An algebraic operator approach to the analysis of Gerber-Shiu functions

We introduce an algebraic operator framework to study discounted penalty functions in renewal risk models. For inter-arrival and claim size distributions with rational Laplace transform, the usual integral equation is transformed into a boundary value problem, which is solved by symbolic techniques. The factorization of the differential operator can be lifted to the level of boundary value problems, amounting to iteratively solving first-order problems. This leads to an explicit expression for the Gerber-Shiu function in terms of the penalty function.

Segmentation and grid generation for numerical simulations of vad connections with patient-specific data

Objectives: We are interested in the numerical simulation of the anastomotic region comprised between outflow canula of LVAD and the aorta. Segmenta¬tion, geometry reconstruction and grid generation from patient-specific data remain an issue because of the variable quality of DICOM images, in particular CT-scan (e.g. metallic noise of the device, non-aortic contrast phase). We pro¬pose a general framework to overcome this problem and create suitable grids for numerical simulations.Methods: Preliminary treatment of images is performed by reducing the level window and enhancing the contrast of the greyscale image using contrast-limited adaptive histogram equalization. A gradient anisotropic diffusion filter is applied to reduce the noise. Then, watershed segmentation algorithms and mathematical morphology filters allow reconstructing the patient geometry. This is done using the InsightToolKit library (www.itk.org). Finally the Vascular Model¬ing ToolKit (www.vmtk.org) and gmsh (www.geuz.org/gmsh) are used to create the meshes for the fluid (blood) and structure (arterial wall, outflow canula) and to a priori identify the boundary layers. The method is tested on five different patients with left ventricular assistance and who underwent a CT-scan exam.Results: This method produced good results in four patients. The anastomosis area is recovered and the generated grids are suitable for numerical simulations. In one patient the method failed to produce a good segmentation because of the small dimension of the aortic arch with respect to the image resolution.Conclusions: The described framework allows the use of data that could not be otherwise segmented by standard automatic segmentation tools. In particular the computational grids that have been generated are suitable for simulations that take into account fluid-structure interactions. Finally the presented method features a good reproducibility and fast application.

Quantifying rates of landscape evolution and tectonic processes by thermochronology and numerical modeling of crustal heat transport using PECUBE

PECUBE is a three-dimensional thermal-kinematic code capable of solving the heat production-diffusion-advection equation under a temporally varying surface boundary condition. It was initially developed to assess the effects of time-varying surface topography (relief) on low-temperature thermochronological datasets. Thermochronometric ages are predicted by tracking the time-temperature histories of rock-particles ending up at the surface and by combining these with various age-prediction models. In the decade since its inception, the PECUBE code has been under continuous development as its use became wider and addressed different tectonic-geomorphic problems. This paper describes several major recent improvements in the code, including its integration with an inverse-modeling package based on the Neighborhood Algorithm, the incorporation of fault-controlled kinematics, several different ways to address topographic and drainage change through time, the ability to predict subsurface (tunnel or borehole) data, prediction of detrital thermochronology data and a method to compare these with observations, and the coupling with landscape-evolution (or surface-process) models. Each new development is described together with one or several applications, so that the reader and potential user can clearly assess and make use of the capabilities of PECUBE. We end with describing some developments that are currently underway or should take place in the foreseeable future. (C) 2012 Elsevier B.V. All rights reserved.

Error estimate and control in the MSFV method for multiphase flow in porous media

n this paper the iterative MSFV method is extended to include the sequential implicit simulation of time dependent problems involving the solution of a system of pressure-saturation equations. To control numerical errors in simulation results, an error estimate, based on the residual of the MSFV approximate pressure field, is introduced. In the initial time steps in simulation iterations are employed until a specified accuracy in pressure is achieved. This initial solution is then used to improve the localization assumption at later time steps. Additional iterations in pressure solution are employed only when the pressure residual becomes larger than a specified threshold value. Efficiency of the strategy and the error control criteria are numerically investigated. This paper also shows that it is possible to derive an a-priori estimate and control based on the allowed pressure-equation residual to guarantee the desired accuracy in saturation calculation.

Standardisation of 18F by a coincidence method using full solid angle detectors.

