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Surface conformal parameterizations have been widely applied to various tasks in computer graphics. In this paper, we develop a convergent conformal energy minimization (CCEM) iterative algorithm via the line-search gradient descent method with a quadratic approximation for the computation of disk-shaped conformal parameterizations of simply connected open triangular meshes. In addition, we prove the global convergence of the proposed CCEM iterative algorithm. Moreover, under some mild assumptions, we prove the existence of the nontrivial solution, which is a local minimum of the conformal energy with a bijective boundary map. Numerical experiments indicate that the efficiency of the proposed CCEM algorithm is highly improved and the accuracy is competitive with that of state-of-the-art algorithms.

By constructing an infinite graph-directed iterated iterated function system associated with a finite iterated function system, we develop a new approach for proving the differentiability of the $L^q$-spectrum and establishing the multifractal formalism of certain self-similar measures with overlaps, especially those defined by similitudes with different contraction ratios. We apply our technique to a well-known class of self-similar measures of generalized finite type.

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We consider the problem of minimizing the sum of a smooth function h with a bounded Hessian and a nonsmooth function. We assume that the latter function is a composition of a proper closed function P and a surjective linear map M. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: the alternating direction method of multipliers and the proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions h and P are semialgebraic. Furthermore, we give simple sufficient conditions to guarantee boundedness of the sequence generated. These conditions can be satisfied for a wide range of applications including the least squares problem with the 1/2 regularization. Finally, when M is the identity so that the proximal gradient algorithm can be efficiently applied, we show that any cluster point is stationary under a slightly more flexible constant step-size rule than what is known in the literature for a nonconvex h. We illustrate our theoretical finding with a variety of applications such as signal denoising and sparse optimisation problems.