Actividades
Seminario: Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation
Carlos Pérez, Mathematics of Computational Science (MACS), University of Twente.
Miércoles 23 de abril de 2025, 13:40 hrs. (Presencial Auditorio Edificio San Agustín)
ABSTRACT
This talk outlines a novel class of high-order methods for the efficient numerical evaluation of volume potentials (VPs) defined by volume integrals over complex geometries. Inspired by the Density Interpolation Method (DIM) for boundary integral operators, the proposed methodology leverages Green’s third identity and a local polynomial interpolation of the density function to recast a given VP as a linear combination of surface-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated inside and outside the integration domain using existing methods (e.g. DIM), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules to integrate over structured or unstructured domain decompositions without local numerical treatment at and around the kernel singularity. The proposed methodology is flexible, easy to implement, and fully compatible with well-established fast algorithms such as the Fast Multipole Method and H-matrices, enabling VP evaluations to achieve linearithmic computational complexity. To demonstrate the merits of the proposed methodology, we applied it to the Nyström discretization of the Lippmann-Schwinger volume integral equation for frequency-domain Helmholtz scattering problems in piecewise-smooth variable media.
This is joint work with Thomas G. Anderson (Rice University), Luiz Faria (ENSTA Paris), and Marc Bonnet (ENSTA Paris).
Seminario: Convexidad en optimización cuadrática no convexa, y más allá
Fabián Flores Bazán, Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción.
Miércoles 15 de enero de 2025, 13:30 hrs. (Presencial Auditorio Edificio San Agustín)
ABSTRACT
Convexidad es una condición que muchos matemáticos quisieran que el problema que enfrentan lo cumpla. Por otro lado, existen resultados en Análisis Funcional, Cálculo de Variaciones, Pencil de Matrices, Inclusiones Diferenciales, entre otras áreas de la Matemática, las cuales muestran que la convexidad surge de modo natural, ya sea de modo directo o indirectamente. Algunos de aquellos resultados se presentaran con énfasis en el mundo cuadrático; también veremos las consecuencias que traen consigo, particularmente en Optimización Matemática cuando se estudia la validez de la propiedad de dualidad fuerte.
Seminario: OptimUS: an open-source Python library for 3D acoustic wave propagation
Pierre Gélat, Division of Surgery & Interventional Science, University College London (UCL), UK.
Miércoles 6 de noviembre de 2024, 13:40 hrs. (Presencial Auditorio Edificio San Agustín; link Zoom disponible escribiendo a Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.)
ABSTRACT
The Helmholtz equation for harmonic wave propagation is a widely used model for many acoustic phenomena, such as room acoustics, sonar, and biomedical ultrasound. The boundary element method (BEM) is one of the most efficient numerical methods to solve Helmholtz transmission problems and is based on boundary integral formulations that rewrite the volumetric partial differential equations into a representation of the acoustic fields in terms of surface potentials at the material interfaces. OptimUS (https://github.com/optimuslib/optimus) is a Python library centered around the BEM, which offers a user-friendly interface via Jupyter Notebooks, enabling the prediction of acoustic waves in piecewise homogeneous media in the frequency domain.
This talk will provide an overview of the OptimUS interface, with a focus on case studies where objects are large relative to the wavelengths involved. This will include biomedical ultrasound, which has a growing number of therapeutic applications such as the treatment of cancers of the liver and kidney. The modelling of transcranial ultrasound neurostimulation, an emerging modality which may one day treat mental health conditions such as depression, will also be reviewed. Acoustic wave propagation into the uterus at audio range frequencies will be presented to provide awareness of the impact of everyday noise exposure on the developing fetus.
Seminario: Data-driven system analysis: Polynomial optimization meets Koopman
Giovanni Fantuzzi, Division of Dynamics, Control, Machine Learning and Numerics, Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg.
Miércoles 30 de octubre de 2024, 13:40 hrs. (Presencial Auditorio Edificio San Agustín; link Zoom disponible escribiendo a Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.)
ABSTRACT
When studying complex systems that evolve over time, it is often not enough to just predict the future. We also want to know if equilibrium states are stable, how the system will behave on average over a long time, and how uncertainty impacts the dynamics. Techniques from polynomial optimization can be used to make these predictions when we have explicit models for the dynamics based on polynomial equations. But what if the model isn’t based on polynomial equations or, worse, we don't know the model at all?
In this talk, I will discuss how we can analyze dynamical systems using only data from measurements, with no need for an explicit mathematical model. The main idea is to combine polynomial optimization with a data analysis technique called extended dynamic mode decomposition, which is related to the celebrated Koopman operator. I will introduce these tools, show how they work together, and illustrate their effectiveness with examples related to stability and long-term behavior.
Seminario: Propagation Algorithms for Handling Symmetries of Mathematical Programs
Jasper van Doornmalen, Instituto de Ingeniería Matemática y Computacional (IMC), Pontificia Universidad Católica de Chile.
Miércoles 20 de noviembre de 2024, 13:40 hrs. (Presencial Auditorio Edificio San Agustín; link Zoom disponible escribiendo a Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.)
ABSTRACT
Optimization problems are often solved using constraint programming or mixed-integer programming techniques, using enumeration or branch-and-bound techniques. It is well known that if the problem formulation is very symmetric, there may exist many symmetrically equivalent solutions to the problem. Without handling the symmetries, traditional solving methods have to check many symmetric parts of the solution space, which comes at a high computational cost. Handling symmetries in optimization problems is thus essential for devising efficient solution methods.
The presentation will introduce and review common methods for solving mathematical programs, the concepts of symmetries in mathematical programs, and how symmetries can be handled. Then, the main results of Jasper's dissertation are presented, along with computational results. In particular, the presentation focuses on a general framework for symmetry handling that captures many of the already existing symmetry handling methods. While these methods are mostly discussed independently from each other in the literature, the framework allows to apply different symmetry handling methods simultaneously and thus outperform their individual effects. Moreover, most existing symmetry handling methods only apply to binary variables. Our framework allows to easily generalize these methods to general variable types. Numerical experiments confirm that our novel framework is superior to the state-of-the-art methods implemented in the solver SCIP.