A Compact High-Dimensional Yield Analysis Method using Low-Rank Tensor Approximation

“Curse of dimensionality” has become the major challenge for existing high-sigma yield analysis methods. In this article, we develop a meta-model using Low-Rank Tensor Approximation (LRTA) to substitute expensive SPICE simulation. The polynomial degree of our LRTA model grows linearly with the circuit dimension. This makes it especially promising for high-dimensional circuit problems. Our LRTA meta-model is solved efficiently with a robust greedy algorithm and calibrated iteratively with a bootstrap-assisted adaptive sampling method. We also develop a novel global sensitivity analysis approach to generate a reduced LRTA meta-model which is more compact. It further accelerates the procedure of model calibration and yield estimation. Experiments on memory and analog circuits validate that the proposed LRTA method outperforms other state-of-the-art approaches in terms of accuracy and efficiency.

Open Problems Related to Quantum Query Complexity

I offer a case that quantum query complexity still has loads of enticing and fundamental open problems—from relativized QMA versus QCMA and BQP versus IP , to time/space tradeoffs for collision and element distinctness, to polynomial degree versus quantum query complexity for partial functions, to the Unitary Synthesis Problem and more.

Remote sensing of methane plumes: instrument tradeoff analysis for detecting and quantifying local sources at global scale

Methane (CH4) is the second most important anthropogenic greenhouse gas with a significant impact on radiative forcing, tropospheric air quality, and stratospheric water vapor. Remote sensing observations enable the detection and quantification of local methane emissions across large geographical areas, which is a critical step for understanding local flux distributions and subsequently prioritizing mitigation strategies. Obtaining methane column concentration measurements with low noise and minimal surface interference has direct consequences for accurately determining the location and emission rates of methane sources. The quality of retrieved column enhancements depends on the choices of the instrument and retrieval parameters. Here, we studied the changes in precision error and bias as a result of different spectral resolutions, instrument optical performance, and detector exposure times by using a realistic instrument noise model. In addition, we formally analyzed the impact of spectrally complex surface albedo features on retrievals using the iterative maximum a posteriori differential optical absorption spectroscopy (IMAP-DOAS) algorithm. We built an end-to-end modeling framework that can simulate observed radiances from reflected solar irradiance through a simulated CH4 plume over several natural and artificial surfaces. Our analysis shows that complex surface features can alias into retrieved methane abundances, explaining the existence of retrieval biases in current airborne methane observations. The impact can be mitigated with higher spectral resolution and a larger polynomial degree to approximate surface albedo variations. Using a spectral resolution of 1.5 nm, an exposure time of 20 ms, and a polynomial degree of 25, a retrieval precision error below 0.007 mole m−2 or 1.0 % of total atmospheric CH4 column can be achieved for high albedo cases, while minimizing the bias due to surface interference such that the noise is uncorrelated among various surfaces. At coarser spectral resolutions, it becomes increasingly harder to separate complex surface albedo features from atmospheric absorption features. Our modeling framework provides the basis for assessing tradeoffs for future remote sensing instruments and algorithmic designs. For instance, we find that improving the spectral resolution beyond 0.2 nm would actually decrease the retrieval precision, as detector readout noise will play an increasing role. Our work contributes towards building an enhanced monitoring system that can measure CH4 concentration fields to determine methane sources accurately and efficiently at scale.

Dislocation transport using a time-explicit Runge-Kutta discontinuous Galerkin finite element approach

A time-explicit Runge-Kutta discontinuous Galerkin (RKDG) finite element scheme is proposed to solve the dislocation transport initial boundary value problem in 3D. The dislocation density transport equation, which lies at the core of this problem, is a first-order unsteady-state advection-reaction-type hyperbolic partial differential equation; the DG approach is well suited to solve such equations that lack any diffusion terms. The development of the RKDG scheme follows the method of lines approach. First, a space semi-discretization is performed using the DG approach with upwinding to obtain a system of ordinary differential equations in time. Then, time discretization is performed using explicit RK schemes to solve this system. The 3D numerical implementation of the RKDG scheme is performed for the first-order (forward Euler), second-order and third-order RK methods using the strong stability preserving approach. These implementations provide (quasi-)optimal convergence rates for smooth solutions. A slope limiter is used to prevent spurious Gibbs oscillations arising from high-order space approximations (polynomial degree ≥ 1) of rough solutions. A parametric study is performed to understand the influence of key parameters of the RKDG scheme on the stability of the solution predicted during a screw dislocation transport simulation. Then, annihilation of two oppositely signed screw dislocations and the expansion of a polygonal dislocation loop are simulated. The RKDG scheme is able to resolve the shock generated during dislocation annihilation without any spurious oscillations and predict the prismatic loop expansion with very low numerical diffusion. These results demonstrate the robustness of the scheme.