A solution of (18)F was standardised with a 4pibeta-4pigamma coincidence counting system in which the beta detector is a one-inch diameter cylindrical UPS89 plastic scintillator, positioned at the bottom of a well-type 5''x5'' NaI(Tl) gamma-ray detector. Almost full detection efficiency-which was varied downwards electronically-was achieved in the beta-channel. Aliquots of this (18)F solution were also measured using 4pigamma NaI(Tl) integral counting and Monte Carlo calculated efficiencies as well as the CIEMAT-NIST method. Secondary measurements of the same solution were also performed with an IG11 ionisation chamber whose equivalent activity is traceable to the Système International de Référence through the contribution IRA-METAS made to it in 2001; IRA's degree of equivalence was found to be close to the key comparison reference value (KCRV). The (18)F activity predicted by this coincidence system agrees closely with the ionisation chamber measurement and is compatible within one standard deviation of the other primary measurements. This work demonstrates that our new coincidence system can standardise short-lived radionuclides used in nuclear medicine.

A pseudo-spectral method for the simulation of poro-elastic seismic wave propagation in 2D polar coordinates using domain decomposition

We present a novel numerical approach for the comprehensive, flexible, and accurate simulation of poro-elastic wave propagation in 2D polar coordinates. An important application of this method and its extensions will be the modeling of complex seismic wave phenomena in fluid-filled boreholes, which represents a major, and as of yet largely unresolved, computational problem in exploration geophysics. In view of this, we consider a numerical mesh, which can be arbitrarily heterogeneous, consisting of two or more concentric rings representing the fluid in the center and the surrounding porous medium. The spatial discretization is based on a Chebyshev expansion in the radial direction and a Fourier expansion in the azimuthal direction and a Runge-Kutta integration scheme for the time evolution. A domain decomposition method is used to match the fluid-solid boundary conditions based on the method of characteristics. This multi-domain approach allows for significant reductions of the number of grid points in the azimuthal direction for the inner grid domain and thus for corresponding increases of the time step and enhancements of computational efficiency. The viability and accuracy of the proposed method has been rigorously tested and verified through comparisons with analytical solutions as well as with the results obtained with a corresponding, previously published, and independently bench-marked solution for 2D Cartesian coordinates. Finally, the proposed numerical solution also satisfies the reciprocity theorem, which indicates that the inherent singularity associated with the origin of the polar coordinate system is adequately handled.

A pseudo-spectral method for simulating of poro-elastic seismic wave propagation in complex borehole environments

We present a novel numerical approach for the comprehensive, flexible, and accurate simulation of poro-elastic wave propagation in cylindrical coordinates. An important application of this method is the modeling of complex seismic wave phenomena in fluid-filled boreholes, which represents a major, and as of yet largely unresolved, computational problem in exploration geophysics. In view of this, we consider a numerical mesh consisting of three concentric domains representing the borehole fluid in the center, the borehole casing and the surrounding porous formation. The spatial discretization is based on a Chebyshev expansion in the radial direction, Fourier expansions in the other directions, and a Runge-Kutta integration scheme for the time evolution. A domain decomposition method based on the method of characteristics is used to match the boundary conditions at the fluid/porous-solid and porous-solid/porous-solid interfaces. The viability and accuracy of the proposed method has been tested and verified in 2D polar coordinates through comparisons with analytical solutions as well as with the results obtained with a corresponding, previously published, and independently benchmarked solution for 2D Cartesian coordinates. The proposed numerical solution also satisfies the reciprocity theorem, which indicates that the inherent singularity associated with the origin of the polar coordinate system is handled adequately.

