An Approximate Direct Inverse as a Preconditioner for Ill-conditioned Problems.
Abstract not provided.
Lee, Chung Hyun↗
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An Approximate Direct Inverse as a Preconditioner For Ill-Conditioned Problems.
Abstract not provided.
Lee, Chung Hyun↗
Recursive inversion of externally defined linear systems
The approximate inversion of an internally unknown linear system, given by its impulse response sequence, by an inverse system having a finite impulse response, is considered. The recursive least squares procedure is shown to have an exact initialization, based on the triangular Toeplitz structure of the matrix involved. The proposed approach also suggests solutions to the problems of system identification and compensation.
Bach, Ralph E., Jr.↗
Recursive inversion of externally defined linear systems by FIR filters
The approximate inversion of an internally unknown linear system, given by its impulse response sequence, by an inverse system having a finite impulse response, is considered. The recursive least-squares procedure is shown to have an exact initialization, based on the triangular Toeplitz structure of the matrix involved. The proposed approach also suggests solutions to the problem of system identification and compensation.
Bach, Ralph E., Jr.↗
Pre-conditioned BFGS-based uncertainty quantification in elastic full-waveform inversion
SUMMARY Full-waveform inversion has become an essential technique for mapping geophysical subsurface structures. However, proper uncertainty quantification is often lacking in current applications. In theory, uncertainty quantification is related to the inverse Hessian (or the posterior covariance matrix). Even for common geophysical inverse problems its calculation is beyond the computational and storage capacities of the largest high-performance computing systems. In this study, we amend the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm to perform uncertainty quantification for large-scale applications. For seismic inverse problems, the limited-memory BFGS (L-BFGS) method prevails as the most efficient quasi-Newton method. We aim to augment it further to obtain an approximate inverse Hessian for uncertainty quantification in FWI. To facilitate retrieval of the inverse Hessian, we combine BFGS (essentially a full-history L-BFGS) with randomized singular value decomposition to determine a low-rank approximation of the inverse Hessian. Setting the rank number equal to the number of iterations makes this solution efficient and memory-affordable even for large-scale problems. Furthermore, based on the Gauss–Newton method, we formulate different initial, diagonal Hessian matrices as pre-conditioners for the inverse scheme and compare their performances in elastic FWI applications. We highlight our approach with the elastic Marmousi benchmark model, demonstrating the applicability of pre-conditioned BFGS for large-scale FWI and uncertainty quantification.
58 GEOSCIENCES↗
Stability and Convergence of Solutions to Stochastic Inverse Problems Using Approximate Probability Densities
Data-consistent inversion is designed to solve a class of stochastic inverse problems where the solution is a pullback of a probability measure specified on the outputs of a quantities of interest (QoI) map. Here, this work presents stability and convergence results for the case where finite QoI data result in an approximation of the solution as a density. Given their popularity in the literature, separate results are proven for three different approaches to measuring discrepancies between probability measures: f-divergences, integral probability metrics, and L p metrics. In the context of integral probability metrics, we also introduce a pullback probability metric that is well-suited for data-consistent inversion. This fills a theoretical gap in the convergence and stability results for data-consistent inversion that have mostly focused on convergence of solutions associated with approximate maps. Numerical results are included to illustrate key theoretical results with intuitive and reproducible test problems that include a demonstration of convergence in the measure-theoretic "almost" sense.
97 MATHEMATICS AND COMPUTING↗
High absorption photovoltaic material and methods of making the same
A high absorption photovoltaic material and method of making the material for use in a solar cell are disclosed. The photovoltaic material includes a surface modified with a layer of repeating photonic crystal structures. The photonic crystal structures are approximately inverse conically shaped and have a curved sidewall that has an approximately Gaussian shape. The photonic crystal structures generally have a high vertical depth and sidewall angle. The structures also have a gradient refractive index profile and exhibit the parallel-to-interface refraction light trapping effect. An anti-reflective coating is disposed over the photonic crystal structure layer. The photovoltaic material exhibits near unity light absorption over a broad range of visible and near infrared wavelengths and incidence angles, even at reduced thicknesses. The photovoltaic structures are formed via a combined photolithography and reactive-ion etching method at low power with a gas mixture having a high ratio of an etchant component to a passivation component.
