Discrete Parameter Space Research Articles

ZeroGrads: Learning Local Surrogates for Non-Differentiable Graphics

Gradient-based optimization is now ubiquitous across graphics, but unfortunately can not be applied to problems with undefined or zero gradients. To circumvent this issue, the loss function can be manually replaced by a "surrogate" that has similar minima but is differentiable. Our proposed framework, ZeroGrads , automates this process by learning a neural approximation of the objective function, which in turn can be used to differentiate through arbitrary black-box graphics pipelines. We train the surrogate on an actively smoothed version of the objective and encourage locality, focusing the surrogate's capacity on what matters at the current training episode. The fitting is performed online, alongside the parameter optimization, and self-supervised, without pre-computed data or pre-trained models. As sampling the objective is expensive (it requires a full rendering or simulator run), we devise an efficient sampling scheme that allows for tractable run-times and competitive performance at little overhead. We demonstrate optimizing diverse non-convex, non-differentiable black-box problems in graphics, such as visibility in rendering, discrete parameter spaces in procedural modelling or optimal control in physics-driven animation. In contrast to other derivative-free algorithms, our approach scales well to higher dimensions, which we demonstrate on problems with up to 35k interlinked variables.

Exact electronic states with shallow quantum circuits from global optimisation

Quantum computers promise to revolutionise molecular electronic simulations by overcoming the exponential memory scaling. While electronic wave functions can be represented using a product of fermionic unitary operators, the best ansatz for strongly correlated electronic systems is far from clear. In this contribution, we construct universal wave functions from gate-efficient, spin symmetry-preserving fermionic operators by introducing an algorithm that globally optimises the wave function in the discrete ansatz design and continuous parameter spaces. Our approach maximises the accuracy that can be obtained with near-term quantum circuits and provides a practical route for designing ansätze in the future. Numerical simulations for strongly correlated molecules, including water and molecular nitrogen, and the condensed-matter Hubbard model, demonstrate the improved accuracy of gate-efficient quantum circuits for simulating strongly correlated chemistry.

A Discussion of “A Note on Universal Inference” by Tse and Davison

A Discussion of “A Note on Universal Inference” by Tse and Davison

Taming the landscape of effective theories

We introduce a generalized notion of finiteness that provides a structural principle for the set of effective theories that can be consistently coupled to quantum gravity. More concretely, we propose a Tameness Conjecture that states that all valid effective theories are labelled by a definable parameter space and must have scalar field spaces and coupling functions that are definable using the tame geometry built from an o-minimal structure. We give a brief introduction to tame geometry and describe how it restricts sets, manifolds, and functions. We then collect evidence for the Tameness Conjecture by studying various effective theories arising from string theory compactifications by using some recent advances in tame geometry. In particular, we will exploit the fact that coset spaces and period mappings are definable in an o-minimal structure and argue for non-trivial tameness results in higher-supersymmetric theories and in Calabi-Yau compactifications. As strongest evidence for the Tameness Conjecture over a discrete parameter space, we then discuss a recent theorem stating that the locus of self-dual flux vacua of F-theory admits a tame geometry even if one allows for any flux choice satisfying the tadpole constraint. This result implies the finiteness of self-dual flux vacua in F-theory.

Prospects for using soft processors in systems-on-a-chip based on field-programmable gate arrays

Objectives. Developing the element base of field-programmable gate arrays (FPGA) may significantly affect the design of electronic devices due to the enhanced logical capacity of such chips and the general tendency towards increased subsystem integration. The system-on-a-chip (SoC) concept is aimed at combining receiving, processing, and exchange subsystems onto a single chip, as well as at implementing control, diagnostic, and other auxiliary subsystems. The study aimed at developing a method for soft processor applications, i.e., processors based on configurable logical resources, for implementing control functions in an FPGA-based SoC.Methods. A digital system design methodology was used.Results. For soft processors, a unified design route based on selecting architectural parameters qualitatively corresponding to control tasks was considered. In particular, such parameters as instruction set addressness, number of pipeline cycles, and arithmetic logic unit configuration are adjustable at the design stage to allow the optimization of the soft processor in the discrete parameter space. An approach to rapid prototyping of the assembler based on stack-oriented programming language with regular grammar was also considered. The control of digital signal processing hardware as part of an SoC is the promising application area for soft processors. An implementation is considered on the example of an SoC based on Xilinx Virtex-7 FPGA containing several processor cores developed using the proposed methodology.Conclusions. The considered approaches to soft processor design allow the rapid prototyping of the control processor core for operation as part of an FPGA-based SoC.

