A new derivative-free method is developed for solving unconstrained nonsmooth optimization problems. This method is based on the notion of a discrete gradient. It is demonstrated that the discrete gradients can be used to approximate subgradients of a broad class of nonsmooth functions. It is also shown that the discrete gradients can be applied to find descent directions of nonsmooth functions. The preliminary results of numerical experiments with unconstrained nonsmooth optimization problems as well as the comparison of the proposed method with nonsmooth optimization solver DNLP from CONOPT-GAMS and derivative-free optimization solver CONDOR are presented.

Novel digital analysis strategies are developed for the quantification of changes in the cytoskeletal and nuclear morphologies of mesenchymal stem cells cultured on micropillars. Severe deformations of nucleus and distinct conformational changes of cell body ranging from extensive elongation to branching are visualized and quantified. These deformations are caused mainly by the dimensions and hydrophilicity of the micropillars.

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An overview of Hamiltonian systems with noncanonical Poisson structures is given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are presented. Numerical integrators using generating functions, Hamiltonian splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems and Hamiltonian systems with a general Poisson structure. Nambu-Poisson systems and the discrete gradient methods are also presented.

Activator-inhibitor FitzHugh-Nagumo (FHN) equation is an example for reactiondiffusion equations with skew-gradient structure. We discretize the FHN equation using symmetric interior penalty discontinuous Galerkin (SIPG) method in space and average vector field (AVF) method in time. The AVF method is a geometric integrator, i.e. it preserves the energy of the Hamiltonian systems and energy dissipation of the gradient systems. In this work, we show that the fully discrete energy of the FHN equation satisfies the minimaximizer property of the continuous energy for the skew-gradient systems. We present numerical results with traveling fronts and pulses for one dimensional, two coupled FHN equations and three coupled FHN equations with one activator and two inhibitors in skewgradient form. Turing patterns are computed for fully discretized two dimensional FHN equation in the form of spots and labyrinths. Because the computation of the Turing patterns is time consuming for different parameters, we applied model order reduction with the proper orthogonal decomposition (POD). The nonlinear term in the reduced equations is computed using the discrete empirical interpolation (DEIM) with SIPG discretization. Due to the local nature of the discontinuous Galerkin (DG) method, the nonlinear terms can be computed more efficiently than for the continuous finite elements. The reduced solutions are very close to the fully discretized ones. The efficiency and accuracy of the POD and POD-DEIM reduced solutions are shown for the labyrinth-like patterns.