A Polynomial-time Algorithm for Solving Certain Classes of Rank Minimization Problem
We present a non-interior point based, polynomial-time, algorithm for solving certain classes of rank minimization problem.
Rank↗
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Kodiak: An Implementation Framework for Branch and Bound Algorithms
Recursive branch and bound algorithms are often used to refine and isolate solutions to several classes of global optimization problems. A rigorous computation framework for the solution of systems of equations and inequalities involving nonlinear real arithmetic over hyper-rectangular variable and parameter domains is presented. It is derived from a generic branch and bound algorithm that has been formally verified, and utilizes self-validating enclosure methods, namely interval arithmetic and, for polynomials and rational functions, Bernstein expansion. Since bounds computed by these enclosure methods are sound, this approach may be used reliably in software verification tools. Advantage is taken of the partial derivatives of the constraint functions involved in the system, firstly to reduce the branching factor by the use of bisection heuristics and secondly to permit the computation of bifurcation sets for systems of ordinary differential equations. The associated software development, Kodiak, is presented, along with examples of three different branch and bound problem types it implements.
Smith, Andrew P.↗
Retrieval of a wind profile from the Galileo Probe telemetry signal
The zonal wind speed profile within Jupiter's upper troposphere is to be estimated on the basis of its influence on the frequency of the Galileo Probe's telemetry signal; this is achieved by linearizing the relativistic Doppler shift equation about a state of zero wind speed, expanding the zonal wind series in a Legendre polynomial series, and reducing the resulting equations to a linear, least-squares problem. If the error sources have magnitudes not greater than those presently assumed, the retrieved wind speed profiles can place a key constraint on the basic drive for the atmospheric circulation, as well as anchoring winds derived from Galileo Orbiter images to well-defined vertical positions in the atmosphere.
Pollack, James B.↗
Application of an advanced trajectory optimization method to ramjet propelled missiles
The mission performance characteristics of ramjet-propelled missiles are highly dependent upon the trajectory flown. Integration of the trajectory profile with the ramjet propulsion system performance characteristics to achieve optimal missile performance is very complex. Past trajectory optimization methods have been extremely problem dependent and require a high degree of familiarity to achieve success. A general computer code (CTOP) has been applied to ramjet-powered missiles to compute open-loop optimal trajectories. CTOP employs Chebyshev polynomial representations of the states and controls. This allows a transformation of the continuous optimal control problem to one of parameter optimization. With this method, the trajectory boundary conditions are always satisfied. State dynamics and path constraints are enforced via penalty functions. The presented results include solutions to minimum fuel-to-climb, minimum time-to-climb, and minimum time-to-target intercept problems.
Paris, S. W.↗
Evaluating Algorithm Performance Metrics Tailored for Prognostics
Prognostics has taken a center stage in Condition Based Maintenance (CBM) where it is desired to estimate Remaining Useful Life (RUL) of the system so that remedial measures may be taken in advance to avoid catastrophic events or unwanted downtimes. Validation of such predictions is an important but difficult proposition and a lack of appropriate evaluation methods renders prognostics meaningless. Evaluation methods currently used in the research community are not standardized and in many cases do not sufficiently assess key performance aspects expected out of a prognostics algorithm. In this paper we introduce several new evaluation metrics tailored for prognostics and show that they can effectively evaluate various algorithms as compared to other conventional metrics. Specifically four algorithms namely; Relevance Vector Machine (RVM), Gaussian Process Regression (GPR), Artificial Neural Network (ANN), and Polynomial Regression (PR) are compared. These algorithms vary in complexity and their ability to manage uncertainty around predicted estimates. Results show that the new metrics rank these algorithms in different manner and depending on the requirements and constraints suitable metrics may be chosen. Beyond these results, these metrics offer ideas about how metrics suitable to prognostics may be designed so that the evaluation procedure can be standardized. 1
Saxena, Abhinav↗
Rolling Horizon with K-Position Search Method for Strategic Deconfliction of Package Delivery UAS
This research focuses on the strategic deconfliction of unmanned aircraft systems (UAS) in an urban package delivery environment with two depots and multiple drop-off locations. Since the formulated mixed-integer nonlinear programming (MINLP) problem is non-deterministic polynomial-time (NP) hard, a heuristic algorithm called "rolling horizon with k-position search (KPS)" is used to compute the departure sequence and scheduled time of departure (STD) of each UAS at a depot, considering temporal constraints at en-route crossing waypoints and depots for strategic deconfliction. The simulation studies show that an increase in the value of k (local neighborhood search) in the KPS reduces the average ground delay at the cost of an increase in the computation time for a given number of UAS, size of the rolling horizon window, and number of depots involved in the local neighborhood search. The studies also show that for a given rolling horizon window, the computation time increases exponentially with an increase in the total number of UAS flights when serial processing the local neighborhood search of KPS (with k > 1) and drops by an order of magnitude upon performing the local neighborhood search of KPS using parallel processing instead of serial processing. The computation time drops with the reduction in air traffic complexity of a scenario for a given number of flights, k (local neighborhood search), and rolling horizon window.
