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Radio frequency ablation (RFA) has become a popular method for the minimally invasive treatment of liver cancer. However, the success rate of these treatments depends heavily on the amount of experience the clinician possesses. Mathematical modeling can help mitigate this problem by providing an indication of the treatment outcome. Thermal lesions in RFA are affected by the cooling effect of both fine-scale and large-scale blood vessels. The exact model for large-scale blood vessels is advection-diffusion, i.e., a model capable of producing directional effects, which are known to occur in certain cases. In previous research, in situations where directional effects do not occur, the advection term in the blood vessel model has been typically replaced with the Pennes perfusion term, albeit with a higher-than-usual perfusion rate. Whether these values of the perfusion rate appearing in literature are optimal for the particular vessel radii in question, has not been investigated so far. This work aims to address this issue. An attempt has been made to determine, for values of vessel radius between 0.55 mm and 5 mm, best estimates for the perfusion rate which minimize the error in thermal lesion volumes between the perfusion-based model and the advection-based model. The results for the best estimate of the perfusion rate presented may be used in existing methods for fast estimation of RFA outcomes. Furthermore, the possible improvements to the presented methodology have been highlighted.

Calibrating geothermal simulations is a critical step, both in scientific and industrial contexts, with suitable model parameterizations being optimized to reduce discrepancies between simulated and measured temperatures. Here we present a methodology to identify model errors in the calibration and compensate for measurement sparsity. With an application to the Upper Rhine Graben, we demonstrate the essential need for global sensitivity studies to robustly calibrate geothermal models, showing that local studies overestimate the influence of some parameters. We ensure the feasibility of the study through a physics-based machine learning approach (reduced basis method), reducing computation time by several orders of magnitude.

One of the biggest challenges in Computational Geosciences is finding ways of efficiently simulating high-dimensional problems. In this paper, we demonstrate how the RB method can be gainfully exploited to solve problems in the Geosciences. The reduced basis method constructs low-dimensional approximations to (high-dimensional) solutions of parametrized partial differential equations. In contrast to other widely used geoscientific reduction techniques, the reduced basis method reduces the Galerkin approximation space, and not the physical space and is consequently much less restrictive. Another advantage of the method is that for the problems considered in this paper, the method provides a bound to the error in the reduced order approximation, thus permitting an objective evaluation of the approximation quality. Using a geothermal conduction problem, we demonstrate that depending on the model, we obtain a maximum speed-up of three orders of magnitude with an approximation error that is very small in comparison with typical measurement errors. This significant reduction of the cost of the forward simulation allows performing uncertainty quantification, inversions, and parameter studies for larger and more complex models than currently possible.

Environmental earth sciences 81(3), 88 (2022). doi:10.1007/s12665-022-10202-5, Published by Springer, Berlin ; Heidelberg

Geophysical simulations are commonly used in many scientific studies. Especially in complex simulations, it is still common practice to provide a single deterministic outcome. Often, a probabilisti...

In Geosciences, we face the challenge of characterizing uncertainties to provide reliable predictions of the earth surface to allow, for instance, a sustainable and renewable energy management. In order, to address the uncertainties we need a good understanding of our geological models and their associated subsurface processes.Therefore, the essential pre-step for uncertainty analyses are sensitivity studies. Sensitivity studies aim at determining the most influencing model parameters. Hence, we require them to significantly reduce the parameter space to avoid unfeasibly large compute times.We distinguish two types of sensitivity analyses: local and global studies. In contrast, to the local sensitivity study, the global one accounts for parameter correlations. That is the reason, why we employ in this work a global sensitivity study. Unfortunately, global sensitivity studies have the disadvantage that they are computationally extremely demanding. Hence, they are prohibitive even for state-of-the-art finite element simulations.For this reason, we construct a surrogate model by employing the reduced basis method. The reduced basis method is a model order reduction technique that aims at significantly reducing the spatial and temporal degrees of freedom of, for instance, finite element solves. In contrast to other surrogate models, we obtain a surrogate model that preserves the physics and is not restricted to the observation space. As we will show, the reduced basis method leads to a speed-up of five to six orders of magnitude with respect to our original problem while retaining an accuracy higher than the measurement accuracy.In this work, we elaborate on the advantages of global sensitivity studies in comparison to local ones. We use several case studies, from large-scale European sedimentary basins to demonstrate how the global sensitivity studies are used to learn about the influence of transient, such as paleoclimate effects, and stationary effects. We also demonstrate how the results can be used in further analyses, such as deterministic and stochastic model calibrations. Furthermore, we show how we can use the analyses to learn about the subsurface processes and to identify model short comes.

Variational inequalities (VIs) are pervasive in mathematical modelling of equilibrium and optimization problems in engineering and science. Examples of applications include traffic network equilibrium problems, financial equilibrium, obstacle problems, lubrication phenomena and many others. Since these problems are computationally expensive to solve, we focus here on the development of model order reduction techniques, in particular the reduced basis technique. Reduced basis techniques for the approximation of solutions to elliptic VIs have been developed in the last few years. These methods apply to VIs of the so-called first kind, i.e. problems that can be equivalently described by a minimization of a functional over a convex set. However, these recent approaches are inapplicable to VIs of the so-called second kind, i.e. problems that involve minimization of a functional containing non-differentiable terms. In this article, we evaluate the feasibility of using the reduced basis method (RBM) comb...

