TMsim : an algorithmic tool for the parametric and worst-case simulation of systems with uncertainties

This paper presents a general purpose, algebraic tool—named TMsim—for the combined parametric and worst-case analysis of systems with bounded uncertain parameters.The tool is based on the theory of Taylor models and represents uncertain variables on a bounded domain in terms of a Taylor polynomial plus an interval remainder accounting for truncation and round-off errors.This representation is propagated from inputs to outputs by means of a suitable redefinition of the involved calculations, in both scalar and matrix form. The polynomial provides a parametric approximation of the variable, while the remainder gives a conservative bound of the associated error. The combination between the bound of the polynomial and the interval remainder provides an estimation of the overall (worst-case) bound of the variable. After a preliminary theoretical background, the tool (freely available online) is introduced step by step along with the necessary theoretical notions. As a validation, it is applied to illustrative examples as well as to real-life problems of relevance in electrical engineering applications, specifically a quarter-car model and a continuous time linear equalizer

Application of Taylor models to the worst-case analysis of stripline interconnects

This paper outlines a preliminary application of Taylor models to the worst-case analysis of transmission lines with bounded uncertain parameters. Taylor models are an algebraic technique that represents uncertain quantities in terms of a Taylor expansion complemented by an interval remainder encompassing approximation and truncation errors. The Taylor model formulation is propagated from input uncertainties to output responses through a suitable redef nition of the algebraic operations involved in their calculation. While the Taylor expansion def nes an analytical and parametric model of the response, the remainder provides a conservative bound inside which the true value is guaranteed to lie. The approach is validated against the SPICE simulation of a coupled stripline and shows promising accuracy and eff ciency

Statistical crosstalk analysis via probabilistic machine learning surrogates

This paper discusses the application of a probabilistic surrogate modeling technique, based on Gaussian process regression (GPR), to the uncertainty quantification (UQ) of crosstalk. Compared to traditional deterministic surrogate models, the GPR provides a stochastic process that carries an estimate of the model uncertainty. This allows assigning confidence bounds to the model prediction and, in an UQ scenario, to statistical estimates. The advocated method is illustrated through its application to a literature test case

Bayesian estimation of the transmissivity spatial structure from pumping test data

Estimating the statistical parameters (mean, variance, and integral scale) that define the spatial structure of the transmissivity or hydraulic conductivity fields is a fundamental step for the accurate prediction of subsurface flow and contaminant transport. In practice, the determination of the spatial structure is a challenge because of spatial heterogeneity and data scarcity. In this paper, we describe a novel approach that uses time drawdown data from multiple pumping tests to determine the transmissivity statistical spatial structure. The method builds on the pumping test interpretation procedure of Copty et al. (2011) (Continuous Derivation method, CD), which uses the time-drawdown data and its time derivative to estimate apparent transmissivity values as a function of radial distance from the pumping well. A Bayesian approach is then used to infer the statistical parameters of the transmissivity field by combining prior information about the parameters and the likelihood function expressed in terms of radially-dependent apparent transmissivities determined from pumping tests. A major advantage of the proposed Bayesian approach is that the likelihood function is readily determined from randomly generated multiple realizations of the transmissivity field, without the need to solve the groundwater flow equation. Applying the method to synthetically-generated pumping test data, we demonstrate that, through a relatively simple procedure, information on the spatial structure of the transmissivity may be inferred from pumping tests data. It is also shown that the prior parameter distribution has a significant influence on the estimation procedure, given the non-uniqueness of the estimation procedure. Results also indicate that the reliability of the estimated transmissivity statistical parameters increases with the number of available pumping tests.Peer ReviewedPostprint (author's final draft