UTM↗
Minimum fuel coplanar aeroassisted orbital transfer using collocation and nonlinear programming
The fuel optimal control problem arising in coplanar orbital transfer employing aeroassisted technology is addressed. The mission involves the transfer from high energy orbit (HEO) to low energy orbit (LEO) without plane change. The basic approach here is to employ a combination of propulsive maneuvers in space and aerodynamic maneuvers in the atmosphere. The basic sequence of events for the coplanar aeroassisted HEO to LEO orbit transfer consists of three phases. In the first phase, the transfer begins with a deorbit impulse at HEO which injects the vehicle into a elliptic transfer orbit with perigee inside the atmosphere. In the second phase, the vehicle is optimally controlled by lift and drag modulation to satisfy heating constraints and to exit the atmosphere with the desired flight path angle and velocity so that the apogee of the exit orbit is the altitude of the desired LEO. Finally, the second impulse is required to circularize the orbit at LEO. The performance index is maximum final mass. Simulation results show that the coplanar aerocapture is quite different from the case where orbital plane changes are made inside the atmosphere. In the latter case, the vehicle has to penetrate deeper into the atmosphere to perform the desired orbital plane change. For the coplanar case, the vehicle needs only to penetrate the atmosphere deep enough to reduce the exit velocity so the vehicle can be captured at the desired LEO. The peak heating rates are lower and the entry corridor is wider. From the thermal protection point of view, the coplanar transfer may be desirable. Parametric studies also show the maximum peak heating rates and the entry corridor width are functions of maximum lift coefficient. The problem is solved using a direct optimization technique which uses piecewise polynomial representation for the states and controls and collocation to represent the differential equations. This converts the optimal control problem into a nonlinear programming problem which is solved numerically by using a modified version of NPSOL. Solutions were obtained for the described problem for cases with and without heating constraints. The method appears to be more robust than other optimization methods. In addition, the method can handle complex dynamical constraints.
Shi, Yun Yuan↗
Wing Weight Optimization Under Aeroelastic Loads Subject to Stress Constraints
A minimum weight optimization of the wing under aeroelastic loads subject to stress constraints is carried out. The loads for the optimization are based on aeroelastic trim. The design variables are the thickness of the wing skins and planform variables. The composite plate structural model incorporates first-order shear deformation theory, the wing deflections are expressed using Chebyshev polynomials and a Rayleigh-Ritz procedure is adopted for the structural formulation. The aerodynamic pressures provided by the aerodynamic code at a discrete number of grid points is represented as a bilinear distribution on the composite plate code to solve for the deflections and stresses in the wing. The lifting-surface aerodynamic code FAST is presently being used to generate the pressure distribution over the wing. The envisioned ENSAERO/Plate is an aeroelastic analysis code which combines ENSAERO version 3.0 (for analysis of wing-body configurations) with the composite plate code.
