Discrete Entropy Research Articles

The Clinical Application of Remimazolam Benzenesulfonate Combined with Esketamine Intravenous Anesthesia in Endoscopic Retrograde Cholangiopancreatography.

In this project, algorithm-based image processing methods in 3D endoscopic image processing endoscopic retrograde cholangiopancreatography (ERCP) were analyzed. To enhance local information of images, an adaptive histogram equalization method with limited contrast is introduced. The influences of the algorithm on 3D endoscopic image peak signal-to-noise ratio (PSNR), image discrete information entropy (DE), and average mean brightness error (AMBE) of images before and after the optimization before were compared. A total of 92 patients receiving ERCP at Yuhuangding Hospital between December 2019 and December 2021 were selected and divided into the control group (fentanyl+propofol) and the observation group (remimazolam benzenesulfonate+esketamine). Mean arterial pressure heart rate (HR), oxygen saturation (SpO2), and respiratory rate (RR) of the patients at each time point including the entry into the operation room (T0), 2 minutes after the beginning of medication (T1), after endoscopy (T2), endoscopy withdrawal (T3), and postoperative awakening (T4) were recorded. The comparison of MAP between T1, T2, T3, and T4 and T0 among patients in the observation group and the control group showed statistical differences (P < 0.05). Besides, HR and RR at T4 in the observation group were obviously higher than those in the control group (P < 0.05). The comparison of SpO2 at T3 and T4 and that at T0 both showed statistical differences (P < 0.05). Awakening time and VAS scores in the observation group were obviously lower than those in the control group (P < 0.05). The incidence of bradycardia, nausea, vomiting, and chill in the observation group was all lower than that in the control group (P < 0.05). The results indicated that an effective endoscopic image processing method was established based on an image enhancement algorithm, and the combination of remimazolam benzenesulfonate and esketamine showed high safety and efficacy in ERCP.

Maximal overlap discrete wavelet packet transforms-based bipolar neutrosophic cross entropy measure for identification of rotor defects

This study propounds a novel methodology for automatic identification of rotor defects severity, when the machine is operated at constant speed, through maximal overlap discrete wavelet packet transforms (MODWPT) and proposed cross entropy measures of bipolar neutrosophic sets, single valued neutrosophic sets and fuzzy sets respectively. After the 3rd level of decomposition of raw vibration signals into eight frequency bands (called as WPT3 bands) through MODWPT, the energy reading in each WPT3 band is monitored. Thereafter, the lower (here called as truth membership degrees) and upper energy bounds from the training and testing samples of various rotor defect conditions at each WPT3 band are extracted and then normalized to lie in [0,1]. Finally, the energy intervals are constructed and thereafter converted into the desired forms of fuzzy sets, bipolar neutrosophic sets and single valued neutrosophic sets consecutively. The minimum cross entropy measure values between the fuzzy sets (Method 1), single valued neutrosophic sets (Method 2) and bipolar neutrosophic sets (Method 3) of training and testing samples of rotor defect conditions are computed and utilized to identify defect conditions such as angular/parallel misalignment of 10 Mils, unbalance, rub and defect free respectively. The novelty of the proposed rotor defect identification methodology lies in the fact that all the proposed variants of bipolar neutrosophic cross entropy measure have been found skilled enough to identify the desired rotor defects under study. On another hand, the existing cosine similarity measure [1] does not possess the necessary capability of identifying all rotor defects, especially, the rub defect condition. Furthermore, like the existing cosine similarity measure, our proposed variants of bipolar neutrosophic cross entropy measures have been found equally compatible for fairly furnishing the consistent and feasible results under intuitive analysis.

