Numerical And Statistical Methods For Bioengine...
Abstract:Discontinuous Galerkin (DG) methods are specialized finite element methods that utilize discontinuous piecewise polynomial spaces to approximate the solutions of differential equations, with boundary conditions and inter-element continuity weakly imposed through bilinear forms. DG methods have recently gained popularity for many attractive properties. First of all, the methods are locally mass conservative and have small numerical diffusion. In addition, they support general nonconforming spaces including unstructured meshes, nonmatching grids and variable degrees of local approximations, thus allowing efficient h-, p-, and hp-adaptivities. Moreover, DG algorithms treat rough coefficient problems and effectively capture discontinuities in solutions. They have excellent parallel efficiency since data communications are relatively local. For time-dependent problems in particular, their mass matrices are block diagonal, providing a substantial computational advantage if explicit time integrations are used.
Numerical and Statistical Methods for Bioengine...
In this talk, we consider a family of discontinuous Galerkin (DG) applied to reactive transport problems. They are the four primal discontinuous Galerkin schemes for the space discretization: Symmetric Interior Penalty Galerkin (SIPG), Oden-Baumann-Babuska DG formulation (OBB-DG), Nonsymmetric Interior Penalty Galerkin (NIPG) and Incomplete Interior Penalty Galerkin (IIPG) methods. We address a priori error bounds in the L2(H1), L2(L2) and negative norms, and a posteriori error estimates in L2(H1) and L2(L2). Efficient implementation issues are discussed with emphasis on dynamic mesh adaptation strategies. A number of numerical examples are presented to illustrate various features of DG methods including their sharp a posteriori error indicators and effective adaptivities.
Abstract:We describe two regularization techniques based on optimal control for solving two types of ill-posed problems. The two ill-posed problems we consider are in signal processing and parameter identification. Our new mathematical formulations of these ill-posed problems lead to new efficient numerical methods. We compare our numerical results to the same examples using the well-known Tikhonov regularization.
It is the combinatorial properties of the Schur function that ultimately allowed us to develop the first efficient algorithm for computing the hypergeometric function of a matrix argument. As a result, 3D target classification is now possible and efficient, as are multivariate statistical methods in genomics, wireless communications, etc. I will present the key ideas in the development of our algorithm as well as the impact it has had on the above applications.
558. Methods of Analysis in Bioengineering. (3)Presents applied analytical and numerical mathematical methods in the context of biomedical engineering problems. Introduces statistical methods for the design of experiments and analysis of experimental data in research and development activities.
Applied mathematics and statistics are disciplines devoted to the use of mathematical methods and reasoning to solve real-world problems of a scientific or decision-making nature in a wide variety of subjects, principally (but not exclusively) in engineering, medicine, the physical and biological sciences, and the social sciences. Applied mathematical modeling often involves the use of systems of (partial) differential equations to describe and predict the behavior of complex real-world systems that unfold dynamically in time. Statistics, construed broadly, is the study of uncertainty: how to measure it (using ideas and methods in probability theory), and what to do about it (using concepts from statistical inference and decision theory).
The applied mathematics minor is available for students who wish to develop (1) proficiency in modeling real-life problems using mathematics and (2) knowledge of standard, practical analytical and numerical methods for the solution of these models. This minor could be combined with a major in any of the physical, biological, mathematical, or engineering sciences as preparation for a graduate degree in that field or in applied mathematics.
Students in the statistical science program learn to develop and use statistical methods to provide a probabilistic assessment of the variability in different data structures. This knowledge is applied to the quantification of the uncertainties inherent in the discoveries, summaries and conclusions that are drawn from the data analysis. The Ph.D. program provides mastery of fundamental concepts in statistical theory and methods, as well as analytical and computational skills to build modern statistical models, implement them, and effectively communicate their results. Through the process of learning these skills, the students develop the ability to conduct independent research. The M.S. program has its own identity. It places emphasis on the application of statistical methods to the solution of relevant scientific, technological and engineering problems, with the goal of preparing students for professional careers.
Students will obtain a graduate degree (M.S. or Ph.D.) in statistical science. More specifically, students will develop background on statistical theory, methods, computing and applications through the program coursework, with research emphasis on novel methods and applications of Bayesian statistics.
The biomedical signal and image processing (BSIP) program prepares students for careers in the acquisition and analysis of biomedical signals and enables students to apply quantitative methods to extract meaningful information for both clinical and research applications. The program is premised on the fact that a core set of mathematical and statistical methods are held in common across signal acquisition and imaging modalities and across data analyses regardless of their dimensionality. These include signal transduction, characterization and analysis of noise, transform analysis, feature extraction from time series or images, quantitative image processing, and imaging physics. Students have the opportunity to focus their work over a broad range of modalities, including electrophysiology, optical imaging methods, MRI, CT, PET, and other tomographic devices, and/or on the extraction of image features such as organ morphometry or neurofunctional signals, and detailed anatomic/functional feature extraction. Career opportunities for BSIP trainees include medical instrumentation, engineering positions in medical imaging, and research in the application of advanced engineering skills to the study of anatomy and function.
Required Skills: This ESR should have or be close to obtaining an Undergraduate degree in Bioengineering, Mechanical Engineering, Materials Science, Chemistry or a related discipline. Basic knowledge of biomaterials, regenerative medicine, processing of ceramic nanoparticles, chemical/physical characterisation of biomaterials, statistical methods and experience of cell culture is desirable. Chemical laboratory experience is required. The ESR should show highly collaborative attitude, excellent written and verbal communication skills.
Required Skills: This ESR should have or be close to obtaining an Undergraduate degree in Bioengineering, Mechanical Engineering, Materials Science, Chemistry or a related discipline. Basic knowledge of ceramic biomaterials and processing, methodologies for chemical/physical characterisation of biomaterials, statistical methods and experience of cell culture is desirable. Chemical laboratory experience is required. The ESR should show highly collaborative attitude, excellent written and verbal communication skills.
Acquired skills: This ESR will specialise in in silico modelling of the spine. The project will give the ESR the opportunity to acquire high skills in computational modelling, in particular in the modelling of the many different tissues that compose the spine. The ESR will acquire knowledge of the multi-scale anatomy and biomechanics of the spine, and the different orthopaedic interventions available (interspinous implant concepts). In addition, the ESR will get a very complete experience in in vitro testing, and integration of experimental and numerical methods. 041b061a72