Eugene Kashdan

A Fast Explicit Operator Splitting Method for Passive Scalar Advection

The dispersal and mixing of scalar quantities such as concentrations or thermal energy are often modeled by advection-diffusion equations. Such problems arise in a wide variety of engineering, ecological and geophysical applications. In these situations a quantity such as chemical or pollutant concentration or temperature variation diffuses while being transported by the governing flow. In the passive scalar case, this flow prescribed and unaffected by the scalar. Both steady laminar and complex (chaotic, turbulent or random) time-dependent flows are of interest and such systems naturally lead to questions about the effectiveness of the stirring to disperse and mix the scalar. The development of reliable numerical methods for advection-diffusion equations is crucial for understanding their properties, both physical and mathematical. In this paper, we extend a fast explicit operator splitting method, recently proposed in (A. Chertock, A. Kurganov, G. Petrova, Int. J. Numer. Methods Fluids 59:309–332, 2009), for solving deterministic convection-diffusion equations, to the problems with random velocity fields and singular source terms. A superb performance of the method is demonstrated on several two-dimensional examples.

High-order accurate modeling of electromagnetic wave propagation across media: grid conforming bodies

The Maxwell equations contain a dielectric permittivity e that describes the particular media. For homogeneous materials at low temperatures this coefficient is constant within a material. However, it jumps at the interface between different media. This discontinuity can significantly reduce the order of accuracy of the numerical scheme. We solve the Maxwell equations, with an interface between two media, using a fourth-order accurate algorithm. We regularize the discontinuous dielectric permittivity by a continuous function either locally, near the interface, or globally, in the entire domain. We study the effect of this regularization on the order of accuracy for a one-dimensional time-dependent problem. We then implement this for the three-dimensional Maxwell equations in spherical coordinates with appropriate physical and artificial absorbing boundary conditions. We use Fourier filtering of the high frequency modes near the poles to increase the time-step.

Mean field mutation dynamics and the continuous Luria-Delbrück distribution.

The Luria-Delbrück mutation model has a long history and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using some mathematical tools from nonlinear statistical physics. Starting from the classical formulations we derive the corresponding differential models and show that under a suitable mean field scaling they correspond to generalized Fokker-Planck equations for the mutants distribution whose solutions are given by the corresponding Luria-Delbrück distribution. Numerical results confirming the theoretical analysis are also presented.

A High-Order Accurate Method for Frequency Domain Maxwell Equations with Discontinuous Coefficients

Maxwell equations contain a dielectric coefficient ɛ that describes the particular media. For homogeneous materials the dielectric coefficient is constant. There is a jump in this coefficient across the interface between differing media. This discontinuity can significantly reduce the order of accuracy of the numerical scheme. We present an analysis and implementation of a fourth order accurate algorithm for the solution of Maxwell equations with an interface between two media and so the dielectric coefficient is discontinuous. We approximate the discontinuous function by a continuous one either locally or in the entire domain. We study the one-dimensional system in frequency space. We only consider schemes that can be implemented for multidimensional problems both in the frequency and time domains.

Modeling and simulation of a low-grade urinary bladder carcinoma

In this work, we present a mathematical model of the initiation and progression of a low-grade urinary bladder carcinoma. We simulate the crucial processes affecting tumor growth, such as oxygen diffusion, carcinogen penetration, and angiogenesis, within the framework of the urothelial cell dynamics. The cell dynamics are modeled using the discrete technique of cellular automata, while the continuous processes of carcinogen penetration and oxygen diffusion are described by nonlinear diffusion-absorption equations. As the availability of oxygen is necessary for tumor progression, processes of oxygen transport to the tumor growth site seem most important. Our model yields a theoretical insight into the main stages of development and growth of urinary bladder carcinoma with emphasis on the two most common types: bladder polyps and carcinoma in situ. Analysis of histological structure of bladder tumor is important to avoid misdiagnosis and wrong treatment. We expect our model to be a valuable tool in the study of bladder cancer progression due to the exposure to carcinogens and the oxygen dependent expression of genes promoting tumor growth. Our numerical simulations have good qualitative agreement with in vivo results reported in the corresponding medical literature.