Positivity-preserving entropy stable schemes for the 1-D compressible Navier-Stokes equations: High-order flux limiting

This is the second of two companion papers, in which we develop and analyze a new class of positivity-preserving, entropy stable spectral collocation schemes of arbitrary order of accuracy for the 1-D compressible Navier-Stokes equations. The key distinctive property of the proposed methodology is that it is proven to guarantee the pointwise positivity of thermodynamic variables for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method developed in the companion paper [1]. This first-order scheme is discretized on the same Legendre-Gauss-Lobatto points used for the high-order counterpart. As a result, no interpolation or restriction is required between high- and low-order elements which are coupled by using the simultaneous approximation term (SAT) penalty method. To provide positivity preservation and excellent discontinuity-capturing properties, the Navier-Stokes equations are regularized by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The high- and low-order schemes are combined by using a limiting procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of thermodynamic variables, preserves the design–order of accuracy of the corresponding high-order baseline spectral collocation method for smooth solutions, and satisfies the discrete entropy inequality, thus facilitating a rigorous L2-stability proof for the symmetric form of the discretized Navier-Stokes equations. The proposed methodology is very general and can be extended to multiple dimensions and other summation-by-parts SAT-type schemes.

Positivity-preserving entropy stable schemes for the 1-D compressible Navier-Stokes equations: First-order approximation

This is the first of two companion papers that introduce a new class of positivity-preserving, entropy stable schemes for the 1-D compressible Navier-Stokes equations. In this paper, we develop and analyze a novel first–order finite volume scheme that is discretized on Legendre-Gauss-Lobatto grids used for high-order spectral collocation methods. The scheme is constructed by regularizing the Navier-Stokes equations by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The key distinctive property of the proposed scheme is that it is proven to satisfy the discrete entropy inequality and to guarantee the pointwise positivity of density, temperature, pressure, and internal energy for 1–D compressible viscous flows, while using high–order spectral collocation summation-by-parts operators for discretizing the Navier–Stokes viscous terms. To eliminate time step stiffness caused by the high-order approximation of the viscous terms, we develop two new limiters that bound the magnitude of velocity and temperature gradients and preserve the entropy stability and positivity properties of the baseline scheme. Numerical results demonstrating positivity-preserving and discontinuity-capturing capabilities of the new scheme are presented for viscous and inviscid flows with nearly vacuum regions and very strong shocks and contact discontinuities.

On Thermodynamically Compatible Finite Volume Schemes for Continuum Mechanics

In this paper we present a new family of semidiscrete and fully discrete finite volume schemes for overdetermined, hyperbolic, and thermodynamically compatible PDE systems. In the following, we will denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible gasdynamics as well as the more complex Godunov--Peshkov--Romenski (GPR) model of continuum mechanics, which, with the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first-order hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies in the fact that we solve the entropy inequality as a primary evolution equation rather than the usual total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations. For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Godunov's seminal paper [Dokl. Akad. Nauk SSSR, 139(1961), pp. 521--523] exactly at the discrete level. All other terms in the governing equations of the more general GPR model, including nonconservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations. As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction. We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing PDEs.

Variations on a Theme by Massey

In 1994, Jim Massey proposed the guessing entropy as a measure of the difficulty that an attacker has to guess a secret used in a cryptographic system, and established a well-known inequality between entropy and guessing entropy. Over 15 years before, in an unpublished work, he also established a well-known inequality for the entropy of an integer-valued random variable of given variance. In this paper, we establish a link between the two works by Massey in the more general framework of the relationship between discrete (absolute) entropy and continuous (differential) entropy. Two approaches are given in which the discrete entropy (or R&#x00E9;nyi entropy) of an integer-valued variable can be upper bounded using the differential (R&#x00E9;nyi) entropy of some suitably chosen continuous random variable. As an application, lower bounds on guessing entropy and guessing moments are derived in terms of entropy or R&#x00E9;nyi entropy (without side information) and conditional entropy or Arimoto conditional entropy (when side information is available).

