Dimensional Parameter Space Research Articles

On the Use of Saddlepoint Approximations in High Dimensional Inference

Inference in high dimensional parameter space poses many challenges. One of these is the possible use of saddlepoint approximations. Motivated by a recent use of the saddlepoint approximation to construct a conditional test, we argue that the precision is questionable. We illustrate this by an example giving a 50% relative error in the calculation of the p-value. A power study of the underlying test reveals a low power in many situations. As an alternative it is suggested to use the likelihood ratio test.

Commensuration effects in layered nanoparticle solids

We have developed HiNTS, the {\bf Hi}erarchical {\bf N}anoparticle {\bf T}ransport {\bf S}imulator, and adapted it to study commensuration effects in two classes of Nanoparticle (NP) solids: (1) a bilayer NP solid (BNS) with an energy offset, and (2) a BNS as part of a Field-Effect Transistor (FET). HiNTS integrates the ab initio characterization of single NPs with the phonon-assisted tunneling transition model of the NP-NP transitions into a Kinetic Monte Carlo based simulation of the charge transport in NP solids. First, we studied a BNS with an inter-layer energy offset $\Delta$, possibly caused by a fixed electric field. Our results include the following. (1) In the independent energy-offset model, we observed the emergence of commensuration effects when scanning the electron filling factor $FF$ across integer values. These commensuration effects were profound as they reduced the mobility by several orders of magnitude. We analyzed these commensuration effects in a five dimensional parameter space, as a function of the on-site charging energy $E_C$, energy offset $\Delta$, the disorder $D$, the electron filling factor $FF$, and the temperature $k_{B}T$. We demonstrated the complexity of our model by showing that at integer filling factors $FF$ commensuration effects are present in some regions of the parameter space, while they vanish in other regions, thus defining distinct dynamical phases of the model. We determined the phase boundaries between these dynamical phases. (2) Using these results as a foundation, we shifted our focus to the experimentally much-studied NP-FETs. NP-FETs are also characterized by an inter-layer energy offset $\Delta$, which, in contrast to our first model, is set by the gate voltage $V_G$ and thereby related to the electron filling $FF$. We demonstrated the emergence of commensuration effects and distinct dynamical phases in these NP-FETs.

Sparse polynomial surrogates for non-intrusive, high-dimensional uncertainty quantification of aeroelastic computations

This paper is concerned with aircraft aeroelastic interactions and the propagation of parametric uncertainties in numerical simulations using high-fidelity fluid flow solvers. More specifically, the influence of variable operational and structural parameters (random inputs) on the drag performance and deformation (outputs) of a flexible wing in transonic regime, is assessed. Because of the complexity of fluid flow solvers, non-intrusive uncertainty quantification techniques are favored. Polynomial surrogate models based on homogeneous chaos expansions in the random inputs are commonly considered in this respect. The polynomial expansion coefficients are constructed using either structured sampling sets of the input parameters, as Gauss quadrature nodes, or unstructured sampling sets, as in Monte-Carlo methods. In complex systems such as the advanced aeroelastic test case studied here, the output quantities of interest generally depend only weakly on the multiple cross-interactions between the random inputs. Consequently, only low-order polynomials significantly contribute to their surrogates, which thus have a sparse structure in the underlying polynomial bases. This feature prompts to use compressed sensing, or compressive sampling theory for the construction of the polynomial surrogates. The proposed methodology is non-adapted and considers unstructured sampling sets orders of magnitude smaller than the ones required by the usual techniques with structured sampling sets. It is illustrated in the present work for a moderately to high dimensional parametric space.

High dimensional parameter tuning for event generators

Monte Carlo Event Generators are important tools for the understanding of physics at particle colliders like the LHC. In order to best predict a wide variety of observables, the optimization of parameters in the Event Generators based on precision data is crucial. However, the simultaneous optimization of many parameters is computationally challenging. We present an algorithm that allows to tune Monte Carlo Event Generators for high dimensional parameter spaces. To achieve this we first split the parameter space algorithmically in subspaces and perform a Professor tuning on the subspaces with binwise weights to enhance the influence of relevant observables. We test the algorithm in ideal conditions and in real life examples including tuning of the event generators Herwig 7 and Pythia 8 for LEP observables. Further, we tune parts of the Herwig 7 event generator with the Lund string model.