Characterization of Spatial Heterogeneity in Groundwater Applications

La heterogeneidad es una característica saliente de cada formación geológica natural. En las últimas décadas un gran número de trabajos se han centrado en estudiar la influencia de la heterogeneidad en los problemas de flujo y transporte en medios poros. Dichos trabajos han mejorado sustancialmente la comprensión de los mecanismos que gobiernan estos fenómenos, sin embargo aun existen carencias en la caracterización de propiedades importantes de un acuífero. Entre ellas, el flujo preferencial, la conectividad entre dos puntos de un acuífero y la interpretación de ensayos en acuíferos heterogéneos son aspectos importantes que tienen que ser adecuadamente evaluados para obtener modelos predictivos robustos. En este contexto, el objetivo de esta tesis es doble:· mejorar el conocimiento de la influencia de la heterogeneidad en los procesos de flujo y transporte· proporcionar nuevas herramientas para caracterizar la heterogeneidad de los acuíferosEn primer lugar, evaluamos la relación entre dos indicadores de conectividad de flujo y transporte. El indicador de conectividad de flujo utilizado aquí se basa en el tiempo de respuesta hidráulica en un ensayo de bombeo (por ejemplo, el coeficiente de almacenamiento estimado con el método de Cooper-Jacob). Como indicador de conectividad de transporte utilizamos la porosidad estimada de curvas de llegadas (Φest) en ensayos de trazadores con flujo radial. Los resultados obtenidos permiten explicar la poca correlación entre estos dos indicadores ya observada númericamente por Knudby y Carrera (2005).En segundo lugar, se ha desarrollado un marco geoestádistico para delinear mapas de conectividad utilizando un número limitado de medidas hidráulicas. La metodología permite condicionar los resultados a tres tipos de datos medidos a diferentes escalas: (a) tiempos de llegada de ensayos de trazadores en flujo convergente ta, (b) estimaciones del coeficiente de almacenamiento obtenidas de ensayos de bombeo interpretados con el método de Cooper-Jacob, Sest, y (c) mediciones de valores de transmisividad puntuales, T. La capacidad de la metodología para delinear correctamente zonas de captura se evalúa a través de estimaciones (krigeado simple y ordinario) y simulaciones secuenciales gausianas basadas en diferentes conjuntos de medidas.En tercer lugar, se ha desarrollado una metodología, llamada método del doble punto de inflexión (DIP), para la interpretación de pruebas de bombeo en acuíferos semiconfinados. La ventaja de este método (DIP) se observa en acuíferos heterogéneos, cuando se aplica junto a las demás metodologías. En estos casos cada método proporciona información diferente acerca de la heterogeneidad. En particular, la combinación del método DIP y el método de Hantush permite identificar contrastes entre la transmisividad local (cerca del pozo) y la transmisividad equivalente del acuífero.En cuarto lugar, se utiliza el método de Monte Carlo para evaluar el significado de los parámetros hidráulicos estimados de ensayos de bombeo en acuíferos semiconfinados. Dichos parámetros resultan ser espacialmente dependientes y varían en función del método de interpretación, ya que cada método pondera de forma distinta diferentes partes de la curva de descensos. Por último, se muestra que al combinar las estimaciones de los parámetros obtenidos a partir de los diferentes procedimientos de análisis, se puede inferir información sobre la heterogeneidad del sistema.En quinto lugar, se modela el flujo no saturado en un dique de estériles utilizando un modelo basado en funciones de transferencia. La calibración del modelo proporciona información sobre el tiempo característico del flujo a través de la matriz y de la fracción de agua que se canaliza a través de los macroporos. Por último se presenta un análisis de la influencia de la escala en los resultados. A gran escala el comportamiento del sistema tiende al de un sistema-matriz homogéneo equivalente,enmascarando los efectos del flujo preferencial.Heterogeneity is a salient feature of every natural geological formation. In the past decades a large body of literature has focused on the effects of heterogeneity on flow and transport problems. These works have substantially improved the understanding of flow and transport phenomena but still fail to characterize many of the important features of an aquifer. Among them, preferential flows and solute paths, connectivity between two points of an aquifer, and interpretation of hydraulic and tracer tests in heterogeneous media are crucial points that need to be properly assessed to obtain accurate model predictions. In this context, the aim of this thesis is twofold:· to improve the understanding of the effects of heterogeneity on flow and transport phenomena· to provide new tools for characterizing aquifer heterogeneityFirst, we start by theoretically and numerically examine the relationship between two indicators of flow and transport connectivity. The flow connectivity indicator used here is based on the time elapsed for hydraulic response in a pumping test (e.g., the storage coefficient estimated by the Cooper-Jacob method, Sest). Regarding transport, we select the estimated porosity from the observed breakthrough curve (Φ est) in a forced-gradient tracer test. Our results allow explaining the poor correlation between these two indicators, already observed numerically by Knudby and Carrera (2005).Second, a geostatistical framework has been developed to delineate connectivity patterns using a limited and sparse number of measurements. The methodology allows conditioning the results to three types of data measured over different scales, namely: (a) travel times of convergent tracer tests, ta, (b) estimates of the storage coefficient from pumping tests interpreted using the Cooper-a Jacob method, S est, and (c) measurements of transmissivity point values, T. The ability of the methodology to properly delineate capture zones is assessed through estimations (i.e. ordinary cokriging) and sequential gaussian simulations based on different sets of measurements.Third, a novel methodology for the interpretation of pumping tests in leaky aquifer systems, referred to as the double inflection point (DIP) method, is presented. The real advantage of the DIP method comes when it is applied with all the existing methods independently to a test in a heterogeneous aquifer. In this case each method yields parameter values that are weighted differently, and thus each method provides different information about the heterogeneity distribution. In particular, the combination of the DIP method and Hantush method is shown to lead to the identification of contrasts between the local transmissivity in the vicinity of the well and the equivalent transmissivity of the perturbed aquifer volume.Fourth, the meaning of the hydraulic parameters estimated from pumping test performed in leaky aquifers is assessed numerically within a Monte Carlo framework. A synthetic pumping test is interpreted using three existing methods. The resulting estimated parameters are shown to be space dependent and vary with the interpretation method, since each method gives different emphasis to different parts of the timedrawdown data. Finally, we show that by combining the parameter estimates obtained from the different analysis procedures, information about the heterogeneity of the leaky aquifer system may be inferred.Fifth, an unsaturated highly heterogeneous waste rock pile is modeled using a simple linear transfer function (TF) model. The calibration of the parametric model provides information on the characteristic time of the flow through the matrix and on the fraction of the water that, within each section, is channeled through the macropores. An analysis of the influence of the scale on the results is also provided showing that at large scales the behavior of the system tends to that of an equivalent matrix reservoir masking the effects of preferential flow