Optimization and Simulation of Accounting Information Practice Model Assisted by Discrete Dynamic Events

The performance function expressed by the expected value of the discrete event dynamic system usually does not have a closed analytical expression, so it can only be estimated by the Monte-Carlo method. The inherent limitations of the Monte-Carlo method (low simulation efficiency) and the complexity of the system bring a lot to the simulation optimization problem. A huge challenge has come. A reasonable combination of mathematical methods and computer simulation technology is an important direction to study the optimal design of DESD. On the basis of linear systems on maximal algebras, this article studies a class of multimodel systems on maximal algebras. This article presents a non-Markov DEDS time-synchronized parallel simulation algorithm and proves that the common random number method can maximize the simulation efficiency of an existing event-synchronized algorithm. We take advantage of the inherent parallelism of genetic algorithm, and combine the idea of parallel simulation and DEDS order optimization to give an optimization algorithm for discrete parameter space DEDS. This article discusses the relationship between ultimate control, internal control, and accounting information quality. This article analyzes the relationship between ultimate control right, internal control, and accounting information quality. We use the manipulated accruals calculated by the modified Jones model and the earnings response coefficient of the earnings-return model to measure the quality of accounting information, and analyze the impact of the ultimate control ratio and internal control level on the quality of accounting information. Based on the difference in the nature of the controller and the ratio of ultimate control rights, this article examines the difference in the effect of internal control on the quality of accounting information in companies with high and low ratios of ultimate control rights. The robustness tests of the two accounting information measurement models are carried out, respectively, and the results obtained are basically consistent with the empirical research conclusions. The situation is the most important to its own development. Among them, the asset turnover rate has a greater effect on the early warning of corporate financial crisis, and the status of the current asset turnover rate and the total asset turnover rate in the financial early warning is also crucial. This article studies the correlation between the overall level of internal control and the quality of accounting information, avoiding the one-sidedness caused by using a single index to measure the level of internal control.

Non-parametric Estimation of Stochastic Differential Equations from Stationary Time-Series

We study efficiency of non-parametric estimation of diffusions (stochastic differential equations driven by Brownian motion) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is motivated by the definition of drift and diffusion coefficients. These estimators involve time- and space-discretization parameters for computing expected values from discretely-sampled stationary data. Next, we analyze consistency and mean squared error of these estimators depending on computational parameters. We derive relationships between the number of observational points, time- and space-discretization parameters in order to achieve the optimal speed of convergence and minimize computational complexity. We illustrate our approach with numerical simulations.

Strong error estimates for a space-time discretization of the linear-quadratic control problem with the stochastic heat equation with linear noise

Abstract We propose a time-implicit, finite-element-based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear noise of type $[X(t)+\sigma (t)]\,\,\textrm{d}W(t)$ and prove optimal convergence w.r.t. both space and time discretization parameters. In particular, we employ the stochastic Riccati equation as a proper analytical tool to handle the linear noise, and thus extend the applicability of the earlier work by Prohl & Wang (2021, Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation. ESAIM Control Optim. Calc. Var., 27, 54), where the error analysis was restricted to additive noise.

Inference of ventricular activation properties from non-invasive electrocardiography

The realisation of precision cardiology requires novel techniques for the non-invasive characterisation of individual patients’ cardiac function to inform therapeutic and diagnostic decision-making. Both electrocardiography and imaging are used for the clinical diagnosis of cardiac disease. The integration of multi-modal datasets through advanced computational methods could enable the development of the cardiac ‘digital twin’, a comprehensive virtual tool that mechanistically reveals a patient's heart condition from clinical data and simulates treatment outcomes. The adoption of cardiac digital twins requires the non-invasive efficient personalisation of the electrophysiological properties in cardiac models. This study develops new computational techniques to estimate key ventricular activation properties for individual subjects by exploiting the synergy between non-invasive electrocardiography, cardiac magnetic resonance (CMR) imaging and modelling and simulation. More precisely, we present an efficient sequential Monte Carlo approximate Bayesian computation-based inference method, integrated with Eikonal simulations and torso-biventricular models constructed based on clinical CMR imaging. The method also includes a novel strategy to treat combined continuous (conduction speeds) and discrete (earliest activation sites) parameter spaces and an efficient dynamic time warping-based ECG comparison algorithm. We demonstrate results from our inference method on a cohort of twenty virtual subjects with cardiac ventricular myocardial-mass volumes ranging from 74 cm3 to 171 cm3 and considering low versus high resolution for the endocardial discretisation (which determines possible locations of the earliest activation sites). Results show that our method can successfully infer the ventricular activation properties in sinus rhythm from non-invasive epicardial activation time maps and ECG recordings, achieving higher accuracy for the endocardial speed and sheet (transmural) speed than for the fibre or sheet-normal directed speeds.