Support Vector Regression-Based Active Subspace (SVR-AS) Modeling of High-Speed Links for Fast and Accurate Sensitivity Analysis

A methodology based on the joint usage of support vector regression and active subspace is introduced in this paper for accelerated sensitivity analysis of high-speed links through parameter space dimensionality reduction. The proposed methodology uses the gradient directly obtained by support vector regression with Gaussian kernel to generate an active subspace with its application to the high-speed link model. Active subspace generated by this method is defined by the directions that are most influential on the desirable output measure. The resulting reduced-dimensional model is shown to perform well in sensitivity analysis of high-speed links including IBIS-AMI equalization, and is computationally more efficient than Sobol's method.

Optimizing Deep Network for Image Classification with Hyper Parameter Tuning

The deep network model comprises of several processing layers and deep learning techniques help us in representing data with diverse levels of abstraction. Based on the practical importance and the efficiency of machine learning, optimization of deep models are carried out relating to the objective functions and its parameters for a particular problem. The present work focuses on an empirical analysis of the performance of stochastic optimization methods with regard to hyperparameters for the deep Convolution Neural Network (CNN) and to understand the rate of convergence of the optimization methods in high dimensional parameter spaces. Experimentation has been carried out in deep CNN model with different optimization methods viz. SGD, AdaGard, AdaDelta and Adam. The empirical results are evaluated using benchmark CIFAR10 and CIFAR100 datasets. The optimal values of the hyperparameters obtained demonstrates that the optimizer Adam shows the best results compared to other methods viz. SGD, AdaGard, and AdaDelta over the considered datasets. Further, it is noted that classification accuracy can be increased by choosing the best optimization techniques with hyperparameter tuning to get the optimal configuration of the deep CNN model.

Mutually independent feature flow: An approach to produce fixed complexity latent space for improved performance and decreased parameter count

Deep neural networks are integral part of every recent breakthrough in learning systems. Much of the share of this success goes to new and powerful computational devices, which lead to bigger and deeper models with higher learning capacity. Recently, a lot of new approaches have been developed to make it more convenient to train deeper models, and overcome the training problems like exploding and vanishing gradients. In this paper, we use a very unorthodox approach and propose novel mutually independent feature flow method for efficient architecture design. This approach decomposes the traditional bigger and continuously widening graphical model to fixed width graphs. It is achieved by decomposing the wider layers into conditionally independent branches with smaller width. Resulting model is a constrained parametric graph with fixed size and lower depth kernels. The process leads to huge reduction in model parameters in comparison to their traditional counterparts. This approach also provides low dimensional parameter space for gradient flow leading to faster convergence along with improved accuracy. Proposed approach is tested in three different styles of CNN architectures generally used in computer vision community. Performance is tested on four benchmark datasets of different complexity. Proposed approach not only provides huge reduction in model parameters but also leads to increased model accuracy.

Reducing dimension of parametric space based on the approximation of components for interpolation of multidimensional signals

In this paper, we address the problem of parametric space dimension reduction in the interpolation of multidimensional signals task. We develop adaptive parameterized interpolation algorithms for multidimensional signals. We perform a dimension reduction of the parameter space to reduce the complexity of optimizing such algorithms. The dependences of the samples inside the signal sections and between the signal sections are taken into account in various ways to reduce the dimension. We consider the dependencies between the signal sections through the approximation algorithm for the sections. We take into account the sample dependencies inside sections due to an adaptive parameterized interpolation algorithm. As a result, we solve the optimization problem of an adaptive interpolator in the parameter space of lower dimension for each signal section separately. To study the effectiveness of adaptive interpolators, we perform computational experiments using real-world multidimensional signals. Experimental results showed that the proposed interpolator improves the efficiency of the compression method up to 10% compared with the prototype algorithm.