Kuang, Ping↗
Mie forward scattering - Improved semiempirical approximation with application to particle size distribution inversion
The approximation of Penndorf (1962) and Shifrin-Punina (1968) to the Mie solution at forward scattering angles are extended to small size parameters. The proposed semiempirical approximation accurately represents the Mie results down to x = 0.5-1 for refractive index m = 1.33, and to x = 2.0 for larger index values. The implications of the result for the inversion of particle size distribution from single scattering data in the forward direction are discussed.
Fymat, A. L.↗
Second-order p-iterative solution of the Lambert/Gauss problem
An algorithm is presented for efficient p-iterative solution of the Lambert/Gauss orbit-determination problem using second-order Newton iteration. The algorithm is based on a universal transformation of Kepler's time-of-flight equation and approximate inverse solutions of this equation for short-way and long-way flight paths. The approximate solutions provide both good starting values for iteration and simplified computation of the second-order term in the iteration formula. Numerical results are presented which indicate that in many cases of practical significance (except those having collinear position vectors) the algorithm produces at least eight significant digits of accuracy with just two or three steps of iteration.
Boltz, F. W.↗
A thin-shock-layer solution for nonequilibrium, inviscid hypersonic flows in earth, Martian, and Venusian atmospheres
An approximate inverse solution is presented for the nonequilibrium flow in the inviscid shock layer about a vehicle in hypersonic flight. The method is based upon a thin-shock-layer approximation and has the advantage of being applicable to both subsonic and supersonic regions of the shock layer. The relative simplicity of the method makes it ideally suited for programming on a digital computer with a significant reduction in storage capacity and computing time required by other more exact methods. Comparison of nonequilibrium solutions for an air mixture obtained by the present method is made with solutions obtained by two other methods. Additional cases are presented for entry of spherical nose cones into representative Venusian and Martian atmospheres. A digital computer program written in FORTRAN language is presented that permits an arbitrary gas mixture to be employed in the solution. The effects of vibration, dissociation, recombination, electronic excitation, and ionization are included in the program.
Grose, W. L.↗
A Hybrid Parallel Preconditioning Algorithm For CFD
A new hybrid preconditioning algorithm will be presented which combines the favorable attributes of incomplete lower-upper (ILU) factorization with the favorable attributes of the approximate inverse method recently advocated by numerous researchers. The quality of the preconditioner is adjustable and can be increased at the cost of additional computation while at the same time the storage required is roughly constant and approximately equal to the storage required for the original matrix. In addition, the preconditioning algorithm suggests an efficient and natural parallel implementation with reduced communication. Sample calculations will be presented for the numerical solution of multi-dimensional advection-diffusion equations. The matrix solver has also been embedded into a Newton algorithm for solving the nonlinear Euler and Navier-Stokes equations governing compressible flow. The full paper will show numerous examples in CFD to demonstrate the efficiency and robustness of the method.
Barth,Timothy J.↗
Round-off errors in cutting plane algorithms based on the revised simplex procedure
This report statistically analyzes computational round-off errors associated with the cutting plane approach to solving linear integer programming problems. Cutting plane methods require that the inverse of a sequence of matrices be computed. The problem basically reduces to one of minimizing round-off errors in the sequence of inverses. Two procedures for minimizing this problem are presented, and their influence on error accumulation is statistically analyzed. One procedure employs a very small tolerance factor to round computed values to zero. The other procedure is a numerical analysis technique for reinverting or improving the approximate inverse of a matrix. The results indicated that round-off accumulation can be effectively minimized by employing a tolerance factor which reflects the number of significant digits carried for each calculation and by applying the reinversion procedure once to each computed inverse. If 18 significant digits plus an exponent are carried for each variable during computations, then a tolerance value of 0.1 x 10 to the minus 12th power is reasonable.