Accurate and efficient simulation of multiphase flow in a heterogeneous reservoir by using error estimate and control in the multiscale finite-volume framework

The multiscale finite-volume (MSFV) method is designed to reduce the computational cost of elliptic and parabolic problems with highly heterogeneous anisotropic coefficients. The reduction is achieved by splitting the original global problem into a set of local problems (with approximate local boundary conditions) coupled by a coarse global problem. It has been shown recently that the numerical errors in MSFV results can be reduced systematically with an iterative procedure that provides a conservative velocity field after any iteration step. The iterative MSFV (i-MSFV) method can be obtained with an improved (smoothed) multiscale solution to enhance the localization conditions, with a Krylov subspace method [e.g., the generalized-minimal-residual (GMRES) algorithm] preconditioned by the MSFV system, or with a combination of both. In a multiphase-flow system, a balance between accuracy and computational efficiency should be achieved by finding a minimum number of i-MSFV iterations (on pressure), which is necessary to achieve the desired accuracy in the saturation solution. In this work, we extend the i-MSFV method to sequential implicit simulation of time-dependent problems. To control the error of the coupled saturation/pressure system, we analyze the transport error caused by an approximate velocity field. We then propose an error-control strategy on the basis of the residual of the pressure equation. At the beginning of simulation, the pressure solution is iterated until a specified accuracy is achieved. To minimize the number of iterations in a multiphase-flow problem, the solution at the previous timestep is used to improve the localization assumption at the current timestep. Additional iterations are used only when the residual becomes larger than a specified threshold value. Numerical results show that only a few iterations on average are necessary to improve the MSFV results significantly, even for very challenging problems. Therefore, the proposed adaptive strategy yields efficient and accurate simulation of multiphase flow in heterogeneous porous media.

Vessel centerline tracking and boundary sgementation in coronary MRA with minimal manual interaction.

Magnetic resonance angiography (MRA) provides a noninvasive means to detect the presence, location and severity of atherosclerosis throughout the vascular system. In such studies, and especially those in the coronary arteries, the vessel luminal area is typically measured at multiple cross-sectional locations along the course of the artery. The advent of fast volumetric imaging techniques covering proximal to mid segments of coronary arteries necessitates automatic analysis tools requiring minimal manual interactions to robustly measure cross-sectional area along the three-dimensional track of the arteries in under-sampled and non-isotropic datasets. In this work, we present a modular approach based on level set methods to track the vessel centerline, segment the vessel boundaries, and measure transversal area using two user-selected endpoints in each coronary of interest. Arterial area and vessel length are measured using our method and compared to the standard Soap-Bubble reformatting and analysis tool in in-vivo non-contrast enhanced coronary MRA images.

Use of anisotropic modelling in electrical impedance tomography: description of method and preliminary assessment of utility in imaging brain function in the adult human head.

Electrical Impedance Tomography (EIT) is an imaging method which enables a volume conductivity map of a subject to be produced from multiple impedance measurements. It has the potential to become a portable non-invasive imaging technique of particular use in imaging brain function. Accurate numerical forward models may be used to improve image reconstruction but, until now, have employed an assumption of isotropic tissue conductivity. This may be expected to introduce inaccuracy, as body tissues, especially those such as white matter and the skull in head imaging, are highly anisotropic. The purpose of this study was, for the first time, to develop a method for incorporating anisotropy in a forward numerical model for EIT of the head and assess the resulting improvement in image quality in the case of linear reconstruction of one example of the human head. A realistic Finite Element Model (FEM) of an adult human head with segments for the scalp, skull, CSF, and brain was produced from a structural MRI. Anisotropy of the brain was estimated from a diffusion tensor-MRI of the same subject and anisotropy of the skull was approximated from the structural information. A method for incorporation of anisotropy in the forward model and its use in image reconstruction was produced. The improvement in reconstructed image quality was assessed in computer simulation by producing forward data, and then linear reconstruction using a sensitivity matrix approach. The mean boundary data difference between anisotropic and isotropic forward models for a reference conductivity was 50%. Use of the correct anisotropic FEM in image reconstruction, as opposed to an isotropic one, corrected an error of 24 mm in imaging a 10% conductivity decrease located in the hippocampus, improved localisation for conductivity changes deep in the brain and due to epilepsy by 4-17 mm, and, overall, led to a substantial improvement on image quality. This suggests that incorporation of anisotropy in numerical models used for image reconstruction is likely to improve EIT image quality.

Analysis of a finite volume method for a cross-diffusion model in population dynamics

The main goal of this paper is to propose a convergent finite volume method for a reactionâeuro"diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then the standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time $L^1$ compactness argument that mimics the compactness lemma due to Kruzhkov. The proofs of these results are given in the Appendix.