Group on Applied Mathematical Modeling, Statistics, and Optimization
(MATHMODE)
MATHMODE is recognized as a Research Group of Excellence (A) by the Basque Government
TEAM
Inmaculada Arostegui
Julen Alvarez
Judit Muñoz
David Pardo
Magdalena Strugaru
Joseba Makazaga
Irantzu Barrio
Ander Murua
Josu Najera
Javier del Ser
Eli Alberdi
Javier Omella
Josu Doncel
Carlos Gorria
Amaia Iparragirre
Iker Malaina
Urtzi Ayesta
Carlos Uriarte
AREAS OF KNOWLEDGE AND GOALS
The areas of knowledge of the Group on Applied Mathematical Modeling, Statistics, and Optimization (MATHMODE) are:
1) Applied Mathematics, used to develop advanced numerical and statistical methods.
2) Applied Artificial Intelligence (AI), used often in combination with advanced numerical and statistical methods.
3) Scientific Computing, used to efficiently implement those numerical, statistical, and AI algorithms.
The goals of the Group on Applied Mathematical Modeling, Statistics, and Optimization (MATHMODE) are:
1) Develop knowledge on Applied Mathematics, Applied AI, and Scientific Computing.
2) Transfer this mathematical knowledge to the industry and institutions for the benefit of society.
3) Train new researchers and attract and retain talent in the area.
MAIN INITIATIVES SUPPORTED FROM MATHMODE
Joint Research Lab in Applied Artificial Intelligence (JRL-A2I, www.jrl-a2i.science)
Talent Attraction´s Day in Applied Artificial Intelligence
MAIN RESEARCH DIRECTIONS
Applied Numerical Methods
We exploit deep learning concepts to design innovative efficient and robust algorithms able to solve inverse problems arising in health, geophysics, structural health monitoring, and offshore wind energy.
We aim at incorporating Deep Learning (DL) algorithms in the resolution of applications that require a fast inversion of measurements in order to make trustworthy decisions for the prediction of short- and long-term behaviour of technological devices, such as drilling tools and offshore wind energy platforms, among others.
Mathematically, an inverse problem consists of evaluating specific parameters of a partial differential equation (PDE) from given measurements.
We have developed and implemented successful deep learning algorithms for geophysical and structural health monitoring problems. We are currently focused on expanding them to more realistic and complex models, while working on efficient finite element methods for the generation of significant datasets, which should be large enough and include all possible - extreme and intermediate - scenarios.
Simulation-based optimization: new measurements (sensors) are being gradually added to the sensor system in order to increase the discrepancy between the measurements of different structural states.
We exploit deep learning concepts to design innovative efficient and robust algorithms able to solve inverse problems arising in health, geophysics, structural health monitoring, and offshore wind energy.
2) We develop and analyze mathematically highly accurate and robust numerical methods for the solution and inversion, via computer simulations, of challenging multiphysics applications. This is also crucial for the generation of sufficiently big and significant datasets for the training of artificial intelligence algorithms.
3) Advanced numerical methods for time integration of differential equations. This line of research seeks to analyze, design, and implement numerical integration methods for time evolution problems governed by differential equations whose solution cannot be obtained using conventional packages based on multistep methods nor Runge-Kutta schemes.
We design, analyze mathematically and implement efficient finite element methods, able to approximate fast and accurately the solutions of boundary value problems arising from real life applications. In particular, we are interested in optimizing the time and resources required for the generation of significant datasets for the training of deep neural networks. We focus on the following aspects:
Mesh-Adaptive Finite Elements: We work on different strategies for localizing and reducing the error between the exact and approximated solution, in order to decrease the computational cost and time. We exploit unconventional error representations and explicit-time domain methods to design goal-oriented adaptivity methods.
Also, we employ hierarchical h- and p- basis functions with possibly a large number of Dirichlet nodes to support arbitrary hp-meshes. We develop energy-norm and goal-oriented hp-adaptive algorithms that are simple to implement since they do not involve projections, and are expected to provide quasi-optimal hp-meshes for a large variety of multiphysics problems.
Refined Isogeometric Analysis (rIGA): We develop a refining mechanism that consists of reducing the continuity of the solution over local areas of the domain, while keeping an optimal distribution of computational resources (number of degrees of freedom - DoF). Although this strategy increases the problem size, it also breaks the structure of the stiffness matrix properly, which results in a reduction in the direct solver of up to 50, with respect to traditional isogeometric discretizations.
We have recently employed rIGA to generate a meaningful synthetic database composed of 100,000 earth models with the corresponding measurements in about 56 hours using a workstation equipped with two CPUs.