Initializing Cross-Gradients Joint Inversion of Gravity and Magnetic Data with a Bayesian Surrogate Gravity Model

Cross-gradients joint inversion of gravity and magnetic data is the focus of this work. Cross-gradients are introduced as a constraint in the minimization of a least square functional including the misfits of the available data. We propose to initialize the cross gradients iterations with a surrogate density model. The latter is constructed by means of Bayesian estimation in a low dimensional parameter space. To sample from the posterior, an affine invariant MCMC is also introduced. The proposed methodology is successfully tested on synthetic models consisting of isolated sources.

Weyl points in systems of multiple semiconductor-superconductor quantum dots

As an analogy to the Weyl point in k-space, we search for energy levels which close at a single point as a function of a three dimensional parameter space. Such points are topologically protected in the sense that any perturbation which acts on the two level subsystem can be corrected by tuning the control parameters. We find that parameter controlled Weyl points are ubiquitous in semiconductor-superconductor quantum dots and that they are deeply related to Majorana zero modes. In this paper, we present several semiconductor-superconductor quantum dot devices which host parameter controlled Weyl points. Further, we show how these points can be observed experimentally via conductance measurements.

Reliable Multistage Optimization of Antennas for Multiple Performance Figures in Highly Dimensional Parameter Spaces

Design of modern antenna structures needs to account for multiple performance figures and geometrical constraints. Fulfillment of these calls for the development of complex topologies described by a large number of parameters. Electromagnetic (EM)-driven tuning of such designs is mandatory yet challenging. In this letter, a new framework for multistage design optimization of multidimensional antennas w.r.t. several performance criteria is proposed. The approach is founded on the idea of identifying a feasible region of the search space through a series of local optimization runs with appropriate handling of individual objectives and design constraints. To facilitate local optimization, a procedure for finding a good starting point is proposed, which involves sensitivity-based reduction of the problem dimensionality. The proposed design procedure is aided by surrogate modeling techniques. Our methodology is demonstrated using a gain-enhanced dual-band antenna and compared to alternative design approaches.

On the Complexity of Logistic Regression Models.

We investigate the complexity of logistic regression models, which is defined by counting the number of indistinguishable distributions that the model can represent (Balasubramanian, 1997). We find that the complexity of logistic models with binary inputs depends not only on the number of parameters but also on the distribution of inputs in a nontrivial way that standard treatments of complexity do not address. In particular, we observe that correlations among inputs induce effective dependencies among parameters, thus constraining the model and, consequently, reducing its complexity. We derive simple relations for the upper and lower bounds of the complexity. Furthermore, we show analytically that defining the model parameters on a finite support rather than the entire axis decreases the complexity in a manner that critically depends on the size of the domain. Based on our findings, we propose a novel model selection criterion that takes into account the entropy of the input distribution. We test our proposal on the problem of selecting the input variables of a logistic regression model in a Bayesian model selection framework. In our numerical tests, we find that while the reconstruction errors of standard model selection approaches (AIC, BIC, regularization) strongly depend on the sparsity of the ground truth, the reconstruction error of our method is always close to the minimum in all conditions of sparsity, data size, and strength of input correlations. Finally, we observe that when considering categorical instead of binary inputs, in a simple and mathematically tractable case, the contribution of the alphabet size to the complexity is very small compared to that of parameter space dimension. We further explore the issue by analyzing the data set of the "13 keys to the White House," a method for forecasting the outcomes of US presidential elections.

Reliable data-driven modeling of high-frequency structures by means of nested kriging with enhanced design of experiments