Hybrid discrete-continuous model of invasive bladder cancer

Bladder cancer is the seventh most common cancer worldwide. Epidemiological studies and experiments implicated chemical penetration into urothelium (epithelial tissue surrounding bladder) in the etiology of bladder cancer. In this work we model invasive bladder cancer. This type of cancer starts in the urothelium and progresses towards surrounding muscles and tissues, causing metastatic disease. Our mathematical model of invasive BC consists of two coupled sub-models: (i) living cycle of the urothelial cells (normal and mutated) simulated using discrete technique of Cellular Automata and (ii) mechanism of tumor invasion described by the system of reaction-diffusion equations. Numerical simulations presented here are in good qualitative agreement with the experimental results and reproduce in vitro observations described in medical literature.

Petascale self-consistent electromagnetic computations using scalable and accurate algorithms for complex structures

As the size and cost of particle accelerators escalate, high-performance computing plays an increasingly important role; optimization through accurate, detailed computermodeling increases performance and reduces costs. But consequently, computer simulations face enormous challenges. Early approximation methods, such as expansions in distance from the design orbit, were unable to supply detailed accurate results, such as in the computation of wake fields in complex cavities. Since the advent of message-passing supercomputers with thousands of processors, earlier approximations are no longer necessary, and it is now possible to compute wake fields, the effects of dampers, and self-consistent dynamics in cavities accurately. In this environment, the focus has shifted towards the development and implementation of algorithms that scale to large numbers of processors. So-called charge-conserving algorithms evolve the electromagnetic fields without the need for any global solves (which are difficult to scale up to many processors). Using cut-cell (or embedded) boundaries, these algorithms can simulate the fields in complex accelerator cavities with curved walls. New implicit algorithms, which are stable for any time-step, conserve charge as well, allowing faster simulation of structures with details small compared to the characteristic wavelength. These algorithmic and computational advances have been implemented in the VORPAL7 Framework, a flexible, object-oriented, massively parallel computational application that allows run-time assembly of algorithms and objects, thus composing an application on the fly.

A model of thermotherapy treatment for bladder cancer

In this work, we investigate chemo- thermotherapy, a recently clinically-approved post-surgery treatment of non muscle invasive urothelial bladder carcinoma. We developed a mathematical model and numerically simulated the physical processes related to this treatment. The model is based on the conductive Maxwells equations used to simulate the therapy administration and Convection-Diffusion equation for incompressible fluid to study heat propagation through the bladder tissue. The model parameters correspond to the data provided by the thermotherapy device manufacturer. We base our computational domain on a CT image of a human bladder. Our numerical simulations can be applied to further research on the effects of chemo- thermotherapy on bladder and surrounding tissues and for treatment personalization in order to maximize the effect of the therapy while avoiding burning of the bladder.

Simulation of electromagnetic scattering with stationary or accelerating targets

The scattering of electromagnetic waves by an obstacle is analyzed through a set of partial differential equations combining the Maxwells model with the mechanics of fluids. Solitary type EM waves, having compact support, may easily be modeled in this context since they turn out to be explicit solutions. From the numerical viewpoint, the interaction of these waves with a material body is examined. Computations are carried out via a parallel high-order finite-differences code. Due to the presence of a gradient of pressure in the model equations, waves hitting the obstacle may impart acceleration to it. Some explicative 2D dynamical configurations are then studied, enabling the simulation of photon-particle iterations through classical arguments.

Mathematical Methods and Models in Biomedicine

Spatial aspects of HIV infection.- Basic Principles in Modeling Adaptive Regulation and Immunodominance.- Evolutionary Principles In Viral Epitopes.- A Multiscale Approach Leading to Hybrid Mathematical Models for Angiogenesis: the Role of Randomness.- Modeling Tumor Blood Vessel Dynamics.- Influence of Blood Rheology and Outflow Boundary Conditions in Numerical Simulations of Cerebral Aneurysms.- The Steady State of Multicellular Tumour Spheroids: a Modelling Challenge.- Deciphering Fate Decision in Normal and Cancer Stem Cells - Mathematical Models and Their Experimental Verification..- Data Assimilation in Brain Tumor Models.- Optimisation of Cancer Drug Treatments Using Cell Population Dynamics.- Tumor Development under Combination Treatments with Antiangiogenic Therapies.- Saturable Fractal Pharmacokinetics and Its Applications.- A MathematicalModel of Gene Therapy for the Treatment of Cancer.- Epidemiological Models with Seasonality.- Periodic Incidence in a Discrete-Time SIS Epidemic Model.