Applied Artificial Intelligence
To be completed...
Applied Statistics and Optimization
1) We employ statistics to validate and efficiently analyze real data. We promote the transfer of the research in statistics to biomedical and experimental fields through reliable and user-friendly algorithms based on statistical models.
2) We contribute to the advances in real-world industry and healthcare, by solving the arising mathematical problems with the proposed methods.
3) Applied optimization problems. The main objective is to carry out projects with companies, making technology transfer in the fields of optimization, simulation, operational research, and statistics.
Scientific Computing
To be completed...
Applications: Health, Geophysics, and Engineering
Health application: validation and prediction models for deseases
Our goal is to ensure the transfer of statistics research to medical and experimental fields. In particular, we focus on the validation of prediction models for diseases such as chronic obstructive pulmonary disease (COPD), colon cancer and heart diseases, among others.
Recently, we have designed a computer application to predict adverse events (death and intensive care unit or intermediate respiratory care unit admission), based on five predictive variables: age, previous history of long-term home oxygen therapy, altered consciousness, use of accessory inspiratory muscles, and baseline dyspnea.
Screenshot of the application, running under the Android platform. Data for an imaginary subject with complete information displayed as an example.
In the perspective of a health electronic database available to emergency physicians, our software could serve as an instrument for rapid and reliable decisions in emergency situations, thus ensuring the translation of clinical prediction rules into easy-to-use computer tools suitable for clinical practices.
Health application: Ultrasound imaging of the human body
We are also working with the company GE Healthcare to improve ultrasound imaging for women by incorporating advanced applied AI algorithms.
Health application: obstetric models
To be completed...
Health application: disease transmission models
Aside from the mentioned medical areas, members of MATHMODE group participated in the modelization of CoVid19 evolution through the use of a SEIR (Susceptible. Exposed, Infectious and Recovered) epidemiology representation. Their results were validated by the data provided by Osakidetza (Basque Health Service) and served for the anticipation of the number of infected persons that needed basic medical care or admission to intensive care units in the Basque Country.
Geophysics application: Simulation of wave propagation in the Earth´s subsurface
We aim at providing efficient and reliable mathematical tools and computational algorithms to delineate a map of the Earth's subsurface, which is essential for a variety of applications, such as: earthquake prediction and seismic hazard estimation, mining, geothermal energy production, mine detection, underground CO2 storage, among others. We focus on the following aspects:
Simulation and Inversion of Borehole Resistivity Measurement: A real-time interpretation of borehole resistivity measurements is fundamental to perform geosteering, which is the act of correcting the tool trajectory while drilling in order to maximize ulterior hydrocarbon recovery.
In a recent paper, we have demonstrated that a deep neural network can provide a high-quality approximation to a complex, industry-quality forward model for extra-deep electro-magnetic logs used in modern geosteering operations. With a relatively small dataset of 63.122 samples to train a high-dimensional function, we were able to produce a good approximation to the relevant logs acquired during a synthetic geosteering operation.
Simulation and Inversion of Elasto-Acoustic Waves in Porous Rocks: We create, combine and apply advanced numerical methods to generate computer algorithms able to simulate efficiently laboratory and/or in situ measurements.
We have recently designed a method that approximates the P-wave compressional velocity of a porous rock, for frequencies ranging from 10 to 107 Hz. The numerical P-wave velocity of a real rock matches the experimental results in the high-frequency regime and fills the gap of lower frequencies in laboratory experiments, according to theoretical limits and averages.
Engineering application: Structural Health Monitoring of civil infraestructures
Our goal is to create mathematical models - supervised or unsupervised AI learning approaches - that are trained with synthetic and experimental data and generate accurate health structural diagnostics of civil and industrial engineering infrastructures.
We work on the structural health of bridges, viaducts, and offshore wind energy platforms. We aim at reducing the maintenance and repair cost and risk, by designing intelligent algorithms able to learn from experimental measurements such as platform displacements or vibrations, and automatically detect eventual damage (cracks, corrosions, break of mooring lines, among others).
Interpretation of the experimental measurements by a deep neural network provides critical information on possible damages in the offshore wind energy platform.
We are currently focusing on designing such algorithms for the early detection of failures or monitoring of difficult-to-access or expensive-to-sensor components/subsystems, such as: fractures in towers and support structures, gearboxes and blades, among others.
Given the complexity of the structures and of the physical phenomenon affecting their state, our ultimate goal is the design, implementation and validation of intelligent multi-sensor monitoring data from real structures.