Diffusion MRI simulation of realistic neurons with SpinDoctor and the Neuron Module

The diffusion MRI signal arising from neurons can be numerically simulated by solving the Bloch-Torrey partial differential equation. In this paper we present the Neuron Module that we implemented within the Matlab-based diffusion MRI simulation toolbox SpinDoctor. SpinDoctor uses finite element discretization and adaptive time integration to solve the Bloch-Torrey partial differential equation for general diffusion-encoding sequences, at multiple b-values and in multiple diffusion directions. In order to facilitate the diffusion MRI simulation of realistic neurons by the research community, we constructed finite element meshes for a group of 36 pyramidal neurons and a group of 29 spindle neurons whose morphological descriptions were found in the publicly available neuron repository NeuroMorpho.Org. These finite elements meshes range from having 15,163 nodes to 622,553 nodes. We also broke the neurons into the soma and dendrite branches and created finite elements meshes for these cell components. Through the Neuron Module, these neuron and cell components finite element meshes can be seamlessly coupled with the functionalities of SpinDoctor to provide the diffusion MRI signal attributable to spins inside neurons. We make these meshes and the source code of the Neuron Module available to the public as an open-source package.To illustrate some potential uses of the Neuron Module, we show numerical examples of the simulated diffusion MRI signals in multiple diffusion directions from whole neurons as well as from the soma and dendrite branches, and include a comparison of the high b-value behavior between dendrite branches and whole neurons. In addition, we demonstrate that the neuron meshes can be used to perform Monte-Carlo diffusion MRI simulations as well. We show that at equivalent accuracy, if only one gradient direction needs to be simulated, SpinDoctor is faster than a GPU implementation of Monte-Carlo, but if many gradient directions need to be simulated, there is a break-even point when the GPU implementation of Monte-Carlo becomes faster than SpinDoctor. Furthermore, we numerically compute the eigenfunctions and the eigenvalues of the Bloch-Torrey and the Laplace operators on the neuron geometries using a finite elements discretization, in order to give guidance in the choice of the space and time discretization parameters for both finite elements and Monte-Carlo approaches. Finally, we perform a statistical study on the set of 65 neurons to test some candidate biomakers that can potentially indicate the soma size. This preliminary study exemplifies the possible research that can be conducted using the Neuron Module.

Discontinuous Galerkin approximations for an optimal control problem of three-dimensional Navier–Stokes–Voigt equations

We analyze a fully discrete scheme based on the discontinuous (in time) Galerkin approach, which is combined with conforming finite element subspaces in space, for the distributed optimal control problem of the three-dimensional Navier–Stokes–Voigt equations with a quadratic objective functional and box control constraints. The space-time error estimates of order $$O(\sqrt{\tau }+h)$$ , where $$\tau $$ and h are respectively the time and space discretization parameters, are proved for the difference between the locally optimal controls and their discrete approximations.

Formal and Efficient Synthesis for Continuous-Time Linear Stochastic Hybrid Processes

Stochastic processes are expressive mathematical tools for modeling real-world systems that are subject to uncertainty. It is hence crucial to be able to formally analyze the behavior of these processes, especially in safety-critical applications. Most of the existing formal methods are not designed for continuous-time processes, and those that are typically suffering from state explosion in practice. This article introduces a theoretical framework and a scalable computational method for formal analysis and control synthesis for switched diffusions, a class of stochastic models with linear dynamics that are continuous in both time and space domains; the focus is on safety with possible extensions to other properties. The proposed framework first constructs a finite abstraction in the form of an uncertain Markov process through discretization of both time and space domains. The errors caused by the discretization in each domain are formally characterized and cast into the abstraction model. Then, a strategy that maximizes the probability of the safety property and is robust against the errors is synthesized over the abstraction model. Finally, this robust strategy is mapped to a switching strategy for the stochastic processes that guarantees the safety property. The framework is demonstrated in three case studies, including one that illustrates the tradeoff of the error contribution by the time and space discretization parameters.

