Goal-Oriented Adaptivity using Unconventional Error Representations

In Goal-Oriented Adaptivity (GOA), the error in a Quantity of Interest (QoI) is represented using global error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient GOA. These representations can be employed to design novel h, p, and hp energy-norm and goal-oriented adaptive algorithms. While the method can be applied to a variety of problems, in this Dissertation we first focus on one-dimensional (1D) problems, including Helmholtz and steady-state convection-dominated diffusion problems. Numerical results in 1D show that for the Helmholtz problem, it is advantageous to select the Laplace operator for the alternative error representation. Specifically, the upper bounds of the new error representation are sharper than the classical ones used in both energy-norm and goal-oriented adaptive methods, especially when the dispersion (pollution) error is significant. The 1D steady-state convection-dominated diffusion problem with homogeneous Dirichlet boundary conditions exhibits a boundary layer that produces a loss of numerical stability. The new error representation based on the Laplace operator delivers sharper error upper bounds. When applied to a p-GOA, the alternative error representation captures earlier the boundary layer, despite the existing spurious numerical oscillations. We then focus on the two- and three-dimensional (2D and 3D) Helmholtz equation. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones. When using the alternative error indicators, a naive p-adaptive process converges, whereas under the same conditions, the classical method fails and requires the use of the so-called Projection Based Interpolation (PBI) operator or some other technique to regain convergence. We also provide guidelines for finding operators delivering sharp error representation upper bounds.Basque Government Consolidated Research Group Grant IT649-1

Goal-Oriented p-Adaptivity using Unconventional Error Representations for a 1D Steady State Convection-Diffusion Problem

This work proposes the use of an alternative error representation for Goal-Oriented Adaptivity (GOA) in context of steady state convection dominated diffusion problems. It introduces an arbitrary operator for the computation of the error of an alternative dual problem. From the new representation, we derive element-wise estimators to drive the adaptive algorithm. The method is applied to a one dimensional (1D) steady state convection dominated diffusion problem with homogeneous Dirichlet boundary conditions. This problem exhibits a boundary layer that produces a loss of numerical stability. The new error representation delivers sharper error bounds. When applied to a $p$-GOA Finite Element Method (FEM), the alternative error representation captures earlier the boundary layer, despite the existing spurious numerical oscillations.Basque Government Consolidated Research Group Grant IT649-13 Spanish Ministry under Grant No. FPDI- 2013-17098 ICERMAR Project KK-2015/0000097 CYTED 2011 project 712RT0449 FONDECYT project 116077

A painless multi-level automatic goal-oriented hp-adaptive coarsening strategy for elliptic and non-elliptic problems

This work extends an automatic energy-norm $hp$-adaptive strategy based on performing quasi-optimal unrefinements to the case of non-elliptic problems and goal-oriented adaptivity. The proposed approach employs a multi-level hierarchical data structure and alternates global $h$- and $p$-refinements with a coarsening step. Thus, at each unrefinement step, we eliminate the basis functions with the lowest contributions to the solution. When solving elliptic problems using energy-norm adaptivity, the removed basis functions are those with the lowest contributions to the energy of the solution. For non-elliptic problems or goal-oriented adaptivity, we propose an upper bound of the error representation expressed in terms of an inner product of the specific equation, leading to error indicators that deliver quasi-optimal $hp$-unrefinements. This unrefinement strategy removes unneeded unknowns possibly introduced during the pre-asymptotical regime. In addition, the grids over which we perform the unrefinements are arbitrary, and thus, we can limit their size and associated computational costs. We numerically analyze our algorithm for energy-norm and goal-oriented adaptivity. In particular, we solve two-dimensional ($2$D) Poisson, Helmholtz, convection-dominated equations, and a three-dimensional ($3$D) Helmholtz-like problem. In all cases, we observe \revb{exponential} convergence rates. Our algorithm is robust and straightforward to implement; therefore, it can be easily adapted for industrial applications.BERC.2022-202

A Painless Automatic hp-Adaptive Strategy for Elliptic Problems

In this work, we introduce a novel hp-adaptive strategy. The main goal is to minimize the complexity and implementational efforts hence increasing the robustness of the algorithm while keeping close to optimal numerical results. We employ a multi-level hierarchical data structure imposing Dirichlet nodes to manage the so-called hanging nodes. The hp-adaptive strategy is based on performing quasi-optimal unre finements. Taking advantage of the hierarchical structure of the basis functions both in terms of the element size h and the polynomial order of approximation p, we mark those with the lowest contributions to the energy of the solution and remove them. This straightforward unrefi nement strategy does not need from a fi ne grid or complex data structures, making the algorithm flexible to many practical situations and existing implementations. On the other side, we also identify some limitations of the proposed strategy, namely: (a) data structures only support isotropic h-re nements (although p-anisotropic re nements are enabled), (b) we assume certain quasi-orthogonality properties of the basis functions in the energy norm, and (c) in this work, we restrict to symmetric and positive defi nite problems. We illustrate these and other advantages and limitations of the proposed hp-adaptive strategy with several one-, two- and three-dimensional Poisson examples

Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation

In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.V. Darrigrand, A. Rodriguez-Rozas and D. Pardo were partially funded by the Projects of the Spanish Ministry of Economy and Competitiveness with reference MTM2013-40824-P, MTM2016-76329-R (AEI/FEDER, EU), MTM2016-81697-ERC and the Basque Government Consolidated Research Group Grant IT649- 13 on “Mathematical Modeling, Simulation, and Industrial Applications (M2SI)”. A. Rodriguez-Rozas and D.Pardo were also partially funded by the BCAM “Severo Ochoa” accreditation of excellence SEV-2013-0323 and the Basque Government through the BERC2014-2017 program. A. Rodriguez-Rozas acknowledges support from Spanish Ministry under Grant No. FPDI- 2013-17098. I. Muga was partially funded by the FONDECYT project 1160774. The first four authors were also partially funded by the European Union’s Horizon 2020, research and innovation program under the Marie Sklodowska-Curie grant agreement No 644202. Serge Prudhomme is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada

A Simulation Method for the Computation of the E

We propose a set of numerical methods for the computation of the frequency-dependent eff ective primary wave velocity of heterogeneous rocks. We assume the rocks' internal microstructure is given by micro-computed tomography images. In the low/medium frequency regime, we propose to solve the acoustic equation in the frequency domain by a Finite Element Method (FEM). We employ a Perfectly Matched Layer to truncate the computational domain and we show the need to repeat the domain a su cient number of times to obtain accurate results. To make this problem computationally tractable, we equip the FEM with non-fitting meshes and we precompute multiple blocks of the sti ffness matrix. In the high-frequency range, we solve the eikonal equation with a Fast Marching Method. Numerical results con rm the validity of the proposed methods and illustrate the e ffect of density, porosity, and the size and distribution of the pores on the e ective compressional wave velocity

Tumors escape immunosurveillance by overexpressing the proteasome activator PSME3

The success of CD8+ T cell-based cancer immunotherapy emphasizes the importance of understanding the mechanisms of generation of MHC-I peptide ligands and the possible pathways of tumor cell escape from immunosurveillance. Recently, we showed that peptides generated in the nucleus during a pioneer round of mRNA translation (pioneer translation products, or PTPs) are an important source of tumor specific peptides which correlates with the aberrant splicing and transcription events associated with oncogenesis. Here we show that up-regulation of PSME3 proteasome activator in cancer cells results in increased destruction of PTP-derived peptides in the nucleus thus enabling cancer cell to subvert immunosurveillance. These findings unveil a previously unexpected role for PSME3 in antigen processing and identify PSME3 as a druggable target to improve the efficacy of cancer immunotherapy

1D Painless Multi-level Automatic Goal-Oriented h and p Adaptive Strategies Using a Pseudo-Dual Operator

The main idea of our Goal-Oriented Adaptive (GOA) strategy is based on performing global and uniform h- or p-refinements (for h- and p-adaptivity, respectively) followed by a coarsening step, where some basis functions are removed according to their estimated importance. Many Goal-Oriented Adaptive strategies represent the error in a Quantity of Interest (QoI) in terms of the bilinear form and the solution of the direct and adjoint problems. However, this is unfeasible when solving indefinite or non-symmetric problems since symmetric and positive definite forms are needed to define the inner product that guides the refinements. In this work, we provide a Goal-Oriented Adaptive (h- or p-) strategy whose error in the QoI is represented in another bilinear symmetric positive definite form than the one given by the adjoint problem. For that purpose, our Finite Element implementation employs a multi-level hierarchical data structure that imposes Dirichlet homogeneous nodes to avoid the so-called hanging nodes. We illustrate the convergence of the proposed approach for 1D Helmholtz and convection-dominated problems

Model-Based Analysis of Myocardial Contraction Patterns in Ischemic Heart Disease

International audienceObjectives: The objective of the current study is to assess the feasibility of using a left ventricle (LV) model in order to reproduce myocardial strains in the case of ischemic heart disease (IHD). Materials and Methods: The proposed integrated LV model couples a finite element method (FEM) sub-model of the cardiac electrical activity, representing the propagation of the transmembrane potential through the LV, a FEM sub-model of the cardiac mechanical activity, and a lumped-parameter model of the circulatory system that improves the physiological definition of appropriate boundary conditions. Simulations were compared to myocardial strain data obtained from echocardiography, performed on two healthy subjects and two patients diagnosed with chronic IHD. Results: Results show the model ability to simulate jointly the hemodynamic variables (Systolic and diastolic pressures respectively equal to 110 and 60 mmHg) and myocardial strain curves during each phase of the cardiac cycle. A close match is observed between minimum strains obtained from simulations and clinical data from healthy subjects and IHD patient, with a root mean square error around 2,81% and 2,63% respectively. The modifications of regional LV systolic strains observed from IHD patients with respect to healthy cases were partially explained by a decrease of regional cardiac stress. Conclusion: This paper provides a description of a new coupling algorithm between FEM and lumped-parameter models. Results are promising for the analysis of cardiac strains in the context of IHD