Bi Histogram Equalization Based Image Enhancement with Bicubic Interpolation

In image processing, enhancement histogram equalization is the widely used technique for contrast enhancement. However, this technique tends to change the brightness of the image. Here, the contrast and resolution of image were enhanced using the proposed Bi Histogram Equalization Based Image Enhancement with Bicubic Interpolation (BHBI) technique. Bi Histogram for contrast enhancement and bicubic interpolation for resolution enhancement has taken. Bi Histogram Equalization separates the input image's histogram into two based on input mean before equalizing them independently. Bicubic Interpolation can generate bigger or high-resolution image from one or more low resolution or smaller images. The performance of the BHBI method can be compared for some typical image’s cameraman lens that are applied to some existing enhancement methods like Adaptive Gamma Correction with Weighting Distribution (AGCWD), Adaptive Scale Adjustment Design of Unsharp Masking Filters (ASAUMF), Averaging Histogram Equalization (AVGHEQ), and Median-Mean Based Sub-Image-Clipped Histogram Equalization (MMSICHE). The performance of these existing techniques can be evaluated subjectively in terms of person illustration observation and measurably using Discrete Entropy (DE), Image Quality Index (IQI), Normalized Correlation Coefficient (NCC), Contrast Improvement Index (CII), and Absolute Mean Brightness Error (AMBE). The results obtained from the BHBI technique shows better when compared with respect to the various existing techniques.

Energy relaxation approximation for compressible multicomponent flows in thermal nonequilibrium

This work concerns the numerical approximation with an explicit first-order finite volume method of inviscid, nonequilibrium, high-temperature flows in multiple space dimensions. It is devoted to the analysis of the numerical scheme for the approximation of the hyperbolic system in homogeneous form. We derive a general framework for the design of numerical schemes for this model from numerical schemes for the monocomponent compressible Euler equations for a polytropic gas. Under a very simple condition on the adiabatic exponent of the polytropic gas, the scheme for the multicomponent system enjoys the same properties as the one for the monocomponent system: discrete entropy inequality, positivity of the partial densities and internal energies, discrete maximum principle on the mass fractions, and discrete minimum principle on the entropy. Our approach extends the relaxation of energy (Coquel and Perthame in SIAM J. Numer. Anal. 35:2223–2249, 1998) to the multicomponent Euler system. In the limit of instantaneous relaxation we show that the solution formally converges to a unique and stable equilibrium solution to the multicomponent Euler equations. We then use this framework to design numerical schemes from three schemes for the polytropic Euler system: the Godunov exact Riemann solver, and the HLL and pressure relaxation based approximate Riemann solvers. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and convergence of the schemes.

A new discrete information model and its applications for the study of contingency tables

There exists an assortment of parametric and non-parametric discrete and continuous information models but still inevitability arises to enlarge supplementary parametric models to induce flexibility in the system under study. Moreover, there appears well-built relation between information entropy and the field of statistics. The present communiqué deals with the proposal of a new discrete information entropy model along with its applications in statistics. By employing the newly developed entropy model, we have enriched the application region of maximum entropy principle after the learning of contingency tables recurrently employed in statistics.

Dealing with Uncertainty in the MRCPSP/Max Using Discrete Differential Evolution and Entropy

In this paper, we investigate the characterization of MRCPSP/max under uncertainty conditions and emphasize managerial ability to recognize and handle positively disruptive events. This proposition is then demonstrated using the entropy approach to find disruptive events and response time intervals. The problem is solved using a resilient characteristic of the three-stage procedure gauged by schedule robustness and adaptivity; the resulting schedule absorbs the impact of an unexpected event without rescheduling during execution. The use of the differential evolution algorithm, known as DDE, in a discrete manner is proposed and evaluated against the best known optima (BKO). Our findings indicate the DDE is effective overall; moreover, compared against the BKO for every stage, the most significant difference is that the stability of the solutions provided by DDE under the three-stage framework proves to be sufficiently robust when practitioners add response times at certain range levels, in this case from 8% to 15%.

