Volterra integral equation solver

I am looking for good (fast and accurate, with recent state of art methods) numerical (not symbolic) Volterra Integral Equation (1st kind especially) solver. A few solvers available on FEX are only based on very naive and ineffective methods.

Thanks in advance for any help.

Motivation note:

I would like to numerically solve the following inverse (Linear Dynamic System) problem:

x'(t) = Ax(t) + Bu(t)

y(t) = Cx(t); x(0) = x0

where A,B,C are known constant matrices and y(t) is known (measured) noisy output discrete signal. The goal is estimation of input signal u.

There exist an analytical form of solution y(t):

y(t) = Integral_0^t (C*exp(A*(t-s))*B*u(s) ds = Integral_0^t (K(s,t)*u(s) ds) [*]

where kernel is K(s,t) = C*exp(A*(t-s))*B, so the eq [*] has the form of Volterra intergral equatioon of first kind.

Question: Is there any other possibility how to effectivelly and robustly estimate uknown input signal u(t)?

[Torsten]

Do you know of any (also not written in MATLAB) ? I don't.

[Michal]

I just found this Python solver. But I still trying to find something more "robust and effective".

[Torsten]

In the old days, the NAG library was the standard for numerical computations:

https://www.nag.com/numeric/nl/nagdoc_26.1/nagdoc_fl26.1/html/d05/d05intro.html

I don't know if in times of MATLAB and MATHEMATICA, it can still compete.

Of course, MATHEMATICA is another option:

https://www.wolfram.com/language/11/symbolic-and-numeric-calculus/solve-a-volterra-integral-equation.html?product=mathematica

[Michal]

@Torsten NAG Library ... too far from MATLAB implementation

[Torsten]

https://www.nag.com/content/nag-toolbox-matlab

[Michal]

@Torsten NAG has only one function for Volterra Integral Equation (1st kind), which is based on very specific weakly-singular kernel (Abel) type.

[Torsten]

I don't have experience with integral equations, but aren't there more listed here:

https://www.nag.com/numeric/mb/nagdoc_mb/manual_25_1/html/d05/d05conts.html

?

[John D'Errico]

I'm also not an expert at all on the subject of first kind integral equations, having not touched them for nearly 40 years. But my memory does include the words ill-posed. In turn, that suggests finding an off the shelf solver that will be robust, efficient, etc., will be a difficult thing to hope for. I would contact someone with some expertise in the area.