Exponential Model Reduction

expred

When an exponential sum

\[f(t) = \sum_{i=1}^N c_i \mathrm{e}^{-a_i t}, \quad \mathrm{Re}\;a_i > 0\]

is already available, the expred function can find a reduced exponential sum approximation with fewer terms for a given tolerance i.e.

\[\left\|f(t)-\sum_{i=1}^{N'} c_i' \mathrm{e}^{-a_i' t}\right\|<\epsilon, \quad \mathrm{Re}\;a_i' > 0\]

where $N'<N$.

Let a be the vector storing the initial exponents and c be the vector storing the initial coefficients. Here, we consider a sum of 100 terms, which are generated randomly. Note that the real part of a must be non-negative.

er = expred(a, c, 1e-2)
print("Approximation order = ", length(er.coeff), "\n")

t = range(0.0, 50.0, length=100)
err = abs.(er.(t) .- Exponentials(a,c).(t))
println("Root mean square = ", norm(err)/sqrt(N))

er is an instance of the Exponentials type.

When executed, the terminal will display the following.

Approximation order = 10
Root mean square = 0.0003602011932219654

The results are also illustrated below.