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This paper demonstrates continuous optimization on the differentiable manifold formed by continuous constraint functions. The first order tensor geodesic differential equation is solved on the manifold in both numerical and closed analytic form for simple nonlinear programs. Advantages and disadvantages with respect to conventional optimization techniques are discussed.
This paper demonstrates the capability to obtain and dynamically track eigenvalues and eigenvectors using continuous optimization on constraint manifolds.
In recent years, there is growing interest in using quantum computers for solving combinatorial optimization problems. In this work, we developed a generic, machine learning-based framework for mapping continuous-space inverse design problems into surrogate quadratic unconstrained binary optimization (QUBO) problems by employing a binary variational autoencoder and a factorization machine. The factorization machine is trained as a low-dimensional, binary surrogate model for the continuous design space and sampled using various QUBO samplers. Using the D-Wave Advantage hybrid sampler and simulated annealing, we demonstrate that by repeated resampling and retraining of the factorization machine, our framework finds designs that exhibit figures of merit exceeding those of its training set. We showcase the framework’s performance on two inverse design problems by optimizing (i) thermal emitter topologies for thermophotovoltaic applications and (ii) diffractive meta-gratings for highly efficient beam steering. This technique can be further scaled to leverage future developments in quantum optimization to solve advanced inverse design problems for science and engineering applications.
A common optimization problem in the areas of magnetized plasmas and fusion energy is the design of magnets to produce a given three-dimensional magnetic field distribution to high precision. When designing arrays of permanent magnets for stellarator plasma confinement, such problems have tens of thousands of degrees of freedom whose solutions, for practical reasons, should be constrained to discrete spaces. We perform a direct comparison between two algorithms that have been developed previously for this purpose, and demonstrate that composite procedures that apply both algorithms in sequence can produce substantially improved results. One approach uses a continuous, quasi-Newton procedure to optimize the dipole moments of a set of magnets and then projects the solution onto a discrete space. The second uses an inherently discrete greedy optimization procedure that has been enhanced and generalized for this work. Further, the approaches are both applied to design arrays cubic rare-Earth permanent magnets to confine a quasi-axisymmetric plasma with a magnetic field on axis of 0.5 T. The first approach tends to find solutions with higher field accuracy, whereas the second can find solutions with substantially (up to 30%) fewer magnets. When the approaches are combined, they can obtain solutions with magnet quantities comparable to the second approach while matching the field accuracy of the first.
Technologies for a quantum/classical hybrid approach to solving optimization problems is disclosed. In the illustrative embodiment, an optimization problem is decomposed into two sub-problems. The first sub-problem is solved on a classical computer, and a result from the first sub-problem is provided to a quantum computer. The quantum computer then solves the second sub-problem based on the result of the first sub-problem from the classical computer. The quantum computer can then provide a result to the classical computer to re-solve the first problem. The iterative calculation is continued until an end condition is met.
Continuous Liquid Interface Production (CLIP) is poised to bring additive manufacturing to multiple American manufacturing sectors owing to its unique combination of rapid print speeds and material options that resemble injection molding thermoplastics. In spite of these benefits, CLIP is still a maturing process. To improve our understanding and control of the process, this project developed a multi-physics computational model that encompasses the coupled chemical-physical processes of photopolymerization and fluid flow to predict part outcomes. A physically predictive model can enable rapid optimization of CLIP and reduce the current cycle time and waste associated with optimization by four-fold. Ultimately, this effort will help open the U.S. $400B plastic manufacturing industry to the key benefits of additive manufacturing, namely mass customization, unlimited design space, and a cost- and energy-effective path to mainstream manufacturing. In the course of the project we produced a model which coupled fluid-flow, species transport, and photochemistry to model the Carbon CLIP process. As the development proceeded, we validated the code against theory and experiments. The resulting code was transferred to Carbon, and we expect it to be a useful part of their modeling capability. Future work includes improvements to the fluid solver’s robustness and performance, along with the addition of additional physics models.
Quantum computing (QC) has gained popularity due to its unique capabilities that are quite different from that of classical computers in terms of speed and methods of operations. This paper proposes hybrid models and methods that effectively leverage the complementary strengths of deterministic algorithms and QC techniques to overcome combinatorial complexity for solving large-scale mixed-integer programming problems. Four applications, namely the molecular conformation problem, job-shop scheduling problem, manufacturing cell formation problem, and the vehicle routing problem, are specifically addressed. Large-scale instances of these application problems across multiple scales ranging from molecular design to logistics optimization are computationally challenging for deterministic optimization algorithms on classical computers. To address the computational challenges, hybrid QC-based algorithms are proposed and extensive computational experimental results are presented to demonstrate their applicability and efficiency. The proposed QC-based solution strategies enjoy high computational efficiency in terms of solution quality and computation time, by utilizing the unique features of both classical and quantum computers.