Kapania, Rakesh K.↗
Two Solvers for Tractable Temporal Constraints with Preferences
A number of reasoning problems involving the manipulation of temporal information can naturally be viewed as implicitly inducing an ordering of potential local decisions involving time on the basis of preferences. Soft temporal constraints problems allow to describe in a natural way scenarios where events happen over time and preferences are associated to event distances and durations. In general, solving soft temporal problems require exponential time in the worst case, but there are interesting subclasses of problems which are polynomially solvable. We describe two solvers based on two different approaches for solving the same tractable subclass. For each solver we present the theoretical results it stands on, a description of the algorithm and some experimental results. The random generator used to build the problems on which tests are performed is also described. Finally, we compare the two solvers highlighting the tradeoff between performance and representational power.
Rossi, F.↗
Development of Response Surface Models for Rapid Analysis & Multidisciplinary Optimization of Launch Vehicle Design Concepts
Multdisciplinary design optimization (MDO) is an important step in the design and evaluation of launch vehicles, since it has a significant impact on performance and lifecycle cost. The objective in MDO is to search the design space to determine the values of design parameters that optimize the performance characteristics subject to system constraints. Vehicle Analysis Branch (VAB) at NASA Langley Research Center has computerized analysis tools in many of the disciplines required for the design and analysis of launch vehicles. Vehicle performance characteristics can be determined by the use of these computerized analysis tools. The next step is to optimize the system performance characteristics subject to multidisciplinary constraints. However, most of the complex sizing and performance evaluation codes used for launch vehicle design are stand-alone tools, operated by disciplinary experts. They are, in general, difficult to integrate and use directly for MDO. An alternative has been to utilize response surface methodology (RSM) to obtain polynomial models that approximate the functional relationships between performance characteristics and design variables. These approximation models, called response surface models, are then used to integrate the disciplines using mathematical programming methods for efficient system level design analysis, MDO and fast sensitivity simulations. A second-order response surface model of the form given has been commonly used in RSM since in many cases it can provide an adequate approximation especially if the region of interest is sufficiently limited.
Unal, Resit↗
Some easily analyzable convolutional codes
Convolutional codes have played and will play a key role in the downlink telemetry systems on many NASA deep-space probes, including Voyager, Magellan, and Galileo. One of the chief difficulties associated with the use of convolutional codes, however, is the notorious difficulty of analyzing them. Given a convolutional code as specified, say, by its generator polynomials, it is no easy matter to say how well that code will perform on a given noisy channel. The usual first step in such an analysis is to computer the code's free distance; this can be done with an algorithm whose complexity is exponential in the code's constraint length. The second step is often to calculate the transfer function in one, two, or three variables, or at least a few terms in its power series expansion. This step is quite hard, and for many codes of relatively short constraint lengths, it can be intractable. However, a large class of convolutional codes were discovered for which the free distance can be computed by inspection, and for which there is a closed-form expression for the three-variable transfer function. Although for large constraint lengths, these codes have relatively low rates, they are nevertheless interesting and potentially useful. Furthermore, the ideas developed here to analyze these specialized codes may well extend to a much larger class.
Mceliece, R.↗
Multiresolution With Super-Compact Wavelets
The solution data computed from large scale simulations are sometimes too big for main memory, for local disks, and possibly even for a remote storage disk, creating tremendous processing time as well as technical difficulties in analyzing the data. The excessive storage demands a corresponding huge penalty in I/O time, rendering time and transmission time between different computer systems. In this paper, a multiresolution scheme is proposed to compress field simulation or experimental data without much loss of important information in the representation. Originally, the wavelet based multiresolution scheme was introduced in image processing, for the purposes of data compression and feature extraction. Unlike photographic image data which has rather simple settings, computational field simulation data needs more careful treatment in applying the multiresolution technique. While the image data sits on a regular spaced grid, the simulation data usually resides on a structured curvilinear grid or unstructured grid. In addition to the irregularity in grid spacing, the other difficulty is that the solutions consist of vectors instead of scalar values. The data characteristics demand more restrictive conditions. In general, the photographic images have very little inherent smoothness with discontinuities almost everywhere. On the other hand, the numerical solutions have smoothness almost everywhere and discontinuities in local areas (shock, vortices, and shear layers). The wavelet bases should be amenable to the solution of the problem at hand and applicable to constraints such as numerical accuracy and boundary conditions. In choosing a suitable wavelet basis for simulation data among a variety of wavelet families, the supercompact wavelets designed by Beam and Warming provide one of the most effective multiresolution schemes. Supercompact multi-wavelets retain the compactness of Haar wavelets, are piecewise polynomial and orthogonal, and can have arbitrary order of approximation. The advantages of the multiresolution algorithm are that no special treatment is required at the boundaries of the interval, and that the application to functions which are only piecewise continuous (internal boundaries) can be efficiently implemented. In this presentation, Beam's supercompact wavelets are generalized to higher dimensions using multidimensional scaling and wavelet functions rather than alternating the directions as in the 1D version. As a demonstration of actual 3D data compression, supercompact wavelet transforms are applied to a 3D data set for wing tip vortex flow solutions (2.5 million grid points). It is shown that high data compression ratio can be achieved (around 50:1 ratio) in both vector and scalar data set.
