The ultimate scientific goal of MATHMODE group is to bring a significant mathematical contribution to the resolution of real-life problems. To do so, we explore and combine the following research areas:

Deep Learning

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 in evaluating specific parameters of a partial differential equation (PDE) from given measurements

In geosteering, the goal is to determine the materials composing the Earth’s subsurface from electromagnetic measurements acquired by a drilling logging instrument equipped with transmitters and receivers.

Solution of inverse problems is not necessarily unique: different models may produce identical measurements. To account for this, it is necessary to consider properly designed loss functions.

A deep neural network equipped with a traditional loss function cannot recover any of the two branches (solution) of the square root function.

To overcome this drawback, we design deep neural networks based on optimization with respect to the original input, complemented with a training of the forward operator. We therefore employ encoder-decoder and two-step loss functions and explore different error control techniques. Apart from these, we investigate a variety of architectures, model parameterizations, and the reduction of the highly elevated cost for the generation of datasets used to train the network.

We have already 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.

Comparison between actual and predicted (inverted) formation in a geosteering synthetic model, using an encoder-decoder loss function.

We also work on the design of intelligent measurement acquisition systems that ensure uniqueness of the inverse solution for a fixed parameterization.

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.

Through our work, not only do we provide efficient tools for applications driven by inverse problems using DL, but we also contribute to a more profound and comprehensive understanding of the main mathematical algorithms and tools behind DL.


Our goal is to design valid, accurate, reliable and user-friendly algorithms based on statistical models, which are arising as a key issue in many areas of research: medicine, biology, chemistry, ecology, toxicology, and genetics, among others.

Figures of anxiety and depression with the fit to the corresponding beta‐binomial distribution in a study relating anxiety and depressive symptoms to general health outcome and eating disorders

Additionally, we work towards using statistics as a tool for understanding and efficiently analyzing real data. In this way, we provide methodological support in statistics to other research groups in the fields of biomedicine or experimental sciences.

Advanced Finite Element Methods

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.

Sonic logging instrument in an axi-symmetric borehole environment. This model problem was employed to test various p-adaptive algorithms.

Convergence of the classical (red) vs our proposed (blue) p-adaptive algorithms

  • 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.

DoF in IGA (p=3)

DoF in rIGA (p=3)

DoF in FEM (p=3)

Applications: Geophysics

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.

Well trajectory and layer resistivities for a realistic log example in measured depth vs true vertical depth coordinates.

  • 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.

Three profiles of effective P-wave velocity with respect to frequency.

  • Other geophysical-related areas of interest: Simulation and inversion of magnetotelluric measurements, Simulation of marine controlled source electromagnetic (CSEM) measurements.

Applications: Structural Health Monitoring

Our goal is to create mathematical models - supervised or unsupervised learning approaches - that are trained with experimental data and generate accurate health structural diagnostics of civil and industrial engineering infrastructures.

  • Structural health of bridges, viaducts and other civil engineering constructions: We design real-time data-based supervision tools, able to monitor the global behavior of critical components in the structures of interest.

We have recently proposed and validated such an approach, applied to the Beltran bridge, located in kilometer point (KP) 119.5 of the Guadalajara–Colima highway in Mexico. Our procedure combines measurements from four different sensors into one single performance indicator that is only weakly affected by temperature variations after the application of principal component analysis.

Structural profle of the singular Beltran bridge

Bridge section details

We used a data set of around 40.000 measurements per sensor, acquired continuously from July 2013 on. The obtained results prove the ability of the algorithm to respond with an accuracy of 95% to novel measurements and also its low sensitivity to environmental variability.

The proposed algorithm distinguishes between undamaged bearings (first half) and damaged bearing situations (second half).

Our future work includes applying the proposed procedure to other structures, and using deep neural networks to enrich the model with nonlinear correlations between measurements.

  • Structural health of 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 deformations at specific points of the support structure of an offshore wind platform-, 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 have recently designed, implemented and validated an algorithm that detects displacements in the 3-line anchoring system of a buoy. We employed encoder-decoder loss functions and used the reconstruction error as the damage indicator. We trained and tested the algorithm with a synthetic dataset of less than 40.000 measurement-parameter couples. The results prove the algorithm’s efficiency in identifying very accurately the damaged cases and are, therefore, very promising.

The algorithm separates the undamaged and damaged testing subset with a very satisfactory accuracy.

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.

Applications: Health

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.

Aside from the mentioned medical areas, and due to the emerging and rapidly evolving situation of the CoVid19 disease caused by the virus SARS-CoV2, 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.

We are currently investigating the effect of the social and traveling restrictions imposed in the Basque Country on the disease transmission in order to provide support for future decisions.

Other Research Lines

This research line has been discontinued in 2018.

Non-fitting finite element meshes: Non-fitting grids are convenient because they are simpler to generate and handle than fitting grids when the geometry is complex, as it is the case in magnetic field resistivity measurements using 2.5D Maxwell’s equations. We have already shown, for various geophysical applications, that if the finite element matrix coefficients are properly integrated, the accuracy loss due to the use of non-fitting grids is negligible compared to the case where fitting grids are employed.

This research line has been discontinued in 2018.

Dimensionally Adaptive Method: To reduce the computational cost in geophysical applications, we explicitly separate one-dimensional (1D), 1.5D, 1.75D, 2D, 2.5D, 2.75D, and 3D effects based on physical principles. This reduces dramatically the simulation and inversion time, making the method feasible to efficiently invert real multi-physics measurements. The use of proper exponential and/or Hankel functions enable to mix subdomains of different dimensionality within the same computational domain.

This research line has been discontinued in 2016.

Enhanced Variation Image Dehazing (EVID). Images obtained under adverse weather conditions, such as haze or fog, typically exhibit low contrast and faded colors, which may severely limit the visibility within the scene. This parallels in some sense the process suffered by underwater images. In the same way, unveiling the image structure under the haze layer and recovering vivid colors out of a single image remains a challenging task, since the degradation is depth-dependent and conventional methods are unable to overcome this problem. We present experimental results demonstrating that the developed EVID method outperforms other state-of-the-art methods both qualitatively and quantitatively. In particular, when the illuminant is uneven, the EVID method is the only one that recovers realistic colors, avoiding the appearance of strong chromatic artifacts.

This research line has been discontinued in 2016.

Underwater Image Restoration. Underwater images suffer from low contrast as a result of the exponential decay that light suffers as it travels. Additionally, they exhibit a particular colour distortion associated to different wavelengths having different attenuation rates, being the red wavelength the one that attenuates the fastest. To overcome this unbalanced loss of contrast and colour distortion, we have developed a new approach, based on a modified Dark Channel prior. We call this technique the Red Channel method. The Red Channel method is designed to restore the lost contrast while recovering colours associated to short wavelengths.

This research line has been discontinued in 2016.

Multistep methods with numerical damping control. She has collaborated with Juan José Anza in the construction of multistep methods with good stability properties and parametric control of numerical dissipation in the high-frequency range for solving second order ODE systems obtained after semidiscretizing the wave-type partial differential equation (PDE) with the finite element method (FEM).

This research line has been discontinued in 2015.

Numerical modeling of Cold Crucible Induction Melting (CCIM). The CCIM is a process to melt extremely reactive alloys in molten conditions when purity is required. An example of such an alloy is the use of titanium on medical implants. We have developed a computer simulation model using the software COMSOL Multiphysics in order to understand how each parameter of the process affects in the overheating of the liquid.