We propose a novel approach for computing certified reduced-basis approximations to solutions to variational inequalities of the first kind. The proposed approach has three components: ( i ) a slack-based approximation for the solution; ( ii ) a primal approximation for the Lagrange multiplier; and ( iii ) a posteriori bounds for the error in the combined primal-slack variable approximation. The strict feasibility of the primal-slack approximations leads to two significant improvements upon existing methods. First, it provides a posteriori error bounds that are significantly sharper than existing bounds. Second, it enables a full offline–online computational decomposition, in which the online cost to compute the error bound is completely independent of the dimension of the original (high-dimensional) problem. Our numerical results allow us to compare the performance of the proposed and existing approaches.

Motivated by thermal ablation treatments for prostate cancer, the current work investigates the optimal delivery of heat in tissue. The problem is formulated as an optimal control problem constrained by a parametrized partial differential equation (PDE) which models the heat diffusion in living tissue. Geometry and material parameters as well as a parameter entering through the boundary condition are considered. Since there is a need for real-time solution of the treatment planning problem, we introduce a reduced order approximation of the optimal control problem using the reduced basis method. Numerical results are presented that highlight the accuracy and computational efficiency of our reduced model.

International audience; We propose a certified reduced basis approach for the strong-and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error. Our main contribution is the development of efficiently computable a posteriori upper bounds for the error of the reduced basis approximation with respect to the underlying high-dimensional 4D-Var problem. Numerical results are conducted to test the validity of our approach.

In this paper, we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalar coercive elliptic partial differential equations. We consider the standard linear-quadratic problem setting with distributed control and unilateral control constraints. For this problem class, we propose two different reduced basis approximations and associated error estimation procedures. In our first approach, we directly consider the resulting optimality system, introduce suitable reduced basis approximations for the state, adjoint, control, and Lagrange multipliers, and use a projection approach to bound the error in the reduced optimal control. For our second approach, we first reformulate the optimal control problem using a slack variable, then develop a reduced basis approximation for the slack problem by suitably restricting the solution space, and finally derive error bounds for the slack based optimal control. We discuss benefits and drawbacks of b...

In this paper, we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by elliptic partial differential equations. We consider the standard linear-quadratic problem setting with distributed control and two-sided control constraints, which play an important role in many industrial and economical applications. For this problem class, we propose two different reduced basis approximations and associated error estimation procedures. In our first approach, we directly consider the resulting optimality system, introduce suitable reduced basis approximations for the state, adjoint, control, and Lagrange multipliers, and use a projection approach to bound the error in the reduced optimal control. For our second approach, we first reformulate the optimal control problem using two slack variables, we then develop a reduced basis approximation for both slack problems by suitably restricting the solution space, and derive error bounds for the slack based optimal control. We discuss benefits and drawbacks of both approaches and substantiate the comparison by presenting numerical results for a model problem.

We present reduced basis approximations and associated rigorous a posteriori error bounds for the instationary Stokes equations. The proposed method is an extension of the penalty approach introduced in Gerner and Veroy (2011b) to the time-dependent setting: The introduction of a penalty term enables us to develop a posteriori error bounds that do not rely on the expensive calculation of inf-sup stability constants and that are thus computationally very efficient. Numerical results show the rapid convergence of reduced basis approximations and demonstrate the effects of the penalty parameter.

We introduce an ecient level set framework to parameter estimation problems governed by parametrized partial dierential equations. The main ingredients are: (i) an \admissible region" approach to parameter estimation; (ii) the certied reduced basis method for ecient and reliable solution of parametrized partial dierential equations; and (iii) a parameter-space level set method for construction of the admissible region. The method can handle nonconvex and multiply connected regions. Numerical results for two examples in design and inverse problems illustrate the versatility of the approach. To cite this article: M.A. Grepl, K. Veroy, C. R. Acad. Sci. Paris, Ser. I XXX (2011). R esum e

We present reduced basis approximations and associated rigorous a posteriori error bounds for the Stokes equations in parametrized domains. The method, built upon the penalty formulation for saddle point problems, provides error bounds not only for the velocity but also for the pressure approximation, while simultaneously admitting affine geometric variations with relative ease. The essential ingredients are: (i) dimension reduction through Galerkin projection onto a low-dimensional reduced basis space; (ii) stable, good approximation of the pressure through supremizer-enrichment of the velocity reduced basis space; (iii) optimal and numerically stable approximations identified through an efficient greedy sampling method; (iv) certainty, through rigorous a posteriori bounds for the errors in the reduced basis approximation; and (v) efficiency, through an offline-online computational strategy. The method is applied to a flow problem in a two-dimensional channel with a (parametrized) rectangular obstacle. Numerical results show that the reduced basis approximation converges rapidly, the effectivities associated with the (inexpensive) rigorous a posteriori error bounds remain good even for reasonably small values of the penalty parameter, and that the effects of the penalty parameter are relatively benign.

