Scattering problems for periodic structures have been studied a lot in the past few years. A main idea for numerical solution methods is to reduce such problems to one periodicity cell. In contrast to periodic settings, scattering from locally perturbed periodic surfaces is way more challenging. In this paper, we introduce and analyze a new numerical method to simulate scattering from locally perturbed periodic structures based on the Bloch transform. As this transform is applied only in periodic domains, we firstly rewrite the scattering problem artificially in a periodic domain. With the help of the Bloch transform, we secondly transform this problem into a coupled family of quasiperiodic problems posed in the periodicity cell. A numerical scheme then approximates the family of quasiperiodic solutions (we rely on the finite element method) and backtransformation provides the solution to the original scattering problem. In this paper, we give convergence analysis and error bounds for a Galerkin discretization in the spatial and the quasiperiodicity's unit cells. We also provide a simple and efficient way of implementation that does not require numerical integration in the quasiperiodicity, together with numerical examples for scattering from locally perturbed periodic surfaces computed by this scheme.

In this paper, we will introduce a high order numerical method to solve the scattering problems with non-periodic incident fields and (locally perturbed) periodic surfaces. For the problems we are considering, the classical methods to treat quasi-periodic scattering problems no longer work, while a Bloch transform based numerical method was proposed in [LZ17b]. This numerical method, on one hand, is able to solve this kind of problems convergently; on the other hand, it takes up a lot of time and memory during the computation. The motivation of this paper is to improve this numerical method, from the regularity results of the Bloch transform of the total field, which have been studied in [Zha17]. As the set of the singularities of the total field is discrete in R, and finite in one periodic cell, we are able to improve the numerical method by designing a proper integration contour with special conditions at the singularities. With a good choice of the transformation, we can prove that the new numerical method could possess a super algebraic convergence rate. This new method improves the efficient significantly. At the end of this paper, several numerical results will be provided to show the fast convergence of the new method. The method also provides a possibility to solve more complicated problems efficiently, e.g., three dimensional problems, or electromagnetic scattering problems.

Recently, a new eigenvalue problem, called the transmission eigenvalue problem, has attracted many researchers. The problem arose in inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing. The problem is numerically challenging because it is non-selfadjoint and nonlinear. In this paper, we propose a recursive integral method for computing transmission eigenvalues from a finite element discretization of the continuous problem. The method, which overcomes some difficulties of existing methods, is based on eigenprojectors of compact operators. It is self-correcting, can separate nearby eigenvalues, and does not require an initial approximation based on some a priori spectral information. These features make the method well suited for the transmission eigenvalue problem whose spectrum is complicated. Numerical examples show that the method is effective and robust.

Periodic surface structures are nowadays standard building blocks of optical devices. If such structures are illuminated by aperiodic time-harmonic incident waves as, e.g., Gaussian beams, the resulting surface scattering problem must be formulated in an unbounded layer including the periodic surface structure. An obvious recipe to avoid the need to discretize this problem in an unbounded domain is to set up an equivalent system of quasiperiodic scattering problems in a single (bounded) periodicity cell via the Floquet-Bloch transform. The solution to the original surface scattering problem then equals the inverse Floquet-Bloch transform applied to the family of solutions to the quasiperiodic problems, which simply requires to integrate these solutions in the quasiperiodicity parameter. A numerical scheme derived from this representation hence completely avoids the need to tackle differential equations on unbounded domains. In this paper, we provide rigorous convergence analysis and error bounds for such a scheme when applied to a two-dimensional model problem, relying upon a quadrature-based approximation to the inverse Floquet-Bloch transform and finite element approximations to quasiperiodic scattering problems. Our analysis essentially relies upon regularity results for the family of solutions to the quasiperiodic scattering problems in suitable mixed Sobolev spaces. We illustrate our error bounds as well as efficiency of the numerical scheme via several numerical examples.

Dermatofibrosarcoma protuberans (DFSP) is a rare, plaque-like tumor of the cutaneous tissue occurring more on the trunk than the extremities and neck. More than 95% of DFSP present anomalies on the 17q22 and 22q13 chromosomal regions leading to the fusion of COL1A1 and PDGFB genes. Surgery is the optimal treatment for DFSP, but less effective in locally advanced or metastatic patients, as is the case with chemotherapy and radiotherapy. The aim of this study was to assess retrospectively the therapeutic activity and safety of imatinib on 22 Chinese patients with locally inoperative or metastatic DFSP at a single institution.In the collected data of 367 Chinese patients with DFSP, we analyzed retrospectively 22 patients with locally advanced or metastatic DFSP, all of whom received imatinib therapy at 1 center from January 2009 to October 2014. Patients were administered with imatinib at an initial dose of 400 mg and escalated to 800 mg daily after they developed imatinib resistance. The median follow-up time was 36 months, and the median treatment time was 15 months.The results showed that 10 locally advanced DFSP patients and 12 metastatic DFSP patients received imatinib therapy. Apart from 1 patient who developed primary imatinib resistance, 15 patients achieved partial remission (PR), and 6 patients achieved stable disease (SD). Both fibrosarcomatous DFSP and classic DFSP patients demonstrated similar response to imatinib. Median PFS was estimated to be 19 months. Median overall survival (OS) has not been reached, and estimated 1- and 3-year OS rates were 95.5% (21/22) and 77.3% (17/22), respectively. Four out of 10 patients with primarily unresectable DFSP received complete surgical resection after neoadjuvant treatment of imatinib.Imatinib therapy is well tolerated with a safety profile and is the therapy of choice in locally inoperative or metastatic DFSP. Neoadjuvant treatment of locally advanced or metastatic DFSP with imatinib improves surgical outcomes and may facilitate resection of difficult tumors.