High-frequency asymptotic compression of dense BEM matrices for general geometries without ray tracing

Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green's function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green's function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the ...Comment: 24 pages, 13 figure

On the eigenmodes of periodic orbits for multiple scattering problems in 2D

Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the oscillatory properties of the solution. However, the high-frequency wave pattern becomes very complicated in the presence of multiple scattering obstacles. We consider a boundary integral equation formulation of the Helmholtz equation in two dimensions involving several obstacles, for which ray tracing schemes have been previously proposed. The existing analysis of ray tracing schemes focuses on periodic orbits between a subset of the obstacles. One observes that the densities on each of the obstacles converge to an equilibrium after a few iterations. In this paper we present an asymptotic approximation of the phases of those densities in equilibrium, in the form of a Taylor series. The densities represent a full cycle of reflections in a periodic orbit. We initially exploit symmetry in the case of two circular scatterers, but also provide an explicit algorithm for an arbitrary number of general 2D obstacles. The coefficients, as well as the time to compute them, are independent of the wavenumber and of the incident wave. The results may be used to accelerate ray tracing schemes after a small number of initial iterations.Comment: 24 pages, 9 figures and the implementation is available on https://github.com/popsomer/asyBEM/release

Endogenous post-stratification in surveys: classifying with a sample-fitted model

Post-stratification is frequently used to improve the precision of survey estimators when categorical auxiliary information is available from sources outside the survey. In natural resource surveys, such information is often obtained from remote sensing data, classified into categories and displayed as pixel-based maps. These maps may be constructed based on classification models fitted to the sample data. Post-stratification of the sample data based on categories derived from the sample data (``endogenous post-stratification'') violates the standard post-stratification assumptions that observations are classified without error into post-strata, and post-stratum population counts are known. Properties of the endogenous post-stratification estimator are derived for the case of a sample-fitted generalized linear model, from which the post-strata are constructed by dividing the range of the model predictions into predetermined intervals. Design consistency of the endogenous post-stratification estimator is established under mild conditions. Under a superpopulation model, consistency and asymptotic normality of the endogenous post-stratification estimator are established, showing that it has the same asymptotic variance as the traditional post-stratified estimator with fixed strata. Simulation experiments demonstrate that the practical effect of first fitting a model to the survey data before post-stratifying is small, even for relatively small sample sizes.Comment: Published in at http://dx.doi.org/10.1214/009053607000000703 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Postpartum uterine diseases in dairy cows : a review with emphasis on subclinical endometritis

In this review, updated and precise definitions of the most common postpartum uterine diseases in dairy cows are provided. An aberrant uterine environment at inappropriate stages of the reproductive cycle inflicts damage to gametes and zygotes, impairing the reproductive performance of dairy cows. This involves major economic losses for the milk production unit. Consequently, an accurate diagnosis of postpartum uterine diseases is indispensable for practitioners to set up a prompt and efficient treatment. This review furthermore emphasizes on the new perspectives regarding diagnosis and treatment of subclinical endometritis, a highly prevalent uterine disease that is often overlooked by practitioners while causing major reproductive problems. Based on a more profound clinical understanding of the postpartum uterine disease complex, practitioners will be able to better use the available diagnostic tools and therefore apply a more efficient therapeutic approach

Bootstrapping for penalized spline regression.

We describe and contrast several different bootstrapping procedures for penalized spline smoothers. The bootstrapping procedures considered are variations on existing methods, developed under two different probabilistic frameworks. Under the first framework, penalized spline regression is considered an estimation technique to find an unknown smooth function. The smooth function is represented in a high dimensional spline basis, with spline coefficients estimated in a penalized form. Under the second framework, the unknown function is treated as a realization of a set of random spline coefficients, which are then predicted in a linear mixed model. We describe how bootstrapping methods can be implemented under both frameworks, and we show in theory and through simulations and examples that bootstrapping provides valid inference in both cases. We compare the inference obtained under both frameworks, and conclude that the latter generally produces better results than the former. The bootstrapping ideas are extended to hypothesis testing, where parametric components in a model are tested against nonparametric alternatives.Methods; Framework; Regression; Linear mixed model; Mixed model; Model; Theory; Simulation; Hypothesis testing;

Resilience in middle-aged partners of patients diagnosed with incurable cancer : a thematic analysis

Background : Providing care for patients with advanced cancer is often the responsibility of the partner. Being confronted with an incurable cancer diagnosis can be highly disruptive for the patient's partner and can be considered a potentially traumatic event. However, most caregivers seem to adapt well during the process of providing care. This finding is in line with the concept of resilience in literature: a dynamic process of adapting well, resulting from the interplay between intrinsic and extrinsic resources and risks. Resilience is age-related, with the elderly population being higher in resilience as compared to the younger generation. However, resilience has been understudied in middle-aged caregivers. Aim : To explore what intrinsic and extrinsic resources facilitate or hamper resilience in the middle-aged partner of a patient with incurable cancer. Methods : Nine middle-aged partners of patients who died at home of cancer were selected and interviewed in depth within the first year following the death of their partner. A thematic analysis utilizing an inductive approach was conducted. Findings : Resilience was challenged by the partner's diagnosis of incurable cancer. All participants made use of a set of interacting, caregiver-specific and context-related resources, facilitating a resilient process and leading to positive feelings and even personal growth. The partners demonstrated individual competences: adaptive flexibility, positivism, a sense of self-initiative and adaptive dependency. Furthermore, they relied on their context: cancer-related professionals and relatives. Context and situation interact continuously. The resulting dynamics were based on the context-availability, meaningful relationships and the patient's role. Conclusion : A resilient trajectory results from an interplay between individual and contextual resources. To build resilience in middle-aged partners of patients with incurable cancer, health care professionals should address all available resources. Moreover, they should be aware of being part of the caregiver's context, a complex adaptive system that can be either resilience-supporting or -threatening