This paper is a follow-up of the work [Chen, J.-S.: J. Optimiz. Theory Appl., Submitted for publication (2004)] where an NCP-function and a descent method were proposed for the nonlinear complementarity problem. An unconstrained reformulation was formulated due to a merit function based on the proposed NCP-function. We continue to explore properties of the merit function in this paper. In particular, we show that the gradient of the merit function is globally Lipschitz continuous which is important from computational aspect. Moreover, we show that the merit function is <i>SC</i> ¹ function which means it is continuously differentiable and its gradient is semismooth. On the other hand, we provide an alternative proof, which uses the new properties of the merit function, for the convergence result of the descent method considered in [Chen, J.-S.: J. Optimiz. Theory Appl., Submitted for publication (2004)].

We study the effective fronts of first order front propagations in two dimensions (n=2) in the periodic setting. Using PDE-based approaches, we show that for every α∈(0,1), the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts for given front speeds in C^{1,α}(T^2,(0,∞)). This result can also be formulated in the language of stable norms corresponding to periodic metrics in T^2. Similar results were known long time ago when n≥3 for front speeds in C^∞(T^n,(0,∞)). Due to topological restrictions, the two dimensional case is much more subtle. In fact, the effective front is C^1, which cannot be a polygon, for given C^{1,1}(T^2,(0,∞)) front speeds. Our regularity requirements on front speeds are hence optimal. To the best of our knowledge, this is the first time that polygonal effective fronts have been constructed in two dimensions.

This paper is devoted to the theoretical and numerical investigation of the local minimum problem in an inverse boundary value problem. We provide a mathematical analysis for the objective function of the optimal transportation type in the context of seismic full waveform inversion. In particular, we prove that the gradient obtained using the adjoint-state method does not depend on the specific choice of the Kantorovich potentials. Moreover, our frequency analysis results show a decreasing sensitivity of the reconstruction as the data misfit is concentrated in the high-frequency part. This confirms previous observations in many numerical experiments. We also propose a new method using the softplus encoding, which maps the seismic data into probability measures and, therefore, the Wasserstein metric can be applied. The softplus encoding retains the convexity of the data misfit with respect to translations and provides a parameter to tune the landscape of the objective function. The effectiveness of the proposed method is demonstrated numerically on an inversion task with the benchmark Marmousi model.

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