An optimized morphology transform‐based diagnostic computed tomography image enhancement using edge map

AbstractIn computed tomography (CT) imaging, a low‐quality CT scan, or more precisely, a noisy low contrast CT image, may provide insufficient information for visual interpretation of impacted regions. The purpose of this article is to propose a novel optimal morphological transform‐based method for improving the noise and contrast of CT images in the wavelet domain. The low‐quality CT image is first transformed using the optimized morphology transform method, which determines the optimal value of the parameter included in the fundamental morphology transformation equation. The parameter's optimum value is determined using the particle swarm optimization (PSO) method. The proposed approach then employs the discrete wavelet transform (DWT) to decompose the input and transformed CT image into higher and lower sub‐bands. Following that, the lower sub‐bands are modified by applying the correction factor via singular value decomposition (SVD). The input image's higher sub‐band (HH) is denoised using an edge map (EM)‐based method. This produces enhanced CT images with modified lower and higher sub‐band. Finally, the modified sub‐bands and remaining unprocessed higher frequency sub‐bands are processed using the inverse discrete wavelet transform (IDWT), resulting in an enhanced CT image. Experiments and validations are conducted on a CT images obtained from a publicly available database to assess the proposed scheme's effectiveness qualitatively and quantitatively. Significant quantitative study shows that the proposed approach surpasses existing image enhancement approaches in terms of signal‐to‐noise ratio, discrete entropy, enhancement measurement, and contrast ratio. The proposed method generates higher‐quality CT images, which is advantageous for disease inspection and diagnosis.

Entropy Stable Flux Correction for Scalar Hyperbolic Conservation Laws

It is known that Flux Corrected Transport algorithms can produce entropy-violating solutions of hyperbolic conservation laws. Our purpose is to design flux correction with maximal antidiffusive fluxes to obtain entropy solutions of scalar hyperbolic conservation laws. To do this we consider a hybrid difference scheme that is a linear combination of a monotone scheme and a scheme of high-order accuracy. The flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the monotone scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution of scalar hyperbolic conservation laws. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. We present examples that show that not all numerical entropy fluxes guarantee to single out a physically correct solution of scalar hyperbolic conservation laws.

Entropy-stable schemes in the low-Mach-number regime: Flux-preconditioning, entropy breakdowns, and entropy transfers

Entropy-Stable (ES) schemes, specifically those built from Tadmor (1987) [54], have been gaining interest over the past decade, especially in the context of under-resolved simulations of compressible turbulent flows using high-order methods. These schemes are attractive because they can provide stability in a global and nonlinear sense (consistency with thermodynamics). However, fully realizing the potential of ES schemes requires a better grasp of their local behavior. Entropy-stability itself does not imply good local behavior (Gouasmi et al., 2019 [17], Gouasmi et al., 2020 [18]). In this spirit, we studied ES schemes in problems where global stability is not the core issue. In the present work, we consider the accuracy degradation issues typically encountered by upwind-type schemes in the low-Mach-number regime (Turkel, 1999 [59]) and their treatment using Flux-Preconditioning (Turkel, 1987 [58], Miczek et al., 2015 [41]). ES schemes suffer from the same issues and Flux-Preconditioning can improve their behavior without interfering with entropy-stability. This is first demonstrated analytically: using similarity and congruence transforms we were able to establish conditions for a preconditioned flux to be ES, and introduce the ES variants of the Miczek's and Turkel's preconditioned fluxes. This is then demonstrated numerically through first-order simulations of two simple test problems representative of the incompressible and acoustic limits, the Gresho Vortex and a right-moving acoustic wave. The results are overall consistent with previous studies. For instance, we observe that Turkel's preconditioner improves accuracy in the incompressible limit with the downside of overly damping acoustic waves (Bruel et al., 2019 [4]). For Miczek's matrix however, we came across unexpected spurious transients in both problems (a small left-moving acoustic wave in the latter), motivating further analysis. We revisited the pressure fluctuation argument of Guillard and Viozat (1999) [26] in terms of entropy, showing how the standard ES dissipation operator (Ismail and Roe, 2009 [30]) can introduce inconsistent discrete entropy fluctuations in space. These analytical results are achieved by introducing mode-by-mode decompositions of the dissipation operator, similar to Roe and Pike (1984) [51], Tadmor (2003) [55], leading to what we call discrete Entropy Production Breakdowns (EPBs) in space. These EPBs outline the contributions of convective and acoustic modes to the discrete entropy production, both locally and globally. Ultimately, these EPBs enable us to single-out in the ES Miczek flux a skew-symmetric matrix component which we believe causes entropy transfers between acoustic waves. Removing this contribution eliminates the spurious transients without interfering with entropy-stability. This conjecture is explored numerically and analytically.