Lee, Dohyung↗
Large time-step stability of explicit one-dimensional advection schemes
There is a wide-spread belief that most explicit one-dimensional advection schemes need to satisfy the so-called 'CFL condition' - that the Courant number, c = udelta(t)/delta(x), must be less than or equal to one, for stability in the von Neumann sense. This puts severe limitations on the time-step in high-speed, fine-grid calculations and is an impetus for the development of implicit schemes, which often require less restrictive time-step conditions for stability, but are more expensive per time-step. However, it turns out that, at least in one dimension, if explicit schemes are formulated in a consistent flux-based conservative finite-volume form, von Neumann stability analysis does not place any restriction on the allowable Courant number. Any explicit scheme that is stable for c is less than 1, with a complex amplitude ratio, G(c), can be easily extended to arbitrarily large c. The complex amplitude ratio is then given by exp(- (Iota)(Nu)(Theta)) G(delta(c)), where N is the integer part of c, and delta(c) = c - N (less than 1); this is clearly stable. The CFL condition is, in fact, not a stability condition at all, but, rather, a 'range restriction' on the 'pieces' in a piece-wise polynomial interpolation. When a global view is taken of the interpolation, the need for a CFL condition evaporates. A number of well-known explicit advection schemes are considered and thus extended to large delta(t). The analysis also includes a simple interpretation of (large delta(t)) total-variation-diminishing (TVD) constraints.
Leonard, B. P.↗
Spectral Methods for Computational Fluid Dynamics
As a tool for large-scale computations in fluid dynamics, spectral methods were prophesized in 1944, born in 1954, virtually buried in the mid-1960's, resurrected in 1969, evangalized in the 1970's, and catholicized in the 1980's. The use of spectral methods for meteorological problems was proposed by Blinova in 1944 and the first numerical computations were conducted by Silberman (1954). By the early 1960's computers had achieved sufficient power to permit calculations with hundreds of degrees of freedom. For problems of this size the traditional way of computing the nonlinear terms in spectral methods was expensive compared with finite-difference methods. Consequently, spectral methods fell out of favor. The expense of computing nonlinear terms remained a severe drawback until Orszag (1969) and Eliasen, Machenauer, and Rasmussen (1970) developed the transform methods that still form the backbone of many large-scale spectral computations. The original proselytes of spectral methods were meteorologists involved in global weather modeling and fluid dynamicists investigating isotropic turbulence. The converts who were inspired by the successes of these pioneers remained, for the most part, confined to these and closely related fields throughout the 1970's. During that decade spectral methods appeared to be well-suited only for problems governed by ordinary diSerential eqllations or by partial differential equations with periodic boundary conditions. And, of course, the solution itself needed to be smooth. Some of the obstacles to wider application of spectral methods were: (1) poor resolution of discontinuous solutions; (2) inefficient implementation of implicit methods; and (3) drastic geometric constraints. All of these barriers have undergone some erosion during the 1980's, particularly the latter two. As a result, the applicability and appeal of spectral methods for computational fluid dynamics has broadened considerably. The motivation for the use of spectral methods in numerical calculations stems from the attractive approximation properties of orthogonal polynomial expansions.
