We present two methods for the implicit integration of nonlinear stiff systems. Direct application of the Newton method to backward Euler discretization of such systems may diverge. We observe that the solution is recovered by smoothing out certain eigenvalues in the Jacobian matrix. To this end, we introduce a solution-dependent matrix-weighted combination of backward and forward Euler methods. The weight is tuned on each Newton iteration to reproduce the solution with an exponential integrator, whereby a weight function for smoothing eigenvalues is obtained. We apply the proposed techniques, namely quasi-Newton backward Euler and matrix-weighted Euler, to several stiff systems, including Lotka–Volterra, Van der Pol’s, and a blood coagulation cascade.

In this work, we develop numerical methods for the solution of blood flow and coagulation on dynamic adaptive moving meshes. We consider the blood flow as a flow of incompressible Newtonian fluid governed by the Navier–Stokes equations. The blood coagulation is introduced through the additional Darcy term, with a permeability coefficient dependent on reactions. To this end, we introduce moving mesh collocated finite‐volume methods for the Navier–Stokes equations, advection–diffusion equations, and a method for the stiff cascade of reactions. A monolithic nonlinear system is solved to advance the solution in time. The finite volume method for the Navier–Stokes equations features collocated arrangement of pressure and velocity unknowns and a coupled momentum and mass flux. The method is conservative and inf‐sup stable despite the saddle point nature of the system. It is verified on a series of analytical problems and applied to the blood flow problem in the deforming domain of the right ventricle, reconstructed from a time series of computed tomography scans. At last, we demonstrate the ability to model the coagulation process in deforming microfluidic capillaries.

We propose a novel approach to the problem of mutual information (MI) estimation via introducing normalizing flows-based estimator. The estimator maps original data to the target distribution with known closed-form expression for MI. We demonstrate that our approach yields MI estimates for the original data. Experiments with high-dimensional data are provided to show the advantages of the proposed estimator.