Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D

We present an efficient and novel numerical algorithm for inversion of transforms arising in imaging modalities such as ultrasound imaging, thermoacoustic and photoacoustic tomography, intravascular imaging, non-destructive testing, and radar imaging with circular acquisition geometry. Our algorithm is based on recently discovered explicit inversion formulas for circular and elliptical Radon transforms with radially partial data derived by Ambartsoumian, Gouia-Zarrad, Lewis and by Ambartsoumian and Krishnan. These inversion formulas hold when the support of the function lies on the inside (relevant in ultrasound imaging, thermoacoustic and photoacoustic tomography, non-destructive testing), outside (relevant in intravascular imaging), both inside and outside (relevant in radar imaging) of the acquisition circle. Given the importance of such inversion formulas in several new and emerging imaging modalities, an efficient numerical inversion algorithm is of tremendous topical interest. The novelty of our non-iterative numerical inversion approach is that the entire scheme can be pre-processed and used repeatedly in image reconstruction, leading to a very fast algorithm. Several numerical simulations are presented showing the robustness of our algorithm.

Towards adaptive kinetic-fluid simulations of weakly ionized plasmas

In Systems Biology, system identification, which infers regulatory network in genetic system and metabolic pathways using experimentally observed time-course data, is one of the hottest issues. The efficient numerical optimization algorithm to estimate more than 100 real-coded parameters should be developed for this purpose. New real-coded genetic algorithm (RCGA), the combination of AREX (adaptive real-coded ensemble crossover) with JGG (just generation gap), have applied to the inference of genetic interactions involving more than 100 parameters related to the interactions with using experimentally observed time-course data. Compared with conventional RCGA, the combination of UNDX (unimodal normal distribution crossover) with MGG (minimal generation gap), new algorithm has shown the superiority with improving early convergence in the first stage of search and suppressing evolutionary stagnation in the last stage of search.

This paper presents a new, more efficient numerical algorithm that has been developed to predict thermal and microstructure fields during quenching of steel rods. The present algorithm solves the full nonlinear heat conduction equation using a central finite-difference scheme coupled with a fourth-order Runge-Kutta nonlinear solver. Numerical results, obtained using the present algorithm, have been validated using experimental data and numerical results available in the literature. In addition to its accurate predictions, the present algorithm does not require iteration; hence, it is computationally more efficient than previous numerical algorithms.

Very recently, an efficient computational algorithm (DETQPT algorithm) for the determinant evaluation of general cyclic pentadiagonal Toeplitz matrices has been proposed by Y.L. Jiang and J.T. Jia (J. Math. Chem. 51: 2503-2513, 2013). In this paper, an explicit formula for the determinant of a cyclic pentadiagonal Toeplitz matrix is derived at first. Then, we present a more efficient numerical algorithm with the cost of 7n+O(logn)$7n+O(\log \ n)$ for evaluating n-th order cyclic pentadiagonal Toeplitz determinants. The algorithm is based on the use of a certain type of matrix reordering and matrix partition, and equalities involving the products of special kind of relevant matrices. Three numerical examples demonstrate the performance and effectiveness of the proposed algorithm and its competitiveness with other already existing algorithms.

An efficient three-dimensional damage-based nonlocal model for dynamic tensile failure in concrete

Domain decomposition parabolic Monge–Ampère approach for fast generation of adaptive moving meshes

Efficient, non-iterative, and second-order accurate numerical algorithms for the anisotropic Allen–Cahn Equation with precise nonlocal mass conservation

Algorithms for Differential Games with Bounded Control and State

An efficient numerical algorithm is presented for the solution of transients in systems consisting of dynamic components with both slow and fast eigenvalues, interconnected at their terminals by an algebraic network. The approach consists of producing algebraic input-output equivalents for the individual components by implicit integration of their differential equations. The adopted implicit integration is by parts leaving the impulse response intact and unaffected by the numerical discretization. The effect of this procedure is excellent convergence with large step sizes, resulting in significant savings in computation. The method has been tested on a conveniently chosen large circuit and compared to alternative algorithms. It is successfully applied to the solution of power system transients problems (dynamic stability, switching transients). Although described for linear systems, the method can be applied for nonlinear systems, linearized after each step around the new operating point.

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

Efficient numerical algorithms are developed for use with two finite temperature semiclassical approximations to quantum dynamics both of which require trajectories generated on potentials of mean force derived from the path integral expression for the density matrix. The numerical algorithms are formed from the combination of a classical adiabatic relation similar to that used in the Car–Parrinello method and an efficient path integral molecular dynamics scheme. Results on model, an anharmonic oscillator and a realistic, fluid para-hydrogen, problem indicate that semiclassical dynamics can be obtained for virtually the same computational cost as structure and thermodynamics.

A number of successful variational models for processing planar images have recently been generalized to three-dimensional (3D) surface processing. With this new dimensionality, the amount of numerical computations to solve the minimization of such new 3D formulations naturally grows up dramatically. Though the need of computationally fast and efficient numerical algorithms able to process high resolution surfaces is high, much less work has been done in this area. Recently, a two-step algorithm for the fast solution of the total curvature model was introduced in Tasdizen, Whitaker, Burchard and Osher [Geometric surface processing via normal maps, ACM Trans. Graph. 22(4) (2003), pp. 1012–1033]. In this paper, we generalize and modify this algorithm to the solution of analogues of the mean curvature model of Droske and Martin Rumpf [A level set formulation for Willmore flow, Interfaces Free Bound. 6(3) (2004), pp. 361–378] and the Gaussian curvature model of Elsey and Esedoḡlu [Analogue of the total variation denoising model in the context of geometry processing, Multiscale Model. Simul. 7(4) (2009), pp. 1549–1573]. Numerical experiments are shown to illustrate the good performance of the algorithms and test results.

Large-Eddy-Simulation of Complex Flows Using the Immersed Boundary Method

Exact wave propagation analysis of moderately thick Levy-type plate with piezoelectric layers using spectral element method