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Minimum average cost ordering policies for two product inventory systems with joint setup costs
Low thrust trajectories for minimum time rendezvous between continuous thrust interceptor and passive target vehicle, comparing calculus of variations and steepest ascent analyses
During the last decades thin film ferroelectrics have been investigated in great detail because of their versatility and advantages for tunable devices. Many reports on tunable components such as resonators, filters, oscillators and phase shifters have been published by researchers all around world. These efforts have shed light on the primary advantages as well as the current shortcomings of these materials that should be considered for the development of practical microwave devices and systems. In this paper, we intend to discuss the most relevant outcomes of current collaborations taking place at the NASA Glenn Research Center aimed at improving the quality of ferroelectric microwave components (e.g., phase shifters) via careful processing and evaluation of the materials properties of the thin films. To the extent that our applications allow, we have worked with fixed circuit configurations to determine which material modifications are of most impact to the performance of a given circuit. Examples on the improvements on the performance of these circuits will be provided.
Continuous parameter stochastic optimization principle using abstract variational theory
Stable high radiance in visible and near-ultraviolet wavelengths is desirable for radiometric calibration sources. In this work, newly available electrodeless radio-frequency (RF) driven plasma light sources were combined with research grade, low-noise power supplies and coupled to an integrating sphere to produce a uniform radiance source. The stock light sources consist of a 28 VDC power supply, RF driver, and a resonant RF cavity. The RF cavity includes a small bulb with a fill gas that is ionized by the electric field and emits light. This assembly is known as the emitter. The RF driver supplies a source of RF energy to the emitter. In commercial form, embedded electronics within the RF driver perform a continual optimization routine to maximize energy transfer to the emitter. This optimization routine continually varies the light output sinusoidally by approximately 2% over a several-second period. Modifying to eliminate this optimization eliminates the sinusoidal variation but allows the output to slowly drift over time. This drift can be minimized by allowing sufficient warm-up time to achieve thermal equilibrium. It was also found that supplying the RF driver with a low-noise source of DC electrical power improves the stability of the lamp output. Finally, coupling the light into an integrating sphere reduces the effect of spatial fluctuations, and decreases noise at the output port of the sphere.
Process optimization in the latent space of functions via variational autoencoder (VAE) and Bayesian Optimization (BO). We demonstrate this to optimize the curl of a kinetic ferroelectric model.
A Proximate Time-Optimal Servo (PTOS) is developed, along with conditions for its stability. An algorithm is proposed for adapting the PTOS (APTOS) to improve performance in the face of uncertain plant parameters. Under ideal conditions APTOS is shown to be uniformly asymptotically stable. Simulation results demonstrate the predicted performance.
The paper examines some questions relating to the optimal design of continuous one-dimensional structures driven by harmonically oscillating loads. Optimal-control methods are applied to a cantilever bar driven sinusoidally by an axial force at its tip to illustrate the minimum-weight design of one-dimensional structures under dynamic excitation. Realistic constraints are imposed during the optimizations, including a maximum allowable stress amplitude at any point along the bar and a minimum cross-sectional area. It is shown that in the absence of damping, the design space may contain many disjoint feasible regions, and multiple optima can exist. Detailed solutions are obtained for continuous bars with an excitation frequency less than, and then greater than, the fundamental free-vibration frequency. It is found that above a certain excitation frequency, two or more arcs with different constraints characterize the optimal designs. It is concluded that when more than two different constrained arcs characterize the optimal solution, the continuum approach may be impractical, and finite-element approximations may offer the only alternative.
A continuation method is applied to a singular perturbation parameter to obtain a new numerical method for computing optimal trajectories. This method allows one to use simply calculated reduced order approximations as starting solutions and continue the perturbation parameter until the optimal full-order solution is obtained. The method does not require the calculation of higher order correction terms nor does it require the perturbation parameter to be small - thus, it has potentially superior convergence properties compared to conventional asymptotic expansions when the perturbation parameter is large. A simple trajectory optimization problem is considered to illustrate the method.
In this paper, we present a study to select an appropriate alternator voltage of the free-piston Stirling convertors (FPSC) for efficient and light Dynamic Radioisotope Power Systems (DRPS). With a system thermal-to-electrical efficiency 3-4 times greater than radioisotope thermoelectric generator (RTG) systems and a higher power density than Brayton systems in the power range of interest for radioisotope-powered systems, Stirling-based DRPS is uniquely suited to benefit upcoming NASA missions. Much effort has been invested in the design and optimization of the thermal, mechanical, and materials aspects of FPSCs, but the electrical aspect has been more nebulous. Therefore, in this paper, a preliminary study will be presented to select the appropriate Stirling alternator voltage to develop a light and efficient system using available flight components. Power conversion systems face a trade between efficiency and system volume/mass with the optimal trade being determined by the application. Neglecting non-idealities related to insulation thickness and winding packing factor and assuming a constant winding area, alternator efficiency is independent of alternator voltage. Because wire size and alternator current can be traded against turn count and alternator voltage without impacting efficiency, the guidance on the optimal design comes from analysis of the controller power electronics and the remainder of the system. Properties of available flight-qualified electrical components, such as rated voltage/current and on-resistance, typically come in discrete values instead of a continuous range of values. With multiple components being required to form the power conversion stage of the controller, each limited to incremental values, a continuous optimization is of little benefit. Instead of a continuous optimization, a random process using properties of the available components can be used to develop a Pareto design front indicating the optimized trade space. The most advantageous trade between system efficiency and mass for the system at hand can then be selected from the range of feasible designs. In the final paper, the design process, assumptions, and preliminary results will be presented.