Numerical algorithms for spline interpolation on space of probability density functions. (English) Zbl 1524.49032
Summary: This paper addresses the problem of spline interpolations on \(\mathcal{P}\), the space of probability density functions when only a few observations \(p_i\in\mathcal{P}\) are available. Given a finite set of \(n+1\) distinct time instants \(t_i\) and corresponding data points \(p_i\in\mathcal{P}\), we consider the general problem of estimating a spline as a special regularized function \(\gamma\) on \(\mathcal{P}\) with \(\gamma(t_i) = p_i\). In particular, we focus on estimating missing data using smooth temporal splines to overcome the discrete nature of observations. In addition to generalizing splines on \(\mathcal{P}\) with minimal squared-norm of the acceleration, we give numerical schemes for solving \(C^1\) and \(C^2\) splines from data points \(p_i\in\mathcal{P}\). The two solutions are then shown to be computationally efficient, geometrically simpler, extensible, and can be transposed to other spaces and applications.
MSC:
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 65D10 | Numerical smoothing, curve fitting |
| 49K30 | Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
| 41A15 | Spline approximation |
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