Restoration of Tc-99m methyl diphosphonate bone scan image using Richardson-Lucy algorithm.

In this study, the optimal input parameters point spread function (PSF) and the number of iterations of the Richardson-Lucy algorithm were experimentally determined to restore Tc-99 m methyl diphosphonate (MDP) whole-body bone scan images. The experiment was performed on 60 anonymized Tc-99 m MDP whole-body bone scan images. Ten images were used for estimating the optimum value of PSF and the number of iterations to restore scintigraphic images. The remaining 50 images were used for validation of estimated parameters. The image quality of observed and restored images was assessed objectively using blind/referenceless image spatial quality evaluator (BRISQUE), mean brightness (MB), discrete entropy (DE), and edge-based contrast measure (EBCM) image quality metrics. Image quality was subjectively assessed by two nuclear medicine physicians (NMPs) by comparing the restored image quality with observed image quality and assigning a score to each image on the scale of 0-5. Based on BRISQUE, MB, DE, and EBCM scores, the restored images were significantly sharper, less bright, had more detailed information, and had less contrast around edges compared to the input images. The restored images had improved resolution based on visual assessment as well; NMPs assigned an average image quality score of 4.00 to restored images. Maximum resolution enhancement was noticed at PSF (size: 11 pixels, sigma: 1.75 pixels) and the number of iterations = 10. With the increase in the number of iterations, noise also gets amplified along with resolution enhancement and affects the detectability of small lesions; in the case of relatively low noisy input images, the number of iterations = 5 gave better results. Tc-99 m MDP bone scan images were restored to improve image quality using the Richardson-Lucy algorithm. The optimum value of the PSF parameter was found to be of size = 11 pixels and sigma = 1.75 pixels.

Multi-exposure optimized contrast and brightness balance color image enhancement

Color image enhancement has been very challenging for researchers in imaging systems. In this paper, optimal weighted multi-exposure histogram equalization model is proposed for image contrast enhancement. The proposed method divides the input image into two sub-images, specifically low-exposed and high-exposed regions. Then, exposure information obtained from these sub-images is used to divide the input histogram into sub-histograms, and the level of excessive enhancement is controlled by clipping and the optimal weighting process before applying histogram equalization. Next, each sub-histogram is equalized individually and integrated to preserve the naturalness and fine details of the input image. Finally, dual gamma correction is adopted to improve the contrast in dark areas. In the proposed model, a metaheuristic stud krill herd optimization algorithm is employed to determine the optimal constraints to optimize the degree of enhancement based on the objective function defined as discrete entropy or mean square error. Experimental results prove that the proposed technique produces a better visual quality image with an attractive color tone and improved natural appearance without introducing artifacts.

Spatially adaptive multi-scale image enhancement based on nonsubsampled contourlet transform

Low or uneven luminance results in low contrast of near-infrared and optical remote sensing images, making it challenging to analyze their contents. Traditional image enhancement methods cannot simultaneously take detail preservation, contrast enhancement, and brightness improvement into account. In order to cope with this problem, this paper proposes a spatially adaptive multi-scale image enhancement (SAMSIE) scheme, including three key procedures: First, nonsubsampled contourlet transform (NSCT) is employed to decompose a low-contrast image into multi-scale layers. Second, a spatially adaptive Gamma correction strategy based on improved histogram equalization is proposed to enhance the base layer which is used as a guide layer. Third, an adaptive enhancement operator is proposed to enhance fine details. Finally, a high-contrast optical infrared image is obtained by the inverse NSCT with usage of these enhanced layers. The effectiveness of the proposed SAMSIE method is validated by both visualization assess and the evaluation of three quantitative indexes including discrete entropy (DE), contrast gain (CG), and mean brightness improvement (MBI), with comparison of the state-of-the-arts.

