Intrinsic curvature flows can be used to design Riemannian metrics by prescribed curvatures. This chapter presents three discrete curvature flow methods that are recently introduced into the engineering fields: the discrete Ricci flow and discrete Yamabe flow for surfaces with various topology, and the discrete curvature flow for hyperbolic 3-manifolds with boundaries. For each flow, we introduce its theories in both the smooth setting and the discrete setting, plus the numerical algorithms to compute it. We also provide a brief survey on their history and their link to some of the engineering applications in computer graphics, computer vision, medical imaging, computer aided design and others.

We study a time domain decorrelation method of source signal separation from convolutive sound mixtures based on an infinite impulse response (IIR) model. The IIR model uses fewer parameters to capture the physical mixing process and is useful for finding low dimensional separating solutions. We present inversion formulas to decorrelate the mixture signals and derive filter equations involving second order time lagged statistics of mixtures. We then formulate an l constrained minimization problem and solve it by an iterative method. Numerical experiments on recorded sound mixtures show that our method is capable of sound separation in low dimensional parameter spaces with good perceptual quality and low correlation coefficient comparable to the known infomax method.

Wave propagation problems arise in a wide range of applications. The energy conserving property is one of the guiding principles for numerical algorithms, in order to minimize the phase or shape errors after long time integration. In this paper, we develop and analyze a local discontinuous Galerkin (LDG) method for solving the wave equation. We prove optimal error estimates, superconvergence toward a particular projection of the exact solution, and the energy conserving property for the semi-discrete formulation. The analysis is extended to the fully discrete LDG scheme, with the centered second-order time discretization (the leap-frog scheme). Our numerical experiments demonstrate optimal rates of convergence and superconvergence. We also show that the shape of the solution, after long time integration, is well preserved due to the energy conserving property.

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