Computation of Polynomial Degree-Based Topological Descriptors of Indu-Bala Product of Two Paths

Cheminformatics is entirely a newly coined term that encompasses a field that includes engineering computer sciences along with basic sciences. As we all know, vertices and edges form a network whereas vertex and its degrees contribute to joining edges. The degree of vertex is very much dependent on a reasonable proportion of network properties. There is no doubt that a network has to have a reliance of different kinds of hub buses, serials, and other connecting points to constitute a system that is the backbone of cheminformatics. The Indu-Bala product of two graphs G 1 and G 2 has a special notation as described in Section 2. The attainment of this product is very much due to related vertices at to different places of G 1 ∨ G 2 . This study states we have found M-polynomial and degree-based topological indices for Indu-Bala product of two paths P k and P j for j , k ≥ 2 . We also give some graphical representation of these indices and analyzed them graphically.

A sharp relative-error bound for the Helmholtz h-FEM at high frequency

For the h-finite-element method (h-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth h must decrease with the frequency k to maintain accuracy as k increases has been studied since the mid 80’s. Nevertheless, there still do not exist in the literature any k-explicit bounds on the relative error of the FEM solution (the measure of the FEM error most often used in practical applications), apart from in one dimension. The main result of this paper is the sharp result that, for the lowest fixed-order conforming FEM (with polynomial degree, p, equal to one), the condition “$$h^2 k^3$$ h 2 k 3 sufficiently small" is sufficient for the relative error of the FEM solution in 2 or 3 dimensions to be controllably small (independent of k) for scattering of a plane wave by a nontrapping obstacle and/or a nontrapping inhomogeneous medium. We also prove relative-error bounds on the FEM solution for arbitrary fixed-order methods applied to scattering by a nontrapping obstacle, but these bounds are not sharp for $$p\ge 2$$ p ≥ 2 . A key ingredient in our proofs is a result describing the oscillatory behaviour of the solution of the plane-wave scattering problem, which we prove using semiclassical defect measures.

Polymath: Low-Latency MPC via Secure Polynomial Evaluations and Its Applications

While the practicality of secure multi-party computation (MPC) has been extensively analyzed and improved over the past decade, we are hitting the limits of efficiency with the traditional approaches of representing the computed functionalities as generic arithmetic or Boolean circuits. This work follows the design principle of identifying and constructing fast and provably-secure MPC protocols to evaluate useful high-level algebraic abstractions; thus, improving the efficiency of all applications relying on them. We present Polymath, a constant-round secure computation protocol suite for the secure evaluation of (multi-variate) polynomials of scalars and matrices, functionalities essential to numerous data-processing applications. Using precise natural precomputation and high-degree of parallelism prevalent in the modern computing environments, Polymath can make latency of secure polynomial evaluations of scalars and matrices independent of polynomial degree and matrix dimensions. We implement our protocols over the HoneyBadgerMPC library and apply it to two prominent secure computation tasks: privacy-preserving evaluation of decision trees and privacy-preserving evaluation of Markov processes. For the decision tree evaluation problem, we demonstrate the feasibility of evaluating high-depth decision tree models in a general n-party setting. For the Markov process application, we demonstrate that Poly-math can compute large powers of transition matrices with better online time and less communication.

Performances of Hybridized-, Embedded-, and Weighted-Interior Penalty Discontinuous Galerkin Methods for Heterogeneous and Anisotropic Diffusion Problems

The present paper discusses families of Interior Penalty Discontinuous Galerkin (IP) methods for solving heterogeneous and anisotropic diffusion problems. Specifically, we focus on distinctive schemes, namely the Hybridized-, Embedded-, and Weighted-IP schemes, leading to final matrixes of different sizes and sparsities. Both the Hybridized- and Embedded-IP schemes are eligible for static condensation, and their degrees of freedom are distributed on the mesh skeleton. In contrast, the unknowns are located inside the mesh elements for the Weighted-IP variant. For a given mesh, it is well-known that the number of degrees of freedom related to the standard Discontinuous Galerkin methods increases more rapidly than those of the skeletal approaches (Hybridized- and Embedded-IP). We then quantify the impact of the static condensation procedure on the computational performances of the different IP classes in terms of robustness, accuracy, and CPU time. To this aim, numerical experiments are investigated by considering strong heterogeneities and anisotropies. We analyze the fixed error tolerance versus the run time and mesh size to guide our performance criterion. We also outlined some relationships between these Interior Penalty schemes. Eventually, we confirm the superiority of the Hybridized- and Embedded-IP schemes, regardless of the mesh, the polynomial degree, and the physical properties (homogeneous, heterogeneous, and/or anisotropic).

Stable broken 𝐻(𝑐𝑢𝑟𝑙) polynomial extensions and 𝑝-robust a posteriori error estimates by broken patchwise equilibration for the curl–curl problem

We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken H ( curl ) \boldsymbol H(\boldsymbol {\operatorname {curl}}) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl–curl problem discretized with Nédélec finite elements of arbitrary order. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and inexpensive. They are constructed by a broken patchwise equilibration which, in particular, does not produce a globally H ( curl ) \boldsymbol H(\boldsymbol {\operatorname {curl}}) -conforming flux. The equilibration is only related to edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The error estimates become guaranteed when the regularity pick-up constant is explicitly known. Numerical experiments illustrate the theoretical findings.