Efficient Implementation of the Vector-Valued Kernel Ridge Regression for the Uncertainty Quantification of the Scattering Parameters of a 2-GHz Low-Noise Amplifier

This paper focuses on the application of an efficient implementation of the vector-valued kernel Ridge regression (KRR) to the uncertainty quantification (UQ) of the scattering parameters of a low-noise amplifier (LNA). Specifically, the performance of the proposed technique have been investigated for the statistical assessment of the mean value, variance and probability density function (PDF) of the S11 and S21 parameters of a 2-GHz LNA induced by 25 stochastic input parameters and compared with the corresponding reference results computed via a plain Monte Carlo (MC) simulation

A Perturbative Stochastic Galerkin Method for the Uncertainty Quantification of Linear Circuits

This paper presents an iterative and decoupled perturbative stochastic Galerkin (SG) method for the variability analysis of stochastic linear circuits with a large number of uncertain parameters. State-of-the-art implementations of polynomial chaos expansion and SG projection produce a large deterministic circuit that is fully coupled, thus becoming cumbersome to implement and inefficient to solve when the number of random parameters is large. In a perturbative approach, component variability is interpreted as a perturbation of its nominal value. The relaxation of the resulting equations and the application of a SG method lead to a decoupled system of equations, corresponding to a modified equivalent circuit in which each stochastic component is replaced by the nominal element equipped with a parallel current source accounting for the effect of variability. The solution of the perturbation problem is carried out in an iterative manner by suitably updating the equivalent current sources by means of Jacobi- or Gauss-Seidel strategies, until convergence is reached. A sparse implementation allows avoiding the refinement of negligible coefficients, yielding further efficiency improvement. Moreover, for time-invariant circuits, the iterations are effectively performed in post-processing after characterizing the circuit in time or frequency domain by means of a limited number of simulations. Several application examples are used to illustrate the proposed technique and highlight its performance and computational advantages

Inferring spatial distribution of the radially integrated transmissivity from pumping tests in heterogeneous confined aquifers

Hydrologists routinely analyze pumping test data using conventional interpretation methods that are based on the assumption of homogeneity and that, consequently, yield single estimates of representative flow parameters. However, natural subsurface formations are intrinsically heterogeneous, and hence, the flow parameters influencing the drawdown vary as the cone of depression expands in time. In this paper a novel procedure for the analysis of pumping tests in heterogeneous confined aquifers is developed. We assume that a given heterogeneous aquifer can be represented by a homogeneous system whose flow parameters evolve in time as the pumping test progresses. At any point in time, the interpreted flow parameters are estimated using the ratio of the drawdown and its derivative observed at that particular time. The procedure is repeated for all times, yielding time‐dependent estimates of transmissivity Ti(t) and storativity, Si(t). Based on the analysis of the sensitivity of drawdown to inhomogeneities in the T field, the time‐dependent interpreted transmissivity values are found to be a good estimate of Tg(r), the geometric mean of the transmissivity values encompassed within a progressively increasing radius r from the well. The procedure is illustrated for Gaussian heterogeneous fields with ln(T) variances up to a value of 2. The impact of the separation distance between the pumping well and observation point on data interpretation is discussed. The results show that information about the spatial variability of the transmissivity field can be inferred from time‐drawdown data collected at a single observation poin

Steady-state analysis of switching converters via frequency-domain circuit equivalents

This brief presents a frequency-domain approach for the steady-state analysis of pulsewidth-modulated converters and switched circuits with nonideal switching behavior. The proposed strategy generalizes recent methodologies based on the Fourier expansion of the steady-state responses of a periodically switching circuit and on the simulation of an augmented linear-time-invariant system. This system is now also given an interpretation in terms of an equivalent circuit, which is simulated at a single frequency point to solve for all the harmonics. The method offers a modular topological approach that is combined with standard tools for circuit analysis and enables the simulation of networks with an arbitrary number of switches and driving mechanisms. Single, multiple, and possibly nonideal commutation events within the switching period are handled in the same framework, without additional complexity. The technique allows for the full frequency-domain characterization of both the functional and the noisy behavior of the circuit responses. The feasibility and strength are demonstrated via comparisons with simulations and measurements on two application examples, i. e., a full-bridge single-phase inverter and a dc-dc boost converter