The kernel matrix K is based on the covariance k(x x x i , x x x j ) between two elements i, j, which leads to inv (K) n,n = k(x x x n , x x x n ) -σ 2 noise I, K *,n = k(x x x * , x x x n ), and K *,* = k(x x x * , x x x * ). This can be used to make a function fit to the new and old data in an optimal way. We prefer a linear modeling of the relationship between HiFi and LoFi data, as originally proposed by Kennedy and O'Hagan, over more flexible nonlinear approaches: