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Noninvasive prenatal testing (NIPT) consists of determining fetal aneuploidies by quantifying copy number alteration from the sequencing of cell-free DNA (cfDNA) from maternal blood. Due to the presence of cfDNA of fetal origin in maternal blood, in silico approaches have been developed to accurately predict fetal aneuploidies. Although NIPT is becoming a new standard in prenatal screening of chromosomal abnormalities, there are no integrated pipelines available to allow rapid, accurate and standardized data analysis in any clinical setting. Several tools have been developed, however often optimized only for research purposes or requiring enormous amount of retrospective data, making hard their implementation in a clinical context. Furthermore, no guidelines have been provided on how to accomplish each step of the data analysis to achieve reliable results. Finally, there is no integrated pipeline to perform all steps of NIPT analysis. To address these needs, we tested several tools for performing NIPT data analysis. We provide extensive benchmark of tools performances but also guidelines for running them. We selected the best performing tools that we benchmarked and gathered them in a computational pipeline. NiPTUNE is an open source python package that includes methods for fetal fraction estimation, a novel method for accurate gender prediction, a principal component analysis based strategy for quality control and fetal aneuploidies prediction. NiPTUNE is constituted by seven modules allowing the user to run the entire pipeline or each module independently. Using two cohorts composed by 1439 samples with 31 confirmed aneuploidies, we demonstrated that NiPTUNE is a valuable resource for NIPT analysis. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analogue of it. The Crank–Nicolson method is a well-known method of order |$2$| but is fully implicit and one may prefer a linearly implicit method like the relaxation method introduced in Besse (1998, Analyse numérique des systèmes de Davey-Stewartson. Ph.D. Thesis , Université Bordeaux) for the cubic nonlinear Schrödinger equation. This method is also an energy-preserving method and numerical simulations have shown that its order is |$2$|⁠. In this paper we give a rigorous proof of the order of this relaxation method and propose a generalized version that allows one to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods. [ABSTRACT FROM AUTHOR]

We present a time-implicit hybridizable discontinuous Galerkin (HDG) method for numerically solving the system of three-dimensional (3D) time-domain Maxwell equations. This method can be seen as a fully implicit variant of classical so-called DGTD (Discontinuous Galerkin Time-Domain) methods that have been extensively studied during the last 10 years for the simulation of time-domain electromagnetic wave propagation. The proposed method has been implemented for dealing with general 3D problems discretized using unstructured tetrahedral meshes. We provide numerical results aiming at assessing its numerical convergence properties by considering a model problem on one hand, and its performance when applied to more realistic problems. We also include some performance comparisons with a centered flux time-implicit DGTD method. [ABSTRACT FROM AUTHOR]

We are concerned here with the numerical simulation of electromagnetic wave propagation in biological media. Because of their water content, these media are dispersive, i.e., their electromagnetic material characteristics depend on the frequency. In the time domain, this translates to a time dependency of these parameters that can be taken into account through an additional (auxiliary) differential equation for, e.g., the electric polarization, which is coupled to the system of Maxwell's equations. From the application point of view, the problems at hand most often involve irregularly shaped structures corresponding to biological tissues. Modeling realistically the interfaces between tissues is particularly important if one is interested in evaluating accurately the impact of field discontinuities at these interfaces. In this paper, we propose and study a locally implicit discontinuous Galerkin time domain method formulated on an unstructured tetrahedral mesh for solving the resulting system of differential equations in the case of Debye-type media. Three-dimensional numerical simulations are presented concerning the exposure of head tissues to a localized source of radiation. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new computational strategy for the simulation of combustion fronts based on adaptive time operator splitting and spatial multiresolution. High-order and dedicated one-step solvers compose the splitting scheme for the reaction, diffusion, and convection subproblems, to independently cope with their inherent numerical difficulties and to properly solve the corresponding temporal scales. Adaptive and thus highly compressed spatial representations for localized fronts originating from multiresolution analysis result in important reductions of memory usage, and hence numerical simulations with sufficiently fine spatial resolution can be performed with standard computational resources. The computational efficiency is further enhanced by splitting time steps established beyond standard stability constraints associated to mesh size or stiff source time scales. The splitting time steps are chosen according to a dynamic splitting technique relying on solid mathematical foundations, which ensures error control of the time integration and successfully discriminates time-varying multi-scale physics. For a given semi-discretized problem, the solution scheme provides dynamic accuracy estimates that reflect the quality of numerical results in terms of numerical errors of integration and compressed spatial representations, for general multi-dimensional problems modeled by stiff PDEs. The strategy is efficiently applied to simulate the propagation of laminar premixed flames interacting with vortex structures, as well as various configurations of self-ignition processes of diffusion flames in similar vortical hydrodynamics fields. A detailed study of the error control is provided and show the potential of the approach. It yields large gains in CPU time, while consistently describing a broad spectrum of space and time scales as well as different physical scenarios. [Copyright &y& Elsevier]

In the present work, the error behaviour of the first-order Lie–Trotter splitting method for nonlinear evolutionary problems is analysed. In particular, a local error representation that is suitable in the presence of unbounded nonlinear operators and critical parameters is deduced. Such local error expansions together with stability bounds are the basic ingredients in the derivation of convergence estimates. Essential tools in the theoretical analysis are an abstract formulation of differential equations on function spaces and the formal calculus of Lie derivatives. In order to illustrate the general approach, the application of the Lie–Trotter splitting method to Schrödinger equations in the semiclassical regime is studied. From numerical computations presented in the literature, it is expected that exponential operator splitting methods are favourable for the time integration of nonlinear Schrödinger equations, provided that the time-step size is suitably chosen depending on the magnitude of the critical parameter. For the least technical example method, the first-order Lie–Trotter splitting method, this is substantiated by theoretical considerations for the time-dependent Gross–Pitaevskii equation and confirmed by numerical examples. Numerical illustrations for higher-order exponential operator splitting methods complement the considerations. [ABSTRACT FROM PUBLISHER]

We tackle the numerical simulation of reaction-diffusion equations modeling multi- scale reaction waves. This type of problem induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. The paper considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is restricted by neither the fastest scales in the source term nor by stability constraints of the diffusive steps but only by the physics of the phenomenon. We aim thus at solving complete models including all time and space scales within a prescribed accuracy, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for two-dimensional spiral waves and three-dimensional scroll waves, as an illustration. Future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are discussed. [ABSTRACT FROM AUTHOR]

Abstract: We discuss the difficulties of the numerical simulation of a stroke, and we describe the numerical methods which we have developed and used to obtain some realistic results. Nowadays, the computations are performed in two-dimensional slices of a brain, but the strategies to obtain full three-dimensional simulations are explored. This paper is written so as to be understandable by non-mathematicians. [Copyright &y& Elsevier]

In this paper we study the approximation by splitting techniques of the ordinary differential equation U+A U+B U=0, U(0)=U0 with A and B two matrices. We assume that we have a stiff problem in the sense that A is ill-conditionned and U0 is a vector which is the discretization of a function with a very high derivative. This situation may appear for example when we study the discretization of a partial differential equation. We prove some error estimates for two general matrices and in the stiff case, where the estimates are independent of U0 and the commutator between A and B. [ABSTRACT FROM AUTHOR]

Considers a nonlinear Schrodinger equation in two dimensions. Proof of the Gronwall lemma; Use of splitting methods; Study on the local error between Z[sup t] and S[sup t]; Definition of Schrodinger.

Let E(t)=exp(−t(A+B)) and let W(t) be the Strang approximation to E(t): W(t)=exp(−tA/2)exp(−tB)exp(−tA/2). In this article, we give a formal Taylor expansion with remainder for W(t), where the derivation operator is replaced by the bracket with one of the operators A or B. We validate this expansion in several functional cases where A and B generate a holomorphic semi-group. They include the case of the Kacˇ transfer operator, and the case A=−MΔ with M a non-necessarily diagonal matrix with spectrum included in {R z>0} and B the multiplication by a spatially dependent matrix. We infer stability estimates and estimates on ∥E(t)−W(t/n)n∥ when n tends to infinity. [Copyright &y& Elsevier]

We investigate the initial-value problem for one-dimensional compressible fluids endowed with internal capillarity. We focus on the isothermal inviscid case with variable capillarity. The resulting equations for the density and the velocity, consisting of the mass conservation law and the momentum conservation with Korteweg stress, are a system of third order nonlinear dispersive partial differential equations. Additionally, this system is Hamiltonian and admits travelling solutions, representing propagating phase boundaries with internal structure. By change of unknown, it roughly reduces to a quasi- linear Schrödinger equation. This new formulation enables us to prove local well-posedness for smooth perturbations of travelling profiles and almost-global existence for small enough perturbations. A blow-up criterion is also derived. [ABSTRACT FROM AUTHOR]