Reinterpretation and extension of entropy correction terms for residual distribution and discontinuous Galerkin schemes: Application to structure preserving discretization

For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/ dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall ((2018) [1]). In this work, the correction terms are characterized as solutions of certain optimization problems and are adapted to the SBP-SAT framework, focusing on discontinuous Galerkin methods. Novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. For all of these optimization problems, explicit solutions are provided. Additionally, the correction approach is applied for the first time to obtain a fully discrete entropy conservative/dissipative RD scheme. Here, the application of the deferred correction (DeC) method for the time integration is essential. This paper can be seen as describing a systematic method to construct structure preserving discretization, at least for the considered example.

A discrete boundedness-by-entropy method for finite-volume approximations of cross-diffusion systems

Abstract An implicit Euler finite-volume scheme for general cross-diffusion systems with volume-filling constraints is proposed and analyzed. The diffusion matrix may be nonsymmetric and not positive semidefinite, but the diffusion system is assumed to possess a formal gradient-flow structure that yields $L^\infty $ bounds on the continuous level. Examples include the Maxwell–Stefan systems for gas mixtures, tumor-growth models and systems for the fabrication of thin-film solar cells. The proposed numerical scheme preserves the structure of the continuous equations, namely the entropy dissipation inequality as well as the non-negativity of the concentrations and the volume-filling constraints. The discrete entropy structure is a consequence of a new vector-valued discrete chain rule. The existence of discrete solutions, their positivity, and the convergence of the scheme is proved. The numerical scheme is implemented for a one-dimensional Maxwell–Stefan model and a two-dimensional thin-film solar cell system. It is illustrated that the convergence rate in space is of order two and the discrete relative entropy decays exponentially.

Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems

The algebraic flux correction (AFC) schemes presented in this work constrain a standard continuous finite element discretization of a nonlinear hyperbolic problem to satisfy relevant maximum principles and entropy stability conditions. The desired properties are enforced by applying a limiter to antidiffusive fluxes that represent the difference between the high-order baseline scheme and a property-preserving approximation of Lax–Friedrichs type. In the first step of the limiting procedure, the given target fluxes are adjusted in a way that guarantees preservation of local and/or global bounds. In the second step, additional limiting is performed, if necessary, to ensure the validity of fully discrete and/or semi-discrete entropy inequalities. The limiter-based entropy fixes considered in this work are applicable to finite element discretizations of scalar hyperbolic equations and systems alike. The underlying inequality constraints are formulated using Tadmor’s entropy stability theory.The proposed limiters impose entropy-conservative or entropy-dissipative bounds on the rate of entropy production by antidiffusive fluxes and Runge–Kutta (RK) time discretizations. Two versions of the fully discrete entropy fix are developed for this purpose. The first one incorporates temporal entropy production into the flux constraints, which makes them more restrictive and dependent on the time step. The second algorithm interprets the final stage of a high-order AFC-RK method as a constrained antidiffusive correction of an implicit low-order scheme (algebraic Lax–Friedrichs in space + backward Euler in time). In this case, iterative flux correction is required, but the inequality constraints are less restrictive and limiting can be performed using algorithms developed for the semi-discrete problem. To motivate the use of limiter-based entropy fixes, we prove a finite element version of the Lax–Wendroff theorem and perform numerical studies for standard test problems. In our numerical experiments, entropy-dissipative schemes converge to correct weak solutions of scalar conservation laws, of the Euler equations, and of the shallow water equations.

Numerical analysis of DDFV schemes for semiconductors energy-transport models

This article addresses the construction and the numerical analysis of implicit Discrete Duality Finite Volume schemes for a semiconductors’ energy-transport model. The considered energy-transport model is presented in its scaled version as well as in a symmetrized form which involves entropy variables. We propose implicit in time numerical schemes for both the original system and its symmetrized form. As in the continuous framework, the numerical analysis is based on the reformulation of the PDE system using the set of entropic variables. The equivalence of both schemes allows to establish a discrete entropy inequality and consequently an priori estimates. As a by-product, existence of solutions to the schemes is proved by means of a Leray–Schauder argument. Numerical evidences allow to compare the performances of both schemes on the test case of a 2D ballistic diode.