Zang, T. A.↗
Curved Finite Elements and Curve Approximation
The approximation of parameterized curves by segments of parabolas that pass through the endpoints of each curve segment arises naturally in all quadratic isoparametric transformations. While not as popular as cubics in curve design problems, the use of parabolas allows the introduction of a geometric measure of the discrepancy between given and approximating curves. The free parameters of the parabola may be used to optimize the fit, and constraints that prevent overspill and curve degeneracy are introduced. This leads to a constrained optimization problem in two varibles that can be solved quickly and reliably by a simple method that takes advantage of the special structure of the problem. For applications in the field of computer-aided design, the given curves are often cubic polynomials, and the coefficient may be calculated in closed form in terms of polynomial coefficients by using a symbolic machine language so that families of curves can be approximated with no further integration. For general curves, numerical quadrature may be used, as in the implementation where the Romberg quadrature is applied. The coefficient functions C sub 1 (gamma) and C sub 2 (gamma) are expanded as polynomials in gamma, so that for given A(s) and B(s) the integrations need only be done once. The method was used to find optimal constrained parabolic approximation to a wide variety of given curves.
Baart, M. L.↗
Multi-Criterion Preliminary Design of a Tetrahedral Truss Platform
An efficient method is presented for multi-criterion preliminary design and demonstrated for a tetrahedral truss platform. The present method requires minimal analysis effort and permits rapid estimation of optimized truss behavior for preliminary design. A 14-m-diameter, 3-ring truss platform represents a candidate reflector support structure for space-based science spacecraft. The truss members are divided into 9 groups by truss ring and position. Design variables are the cross-sectional area of all members in a group, and are either 1, 3 or 5 times the minimum member area. Non-structural mass represents the node and joint hardware used to assemble the truss structure. Taguchi methods are used to efficiently identify key points in the set of Pareto-optimal truss designs. Key points identified using Taguchi methods are the maximum frequency, minimum mass, and maximum frequency-to-mass ratio truss designs. Low-order polynomial curve fits through these points are used to approximate the behavior of the full set of Pareto-optimal designs. The resulting Pareto-optimal design curve is used to predict frequency and mass for optimized trusses. Performance improvements are plotted in frequency-mass (criterion) space and compared to results for uniform trusses. Application of constraints to frequency and mass and sensitivity to constraint variation are demonstrated.
Wu, K. Chauncey↗
Multicriterion Design Methods for Aerospace Systems
An efficient method is presented for multi-criterion preliminary design and demonstrated for a tetrahedral truss platform. The present method requires minimal analysis effort and permits rapid estimation of optimized truss behavior for preliminary design. A 14-m-diameter, 3-ring truss platform represents a candidate reflector support structure for space-based science spacecraft. The truss members are divided into 9 groups by truss ring and position. Design variables are the cross-sectional area of all members in a group, and are either 1, 3 or 5 times the minimum member area. Non-structural mass represents the node and joint hardware used to assemble the truss structure. Taguchi methods are used to efficiently identify key points in the set of Pareto-optimal truss designs. Key points identified using Taguchi methods are the maximum frequency, minimum mass, and maximum frequency-to-mass ratio truss designs. Low-order polynomial curve fits through these points are used to approximate the behavior of the full set of Pareto-optimal designs. The resulting Pareto-optimal design curve is used to predict frequency and mass for optimized trusses. Performance improvements are plotted in frequency-mass (criterion) space and compared to results for uniform trusses. Application of constraints to frequency and mass and sensitivity to constraint variation are demonstrated.
taguchi methods↗
Geodetic constraints on the composition of Mars
The mean density and mean moment of inertia are remotely accessible geodetic parameters which provide two integral constraints on the radial density profile of Mars. The mean density is presently well known, but until the axial precession rate is better determined, the moment of inertia will remain uncertain. Even if the moment were accurately known, a considerable degree of nonuniqueness would still remain concerning the density and composition. This nonuniqueness is illustrated by construction of a family of orthogonal polynomials in the normalized radius which make no contribution to either the mean density or the mean inertial moment. If the mean moment is near to the lower end of its plausible range, the nonuniqueness in density and composition is considerably diminished. In order to avoid unrealistically high densities in the deep interior, the Martian mantle must have a density appreciably lower than the terrestrial mantle and must consequently be significantly depleted in iron.
Bills, Bruce G.↗