Modelling coastal wave trains and wave breaking

The model of coastal waves based on the depth-averaging of the large eddy simulation equations (Kazakova and Richard, 2019) is extended to the case of regular and irregular wave trains. To take into account a stronger turbulence, the third moment of the horizontal velocity is modelled with a gradient-diffusion hypothesis. The effect of this new diffusive term is to smooth and regularize the solutions. An asymptotically equivalent model including the improvement of the dispersive properties is solved with a Discontinuous Galerkin numerical scheme. The model has a low sensitivity to the space discretization parameters. Several classical test cases of wave trains are used to validate the model. In the shoaling zone, the model is similar to the Serre–Green–Naghdi equations but the inclusion of a variable called enstrophy to take into account the large-scale turbulence and the non-uniformity of the mean velocity improves the predictive ability in the inner surf zone. In particular, the turbulent energy of the model is dissipated within one wave cycle and is transported shoreward in the case of waves with a long period whereas, in the case of short periods, it is mostly transported seaward because its dissipation is far from being complete within one period. The case of an irregular wave train propagating over a submerged bar is simulated without any breaking criterion. This benchmark test case validates further the model’s ability in predicting the nonlinear effects due to shoaling, breaking, propagation in a shallow horizontal part and in a deeper region.

Timing and Parameter Optimization for One-time Motion Problem Based on Reinforcement Learning

Baseball hitting, swatter swing and football catching, there are many tasks can be seen as a one-time action, whose goal is to control the timing and parameters of the action to achieve optimal results. Many one-time motion problems are difficult to obtain the optimal policy through model solving, and model-free reinforcement learning has advantages for such problems. However, although reinforcement learning has developed rapidly, there is currently no universal one-time motion problem algorithm architecture. Decomposing the one-time motion problem into the action timing problem and the action parameter problem, we construct a suitable reinforcement learning method for each of them. We design a combination mechanism that allows the two modules to learn simultaneously by passing the estimated value between the two modules while interacting with the environment. We use REINFORCE + DPG to solve the problem of continuous motion parameter space, and use REINFORCE + Q learning to solve the problem of discrete motion parameter space. To testing the algorithm model, we designed and realized an aircraft bombing simulation environment. The test results show that the algorithm can converge quickly and stably, and is robust to different time step and observation errors.

Boolean Kalman filter and smoother under model uncertainty

Partially-observed Boolean dynamical systems (POBDS) are a general class of nonlinear state-space models that provide a rich framework for modeling many complex dynamical systems. The model consists of a hidden Boolean state process, observed through an arbitrary noisy mapping to a measurement space. The optimal minimum mean-square error (MMSE) POBDS state estimators are the Boolean Kalman Filter and Smoother. However, in many practical problems, the system parameters are not fully known and must be estimated. In this paper, for POBDS under model uncertainty, we derive an optimal Bayesian estimator for state and parameter estimation. The exact algorithms are derived for the case of discrete and finite parameter space, and for general parameter spaces, an approximate Markov-Chain Monte-Carlo (MCMC) implementation is introduced. We demonstrate the performance of the proposed methodology by means of numerical experiments with POBDS models of gene regulatory networks observed through noisy measurements.

Solving hyperbolic-elliptic problems on singular mapped disk-like domains with the method of characteristics and spline finite elements

A common strategy in the numerical solution of partial differential equations is to define a uniform discretization of a tensor-product multi-dimensional logical domain, which is mapped to a physical domain through a given coordinate transformation. By extending this concept to a multi-patch setting, simple and efficient numerical algorithms can be employed on relatively complex geometries. The main drawback of such an approach is the inherent difficulty in dealing with singularities of the coordinate transformation.This work suggests a comprehensive numerical strategy for the common situation of disk-like domains with a singularity at a unique pole, where one edge of the rectangular logical domain collapses to one point of the physical domain (for example, a circle). We present robust numerical methods for the solution of Vlasov-like hyperbolic equations coupled to Poisson-like elliptic equations in such geometries. We describe a semi-Lagrangian advection solver that employs a novel set of coordinates, named pseudo-Cartesian coordinates, to integrate the characteristic equations in the whole domain, including the pole, and a finite element elliptic solver based on globally C1 smooth splines (Toshniwal et al., 2017 [27]). The two solvers are tested both independently and on a coupled model, namely the 2D guiding-center model for magnetized plasmas, equivalent to a vorticity model for incompressible inviscid Euler fluids. The numerical methods presented show high-order convergence in the space discretization parameters, uniformly across the computational domain, without effects of order reduction due to the singularity. Dedicated tests show that the numerical techniques described can be applied straightforwardly also in the presence of point charges (equivalently, point-like vortices), within the context of particle-in-cell methods.