We present a technique for the evaluation of linear-functional outputs of parametrized elliptic partial differential equations in the context of deployed (in service) systems. Deployed systems require real-time and certified output prediction in support of immediate and safe (feasible) action. The two essential components of our approach are (i) rapidly, uniformly convergent reduced-basis approximations, and (ii) associated rigorous and sharp a posteriori error bounds; in both components we exploit affine parametric structure and oflline-online computational decompositions to provide real-time deployed response. In this paper we extend our methodology to the parametrized steady incompressible Navier-Stokes equations. We invoke the Brezzi-Rappaz-Raviart theory for analysis of variational approximations of non-linear partial differential equations to construct rigorous, quantitative, sharp, inexpensive a posteriori error estimators

Summarization: We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i ) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space W N spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii ) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive bounds for the error in the outputs of interest; and ( iii ) off-line/on-line computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage – in which, given a new parameter value, we calculate the output of interest and associated error bound – depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. In our earlier work we develop a rigorous a posteriori error bound framework for reduced-basis approximations of elliptic coercive equations. The resulting error estimates are, in some cases, quite sharp: the ratio of the estimated error in the output to the true error in the output, or effectivity , is close to (but always greater than) unity. However, in other cases, the necessary “bound conditioners” – in essence, operator preconditioners that (i ) satisfy an additional spectral “bound” requirement, and (ii ) admit the reduced-basis off-line/on-line computational stratagem – either can not be found, or yield unacceptably large effectivities. In this paper we introduce a new class of improved bound conditioners: the critical innovation is the direct approximation of the parametric dependence of the inverse of the operator (rather than the operator itself); we thereby accommodate higher-order (e.g., piecewise linear) effectivity constructions while simultaneously preserving on-line efficiency. Simple convex analysis and elementary approximation theory suffice to prove the necessary bounding and convergence properties. Presented on: ESAIM: Control, Optimisation and Calculus of Variations

Part 2 of this paper seeks the numerical and experimental procedures for implementing the Morlet wavelet transform for the spectrotemporal analysis of multimode dispersive waves. The sensitivity due to the spectral parameters α and ω0, defining the bandwidth and center frequency of the mother wavelet, is studied through an error analysis of the simulation signals. The algorithm for determining the group delays is extended to multimode signals by introducing a thresholding technique. The ultimate goal of this study is to nondestructively detect and locate discontinuities in a thin plate. This is demonstrated by experimentally measuring the travel distance of a Lamb wave from a single waveform.

In this paper, a wavelet-based time frequency analysis is presented to analyze guided-wave signals for rapid inspection of thin-walled structural members. The overall objective is to detect and locate discontinuities using a single broadband signal. Part 1 of this paper shows how the wavelet transform can be used to analyze a dispersive system. A straightforward procedure is developed to extract group delay information from the computed wavelet transform coefficients. The procedure is demonstrated by a simulation study for single-mode and simple dual-mode dispersion signals. Part 2 is an experimental study of multimode dispersion.

International audience; We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations--Galerkin projection onto a space $W_N$ spanned by solutions of the governing partial differential equation at $N$ selected points in parameter space; (ii) a posteriori error estimation--relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on $N$ (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

In this talk, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations involving constraints on the control. The reduced basis method is used as a low-dimensional surrogate model to solve the optimal control problem. To this end, we introduce reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable and propose rigorous error bounds for the error in the optimal control. The reduced basis optimal control problem and associated a posteriori error bounds can be efficiently evaluated in an offline-online computational procedure, thus making our approach relevant in the many-query or real-time context. We present numerical results for a model problem to show the validity of our approach.

We present reduced basis approximations and associated rigorous a posteriori error bounds for the Stokes equations in parametrized domains. The method is built upon a penalty formulation for saddle point problems and provides error bounds admitting affine geometric variations with relative ease. Essential features are: i) dimension reduction; ii) stable, good approximation of the pressure; iii) optimal and numerically stable approximations identified by an efficient Greedy sampling procedure; iv) certainty, through rigorous a posteriori bounds for the error in the reduced basis approximation; and v) efficiency, through an offline‐online computational strategy. The method is applied to a flow problem in a two‐dimensional channel with a parametrized rectangular obstacle. Numerical results show that the reduced basis approximation converges rapidly, the effectivities associated with the (inexpensive) rigorous a posteriori error bounds remain good even for reasonably small values of the penalty parameter.

We present an application of reduced basis method for Stokes equations in domains with affine parametric dependence. The essential components of the method are (i) the rapid convergence of global reduced basis approximations – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) the off-line/on-line computational procedures decoupling the generation and projection stages of the approximation process. The operation count for the on-line stage – in which, given a new parameter value, we calculate an output of interest – depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. Particular attention is given (i) to the pressure treatment of incompressible Stokes problem; (ii) to find an equivalent inf–sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem; (iii) to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions; (iv) to reduce computational costs in order to allow real-time solution of parametrized problem.