 accept rate: 0%
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