Computer Model Calibration with Confidence and Consistency

SummaryThe paper proposes and examines a calibration method for inexact models. The method produces a confidence set on the parameters that includes the best parameter with a desired probability under any sample size. Additionally, this confidence set is shown to be consistent in that it excludes suboptimal parameters in large sample environments. The method works and the results hold with few assumptions; the ideas are maintained even with discrete input spaces or parameter spaces. Computation of the confidence sets and approximate confidence sets is discussed. The performance is illustrated in a simulation example as well as two real data examples.

Extracting recent short-term glacier velocity evolution over southern Alaska and the Yukon from a large collection of Landsat data

Abstract. The measurement of glacier velocity fields using repeat satellite imagery has become a standard method of cryospheric research. However, the reliable discovery of important glacier velocity variations on a large scale is still problematic because time series span different time intervals and are partly populated with erroneous velocity estimates. In this study we build upon existing glacier velocity products from the GoLIVE dataset (https://nsidc.org/data/golive, last access: 26 February 2019) and compile a multi-temporal stack of velocity data over the Saint Elias Mountains and vicinity. Each layer has a time separation of 32 days, making it possible to observe details such as within-season velocity change over an area of roughly 150 000 km2. Our methodology is robust as it is based upon a fuzzy voting scheme applied in a discrete parameter space and thus is able to filter multiple outliers. The multi-temporal data stack is then smoothed to facilitate interpretation. This results in a spatiotemporal dataset in which one can identify short-term glacier dynamics on a regional scale. The goal is not to improve accuracy or precision but to enhance extraction of the timing and location of ice flow events such as glacier surges. Our implementation is fully automatic and the approach is independent of geographical area or satellite system used. We demonstrate this automatic method on a large glacier area in Alaska and Canada. Within the Saint Elias and Kluane mountain ranges, several surges and their propagation characteristics are identified and tracked through time, as well as more complicated dynamics in the Wrangell Mountains.

Prior Distributions for Objective Bayesian Analysis

We provide a review of prior distributions for objective Bayesian analysis. We start by examining some foundational issues and then organize our exposition into priors for: i) estimation or prediction; ii) model selection; iii) high-dimensional models. With regard to i), we present some basic notions, and then move to more recent contributions on discrete parameter space, hierarchical models, nonparametric models, and penalizing complexity priors. Point ii) is the focus of this paper: it discusses principles for objective Bayesian model comparison, and singles out some major concepts for building priors, which are subsequently illustrated in some detail for the classic problem of variable selection in normal linear models. We also present some recent contributions in the area of objective priors on model space. With regard to point iii) we only provide a short summary of some default priors for high-dimensional models, a rapidly growing area of research.

An artificial bee colony optimization based matching pursuit approach for ultrasonic echo estimation

Ultrasonic echo estimation is important in ultrasonic non-destructive evaluation and material characterization. Matching pursuit is one of the most popular methods for the purpose of estimating ultrasonic echoes. In this paper, an artificial bee colony optimization based matching pursuit approach (ABC-MP) is proposed specifically for ultrasonic signal decomposition by integrating the artificial bee colony algorithm into the matching pursuit method. The optimal atoms are searched from a continuous parameter space over a tailored Gabor dictionary in ABC-MP instead of a discrete parameter space in matching pursuit. As a result, echoes characterized by a set of physical parameters can be estimated accurately and efficiently. The performance of ABC-MP is tested using both simulated signals and real ultrasonic signals, and compared with matching pursuit. Results clearly demonstrate the superior performance of the proposed ABC-MP approach over matching pursuit in ultrasonic echo estimation in terms of the shape and amplitude of the recovered echoes and the reconstructed signal, and the residue signal.