## Computational Mathematics

**Numerical methods for delay differential equations**

In dynamical systems described by delay differential equations, often called Time Delay Systems, the rate of change of the state at a given time t depends on the hystory of the state previous to t, not only on the state at t as in case of ordinary differential equations. Models based on TDS are used in many areas as engineering, control theory, population dynamics, etc, since they are more adherent to the reality than models based on ODEs: For example, TDS models can explain observed real oscillation in situations where ODE models fails to do this. The research topic concerns the development and analysis of numerical methods for TDS, both for simulation and stability/bifurcation analysis. There are several types of TDS, for example state-dependent TDSs, neutral TDs, infinite delay TDSs, vanishing delay TDSs, discontinuos TDEs or stochastic TDSs, for which the numerical analysis is only at a seminal stage. Moreover, also the numerical computation of periodic solutions or of attractors is a subject that has to be well developed and studied.

**Exponential integrators for peridynamic theory**

The peridynamic theory is a nonlocal formulation of the elasticity theory that try to overcome some drawbacks of the classical local theory in the nonlinear case. In this new theory, the deformation satisfies a wave equation where the second local spatial derivate term is replaced by a nonlinear integral term over the whole domain. The research topic concerns the development and analysis of particular numerical integrators, known in literature as exponential integrators, which solve exactly the linear part and approximately the nonlinear integral part.

**Condition numbers for evolution equations**

It is of interest to study how a perturbation in the initial value is propagated along the solution of evolution equations, like Ordinary Differential Equations, Partial Differential Equations or Delay Differential Equations. This is a well-known subject if the perturbation is measured by an absolute errors. But it is also of intereste to study the propagation of the perturbation by using relative errors, rather than absolute errors. It is worthwhile to observe that relative errors describe uncertainties better than the absolute errors, since they are dimensionless and they can give information about the number of significant figures that are known in the numerical data. When an evolution equation is solved, the computed solution obtained is different from the exact solution, since there are uncertainties on the initial value, there are uncertiaintes on the equation parameters and there are errors introduced by the numerical integration. The reaserch topic concerns the relative error analysis of the computed solution with respect to the exact solution. In particular, it is required to define and study what are called the condition numbers, namely the numbers describing the magnification of initial uncertainties along the solution. New and surprising results about the numerical methods are expected.

**Krylov projection method for linear and nonlinear inverse problems**

In the framework of linear ill-posed problems, Krylov projection methods represent an essential tool since their development, which dates back to the early 50's. In recent years, the use of these methods in a hybrid fashion or to solve regularized problems has received a great attention especially for large-scale problems arising in imaging and more generally for problems arising from the discretization of equations involving compact operators. For these kind of problems many Krylov type methods are generally very fast, since they exhibit a convergence rate which is quite close to the decay rate of the singular values of the operator. Moreover, the projective nature of these methods makes them attractive for solving regularized problems, since the projected operators inherits the spectral properties of the underlying one. As consequence, many existing parameter selection strategies can be efficiently employed, without additional computational effort. Important open questions concerns the theoretical properties of Krylov methods when coupled with a Tikhonov-like regularization, even if they are already used to this purpose. Moreover, the use of these methods as a tool for solving nonlinear inverse problems still needs a systematic theoretical and computational analysis.

**Numerical approximation of functions of unbounded operator**

Many numerical methods for solving differential equations of various type, of integer or fractional order, often require the computation of function of unbounded operators after a suitable discretization. Examples include exponential integrators for semilinear IVPs, Caputo's differential equation, anomalous diffusion equations. In this direction the computation of matrix functions related with the matrix exponential and the matrix fractional power plays a crucial role. Nowadays the research is very active in this field, especially for what concern the approximation of these matrix functions through rational forms. In case one is only interested to compute the action of a matrix function, rational Krylov methods constitute an important tool, and, depending on the underlying function, research arguments may include the applicability, the choice of the poles and the subsequent error analysis. In the more general case, rational approximations may arise from the integral representation of the matrix function and the use of suitable quatrature rules.