PurposeA framework for reliable modeling of high-frequency structures by nested kriging with an improved sampling procedure is developed and extensively validated. A comprehensive benchmarking including conventional kriging and previously reported design of experiments technique is provided. The proposed technique is also demonstrated in solving parameter optimization task.Design/methodology/approachThe keystone of the proposed approach is to focus the modeling process on a small region of the parameter space (constrained domain containing high-quality designs with respect to the selected performance figures) instead of adopting traditional, hyper-cube-like domain defined by the lower and upper parameter bounds. A specific geometry of the domain is explored to improve a uniformity of the training data set. In consequence, the predictive power of the model is improved.FindingsBuilding the model in a constrained domain allows for a considerable reduction of a training data set size without a necessity to either narrow down the parameter ranges or to reduce the parameter space dimensionality. Improving uniformity of training data set allocation permits further reduction of the computational cost of setting up the model. The proposed technique can be used to expedite the parameter optimization and enables locating good initial designs in a straightforward manner.Research limitations/implicationsThe developed framework opens new possibilities inaccurate surrogate modeling of high-frequency structures described by a large number of geometry and/or material parameters. Further extensions can be investigated such as the inclusion of the sensitivity data into the model or exploration of the particular geometry of the model domain to further reduce the computational overhead of training data acquisition.Originality/valueThe efficiency of the proposed method has been demonstrated for modeling and parameter optimization of high-frequency structures. It has also been shown to outperform conventional kriging and previous constrained modeling approaches. To the authors’ knowledge, this approach to formulate and handle the modeling process is novel and permits the establishment of accurate surrogates in highly dimensional spaces and covering wide ranges of parameters.

Model order reduction for large-scale structures with local nonlinearities

In solid mechanics, linear structures often exhibit (local) nonlinear behavior when close to failure. For instance, the elastic deformation of a structure becomes plastic after being deformed beyond recovery. To properly assess such problems in a real-life application, we need fast and multi-query evaluations of coupled linear and nonlinear structural systems, whose approximations are not straight forward and often computationally expensive. In this work, we propose a linear–nonlinear domain decomposition, where the two systems are coupled through the solutions on a prescribed linear–nonlinear interface. After necessary sensitivity analysis, e.g. for structures with a high dimensional parameter space, we adopt a non-intrusive method, e.g. Gaussian processes regression (GPR), to solve for the solution on the interface. We then utilize different model order reduction techniques to address the linear and nonlinear problems individually. To accelerate the approximation, we employ again the non-intrusive GPR for the nonlinearity, while intrusive model order reduction methods, e.g. the conventional reduced basis (RB) method or the static-condensation reduced-basis-element (SCRBE) method, are employed for the solution in the linear subdomain. The proposed method is applicable for problems with pre-determined linear–nonlinear domain decomposition. We provide several numerical examples to demonstrate the effectiveness of our method.

A Simplified Crossing Fiber Model in Diffusion Weighted Imaging.

Diffusion MRI (dMRI) is a vital source of imaging data for identifying anatomical connections in the living human brain that form the substrate for information transfer between brain regions. dMRI can thus play a central role toward our understanding of brain function. The quantitative modeling and analysis of dMRI data deduces the features of neural fibers at the voxel level, such as direction and density. The modeling methods that have been developed range from deterministic to probabilistic approaches. Currently, the Ball-and-Stick model serves as a widely implemented probabilistic approach in the tractography toolbox of the popular FSL software package and FreeSurfer/TRACULA software package. However, estimation of the features of neural fibers is complex under the scenario of two crossing neural fibers, which occurs in a sizeable proportion of voxels within the brain. A Bayesian non-linear regression is adopted, comprised of a mixture of multiple non-linear components. Such models can pose a difficult statistical estimation problem computationally. To make the approach of Ball-and-Stick model more feasible and accurate, we propose a simplified version of Ball-and-Stick model that reduces parameter space dimensionality. This simplified model is vastly more efficient in the terms of computation time required in estimating parameters pertaining to two crossing neural fibers through Bayesian simulation approaches. Moreover, the performance of this new model is comparable or better in terms of bias and estimation variance as compared to existing models.

Dimensionality reduction and identification of valid parameter bounds for the efficient calibration of automated driving functions

The industrialization of automated driving functions according to level 3 requires an efficient test and calibration concept to deal with an increased complexity, growing customer demands, and a larger vehicle fleet offered. Therefore, a method for a complexity reduction of the calibration parameter space is presented. In the two-step approach, a qualitative sensitivity analysis is used to identify valid regions in the search space and subsequently decrease dimensionality based on the parameter-specific global influences. The reduced parameter space and sensitivity information can then serve as a starting point for an efficient calibration process on the target hardware. To examine the method’s potential, our approach is applied to the parameter space of an automated driving function. The results expose clear dependencies between parameters and driving scenarios and allow an exclusion of parameter space dimensions based on sensitivity values. The predefined search space can be narrowed down to valid regions using the parameter range identification approach. Finally, the findings are validated with a quantitative variance-based sensitivity analysis. The validation confirms that our method provides equivalent results with a comparably smaller number of system evaluations.

Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra so(4)

This paper is concerned with the topology of the Liouville foliation in the analogue of the Kovalevskaya integrable case on the Lie algebra . The Fomenko-Zieschang invariants (that is, marked molecules) for this foliation are calculated on each nonsingular iso-energy surface. A detailed description of the resulting stratification of the three- dimensional space of parameters of the iso-energy surfaces is given. Bibliography: 23 titles.

A Variability-Aware Robust Design Methodology for Integrated Circuits by Geometric Programming

Process variations have continuously posed significant challenges to the performance and yield of integrated circuits (ICs). The performance modeling and robust optimization method considering process variations has become an important research task in today’s IC design. Aiming at solving the problems of strong nonlinearity and high-dimensional problems in circuit design, this paper proposes a general robust optimization method for ICs by geometric programming. This method first employs regularization sparse models to model a specific performance metric as a posynomial function in terms of design parameters, in order to reduce parameter space dimensionality and to accurately capture the nonlinear relationship between performance perturbations and process variations. Based on the posynomial performance models, this method further uses an uncertainty set to represent the uncertainties of process variations, and formulates the problem of robust optimization under process variations as a general geometric programming model that can be efficiently solved. Experimental results demonstrate that, the proposed method not only enhances the accuracy and efficiency of circuit performance modeling, but also improves the performance yield significantly compared with traditional circuit design methods.

Euclid preparation: II. The EuclidEmulator – a tool to compute the cosmology dependence of the nonlinear matter power spectrum

We present a new power spectrum emulator named EuclidEmulator that estimates the nonlinear correction to the linear dark matter power spectrum. It is based on a spectral decomposition method called polynomial chaos expansion. All steps in the construction of the emulator have been tested and optimized: the large high-resolution N-body simulations carried out with PKDGRAV3 were validated using a simulation from the Euclid Flagship campaign and demonstrated to have converged up to wavenumbers $k\approx 5\,h\,{\rm Mpc}^{-1}$ for redshifts $z\leq 5$. The emulator is constructed using the uncertainty quantification software UQLab and it has been optimized first by creating mock emulators based on Takahashi's HALOFIT. We show that it is possible to successfully predict the performance of the final emulator in this way prior to performing any N-body simulations. We provide a C-code to calculate the nonlinear correction at a relative accuracy of $\sim0.3\%$ with respect to N-body simulations within 50 ms. The absolute accuracy of the final nonlinear power spectrum is comparable to one obtained with N-body simulations, i.e. $\sim 1\%$ for $k\lesssim 1\,h\,{\rm Mpc}^{-1}$ and $z\lesssim 3.5$. This enables efficient forward modeling in the nonlinear regime allowing for maximum likelihood estimation of cosmological parameters. EuclidEmulator has been compared to HALOFIT and CosmicEmu, an alternative emulator based on the Mira-Titan Universe, and shown to be more accurate than these other approaches. This work paves a new way for optimal construction of future emulators that also consider other cosmological observables, use higher resolution input simulations and investigate higher dimensional cosmological parameter spaces.

Crossing exceptional points without phase transition

We show that the theoretical framework linking exceptional points (EPs) to phase transitions in parity-time (PT) symmetric Hamiltonians is incomplete. Particularly, we demonstrate that the application of the squaring operator to a Jx PT lattice dramatically alter the topology of its Riemann surface, eventually resulting in a system that can cross an EP without undergoing a symmetry breaking. We elucidate on these rather surprising results by invoking the notion of phase diagrams in higher dimensional parameter space. Within this perspective, the canonical PT symmetry breaking paradigm arises only along certainprojections of the Riemann surface in the parameter space.