POLSYS GLP: A Parallel General Linear Product Homotopy Code for Solving Polynomial Systems of Equations

Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked, and in the handling of singular solutions, have made probability-one homotopy methods even more practical. POLSYS GLP consists of Fortran 95 modules for nding all isolated solutions of a complex coefficient polynomial system of equations. The package is intended to be used on a distributed memory multiprocessor in conjunction with HOMPACK90 (Algorithm 777), and makes extensive use of Fortran 95 derived data types and MPI to support a general linear product (GLP) polynomial system structure. GLP structure is intermediate between the partitioned linear product structure used by POLSYS PLP (Algorithm 801) and the BKK-based structure used by PHCPACK. The code requires a GLP structure as input, and although nding the optimal GLP structure is a dicult combinatorial problem, generally physical or engineering intuition about a problem yields a very good GLP structure. POLSYS GLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem denition both at a high level and via hand-crafted code. Dierent GLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding

Dynamical problems and phase transitions

Issued as Financial status report, Technical reports [nos. 1-12], and Final report, Project B-06-68

Homotopy methods for constraint relaxation in unilevel reliability based design optimization

Reliability based design optimization is a methodology for finding optimized designs that are characterized with a low probability of failure. The main ob jective in reliability based design optimization is to minimize a merit function while satisfying the reliability constraints. The reliability constraints are constraints on the probability of failure corre- sponding to each of the failure modes of the system or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a relia- bility analysis. During the last few years, a variety of different techniques have been devel- oped for reliability based design optimization. Traditionally, these have been formulated as a double-loop (nested) optimization problem. The upper level optimization loop gen- erally involves optimizing a merit function sub ject to reliability constraints and the lower level optimization loop(s) compute the probabilities of failure corresponding to the failure mode(s) that govern the system failure. This formulation is, by nature, computationally intensive. A new efficient unilevel formulation for reliability based design optimization was developed by the authors in earlier studies. In this formulation, the lower level optimiza- tion (evaluation of reliability constraints in the double loop formulation) was replaced by its corresponding first order Karush-Kuhn-Tucker (KKT) necessary optimality conditions at the upper level optimization. It was shown that the unilevel formulation is computation- ally equivalent to solving the original nested optimization if the lower level optimization is solved by numerically satisfying the KKT conditions (which is typically the case), and the two formulations are mathematically equivalent under constraint qualification and general- ized convexity assumptions. In the unilevel formulation, the KKT conditions of the inner optimization for each probabilistic constraint evaluation are imposed at the system level as equality constraints. Most commercial optimizers are usually numerically unreliable when applied to problems accompanied by many equality constraints. In this investigation an optimization framework for reliability based design using the unilevel formulation is de- veloped. Homotopy methods are used for constraint relaxation and to obtain a relaxed feasible design. A series of optimization problems are solved as the relaxed optimization problem is transformed via a homotopy to the original problem. A heuristic scheme is employed in this paper to update the homotopy parameter. The proposed algorithm is illustrated with example problems

The Impact of Lateral Electron Disequilibrium on Stereotactic Body Radiation Therapy of Lung Cancer

Stereotactic Body Radiation Therapy (SBRT) is an effective treatment option for patients with inoperable early-stage lung cancer. SBRT uses online image-guidance technology [e.g. cone-beam CT (CBCT)] to focus small-fields of high energy x-rays onto a tumour to deliver ablative levels of radiation dose (e.g. 54 Gy) in a few treatment fractions (e.g. 3). For the combination of these treatment parameters and a low density lung, lateral electron disequilibrium (LED) can potentially occur, reducing lung and tumour doses. The goal of this thesis was to determine the impact of LED on stereotactic body radiation therapy for lung cancer. The effect of LED on lung dose distribution was studied using Monte Carlo simulations of a lung slab phantom. The magnitude of lung dose reduction due to LED, and the specific conditions (beam energy, field size, and lung density) that cause the phenomenon, were quantified and could be predicted using a relative depth dose factor (RDDF). The RDDF concept was then used to develop a novel SBRT technique, called LED-optimized SBRT (LED-SBRT), which creates steep dose gradients, caused by intentional LED, to elevate tumour dose, while reducing/maintaining dose levels in healthy lung. Further, the RDDF aided in assessing the accuracy required in CBCT-derived lung density, when applied to adaptive SBRT dose calculations. In this regard, we determined that CBCT image artefacts produced erroneously low lung density, artificially triggering LED, and incorrectly predicting lower lung/tumour dose levels. As a result, CBCT number corrective techniques were developed in order to improve dose calculation accuracy. The results of this thesis provide physicians and physicists with a much better prediction of the radiation dosimetry under disequilibrium conditions, and allow exploration of irradiation conditions that can cause LED. With this knowledge in-mind, competent decisions can be made regarding the choice of dose calculation algorithm, and aid in the design and interpretation of SBRT clinical trials. Furthermore, the outcomes of this work can help launch a new generation of SBRT techniques that exploit LED effects that may offer a dosimetric benefits for selected patients

Message length effects for solving polynomial systems on a hypercube

Polynomial systems of equations frequently arise in solid modelling, robotics, computer vision, chemistry, chemical engineering, and mechanical engineering. Locally convergent iterative methods such as quasi-Newton methods may diverge or fail to find all meaningful solutions of a polynomial system. Recently a homotopy algorithm has been proposed for polynomial systems that is guaranteed globally convergent (always converges from an arbitrary starting point) with probability one, finds all solutions to the polynomial system, and has a large amount of inherent parallelism. For this homotopy algorithm and a given decomposition strategy, the communication overhead for several possible communication stritegies is explored empirically in this paper. The experiments were conducted on an iPSC-32 hypercube.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27982/1/0000415.pd