Moore, J. E.↗
Backed Bending Actuator
Bending actuators of a proposed type would partly resemble ordinary bending actuators, but would include simple additional components that would render them capable of exerting large forces at small displacements. Like an ordinary bending actuator, an actuator according to the proposal would include a thin rectangular strip that would comprise two bonded layers (possibly made of electroactive polymers with surface electrodes) and would be clamped at one end in the manner of a cantilever beam. Unlike an ordinary bending actuator, the proposed device would include a rigid flat backplate that would support part of the bending strip against backward displacement; because of this feature, the proposed device is called a backed bending actuator. When an ordinary bending actuator is inactive, the strip typically lies flat, the tip displacement is zero, and the force exerted by the tip is zero. During activation, the tip exerts a transverse force and undergoes a bending displacement that results from the expansion or contraction of one or more of the bonded layers. The tip force of an ordinary bending actuator is inversely proportional to its length; hence, a long actuator tends to be weak. The figure depicts an ordinary bending actuator and the corresponding backed bending actuator. The bending, the tip displacement (d(sub t)), and the tip force (F) exerted by the ordinary bending actuator are well approximated by the conventional equations for the loading and deflection of a cantilever beam subject to a bending moment which, in this case, is applied by the differential expansion or contraction of the bonded layers. The bending, displacement, and tip force of the backed bending actuator are calculated similarly, except that it is necessary to account for the fact that the force F(sub b) that resists the displacement of the tip could be sufficient to push part of the strip against the backplate; in such a condition, the cantilever beam would be effectively shortened (length L*) and thereby stiffened and, hence, made capable of exerting a greater tip force for a given degree of differential expansion or contraction of the bonded layers. Taking all of these effects into account, the cantilever-beam equations show that F(sub b) would be approximately inversely proportional to d(sup 1/2) for d less than a calculable amount, denoted the transition displacement (dt). For d less than d(sub t), part of the strip would be pressed against the backplate. Therefore, the force F(sub b) would be very large for d at or near zero and would decrease as d increases toward d(sub t). At d greater than d(sub t), none of the strip would be pressed against the backplate and F(sub b) would equal the tip force F of the corresponding ordinary bending actuator. The advantage of the proposal is that a backed bending actuator could be made long to obtain large displacement when it encountered little resistance but it could also exert a large zero-displacement force, so that it could more easily start the movement of a large mass, throw a mechanical switch, or release a stuck mechanism.
Costen, Robert C.↗
Inverse kinematic solution for near-simple robots and its application to robot calibration
This paper provides an inverse kinematic solution for a class of robot manipulators called near-simple manipulators. The kinematics of these manipulators differ from those of simple-robots by small parameter variations. Although most robots are by design simple, in practice, due to manufacturing tolerances, every robot is near-simple. The method in this paper gives an approximate inverse kinematics solution for real time applications based on the nominal solution for these robots. The validity of the results are tested both by a simulation study and by applying the algorithm to a PUMA robot.
Hayati, Samad A.↗
Empirical scaling of the L–H threshold power for metal wall tokamaks using a multi-device database
The empirical scaling for the H-mode power threshold in tokamaks has been revisited using a database with threshold data from machines with a metallic first wall as part of International Tokamak Physics Activity (ITPA) task TC-26. The database contains discharges from ASDEX Upgrade (AUG) (W), JET (Be/W) and Alcator C-Mod (Mo). This was motivated by reports that in like-for-like discharges the power threshold was reduced by approximately 30% after the change from carbon based to metallic first wall materials on AUG (Ryter et al 2013 Nucl. Fusion 53 113003) and JET (Maggi et al 2014 Nucl. Fusion 54 023007). The database contains L–H transition data for all hydrogen isotopes and mixtures, including T and DT from the recent JET campaigns. Compared to the ITPA 2008 scaling (Martin et al 2008 J. Phys.: Conf. Ser. 123 012033), the metal wall scaling has a smaller magnetic field exponent but a larger density exponent. We present an additional parameter to capture the strong dependence of the L–H power threshold (approx. factor 2) on the magnetic configuration in the divertor on JET. The scaling recovers the approximate inverse isotope mass scaling of the threshold power. Alternative scalings involving the plasma current and poloidal magnetic field are explored. Despite the reduction in threshold observed earlier, the scalings based on the metal wall database do not necessarily extrapolate to a lower threshold for ITER compared to the ITPA 2008 scaling, especially at high density. The divertor configuration effect induces the largest uncertainty in the extrapolation.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Advances in Mixed Precision Algorithms: 2021 Edition
Over the last year, the ECP xSDK-multiprecision effort has made tremendous progress in developing and deploying new mixed precision technology and customizing the algorithms for the hardware deployed in the ECP flagship supercomputers. The effort also has succeeded in creating a cross-laboratory community of scientists interested in mixed precision technology and now working together in deploying this technology for ECP applications. In this report, we highlight some of the most promising and impactful achievements of the last year. Among the highlights we present are: Mixed precision IR using a dense LU factorization and achieving a 1.8× speedup on Spock; results and strategies for mixed precision IR using a sparse LU factorization; a mixed precision eigenvalue solver; Mixed Precision GMRES-IR being deployed in Trilinos, and achieving a speedup of 1.4× over standard GMRES; compressed Basis (CB) GMRES being deployed in Ginkgo and achieving an average 1.4× speedup over standard GMRES; preparing hypre for mixed precision execution; mixed precision sparse approximate inverse preconditioners achieving an average speedup of 1.2×; and detailed description of the memory accessor separating the arithmetic precision from the memory precision, and enabling memory-bound low precision BLAS 1/2 operations to increase the accuracy by using high precision in the computations without degrading the performance. We emphasize that many of the highlights presented here have also been submitted to peer-reviewed journals or established conferences, and are under peer-review or have already been published.
97 MATHEMATICS AND COMPUTING↗
Advances in Mixed Precision Algorithms: 2021 Edition
Over the last year, the ECP xSDK-multiprecision effort has made tremendous progress in developing and deploying new mixed precision technology and customizing the algorithms for the hardware deployed in the ECP flagship supercomputers. The effort also has succeeded in creating a cross-laboratory community of scientists interested in mixed precision technology and now working together in deploying this technology for ECP applications. In this report, we highlight some of the most promising and impactful achievements of the last year. Among the highlights we present are • Mixed precision IR using a dense LU factorization and achieving a 1.8× speedup on Spock; • Results and strategies for mixed precision IR using a sparse LU factorization; • A mixed precision eigenvalue solver; • Mixed Precision GMRES-IR being deployed in Trilinos, and achieving a speedup of 1.4× over standard GMRES; • Compressed Basis (CB) GMRES being deployed in Ginkgo and achieving an average 1.4× speedup over standard GMRES; • Preparing hypre for mixed precision execution; • Mixed precision sparse approximate inverse preconditioners achieving an average speedup of 1.2×; • Detailed description of the memory accessor separating the arithmetic precision from the memory precision, and enabling memory-bound low precision BLAS 1/2 operations to increase the accuracy by using high precision in the computations without degrading the performance. We emphasize that many of the highlights presented here have also been submitted to peer-reviewed journals or established conferences, and are under peer-review or have already been published.
97 MATHEMATICS AND COMPUTING↗
Analog Systems for Edge Optimization
Over the past decade, analog computing has the subject of substantial research interest providing a path toward improved computational efficiency in the post-Dennard era. Analog matrix vector multiplication (MVM) accelerators provide a popular approach given the ubiquity of MVM operations in numerous applications. However, historically analog computing systems can struggle with applications requiring high precision due to the inherent susceptibility of these systems to analog non-idealities. Therefore, prior work on analog systems has focused either on applications known to be tolerant of limited precision (e.g., neural network inference), or using expensive techniques to emulate high-precision using many analog MVM operations. In this work, we propose an alternative approach. Motivated by recent advances in inexact nonlinear solvers and optimizers, we explore the potential of co-designing optimization algorithms which can take full advantage of the fundamentally inexact analog MVM operations. To enable these co-designed algorithms we also develop a general mathematical theory of the precision and energy efficiency of analog operations, and a new system architecture for tightly-coupled analog and digital computation. Finally, we examine the applicability of analog computing to a wider class of symmetric positive definite systems and find potential in using analog operations as a sparse approximate inverse preconditioner. With these core innovations, this project provides a path toward effectively implementing optimization algorithms on power-constrained autonomous and semi-autonomous systems.
97 MATHEMATICS AND COMPUTING↗