Accelerating CFD via local POD plus Galerkin projections

Se desarrollan varias técnicas basadas en descomposición ortogonal propia (DOP) local y proyección de tipo Galerkin para acelerar la integración numérica de problemas de evolución, de tipo parabólico, no lineales. Las ideas y métodos que se presentan conllevan un nuevo enfoque para la modelización de tipo DOP, que combina intervalos temporales cortos en que se usa un esquema numérico estándard con otros intervalos temporales en que se utilizan los sistemas de tipo Galerkin que resultan de proyectar las ecuaciones de evolución sobre la variedad lineal generada por los modos DOP, obtenidos a partir de instantáneas calculadas en los intervalos donde actúa el código numérico. La variedad DOP se construye completamente en el primer intervalo, pero solamente se actualiza en los demás intervalos según las dinámicas de la solución, aumentando de este modo la eficiencia del modelo de orden reducido resultante. Además, se aprovechan algunas propiedades asociadas a la dependencia débil de los modos DOP tanto en la variable temporal como en los posibles parámetros de que pueda depender el problema. De esta forma, se aumentan la flexibilidad y la eficiencia computacional del proceso. La aplicación de los métodos resultantes es muy prometedora, tanto en la simulación de transitorios en flujos laminares como en la construcción de diagramas de bifurcación en sistemas dependientes de parámetros. Las ideas y los algoritmos desarrollados en la tesis se ilustran en dos problemas test, la ecuación unidimensional compleja de Ginzburg-Landau y el problema bidimensional no estacionario de la cavidad. Abstract Various ideas and methods involving local proper orthogonal decomposition (POD) and Galerkin projection are presented aiming at accelerating the numerical integration of nonlinear time dependent parabolic problems. The proposed methods come from a new approach to the POD-based model reduction procedures, which combines short runs with a given numerical solver and a reduced order model constructed by expanding the solution of the problem into appropriate POD modes, which span a POD manifold, and Galerkin projecting some evolution equations onto that linear manifold. The POD manifold is completely constructed from the outset, but only updated as time proceeds according to the dynamics, which yields an adaptive and flexible procedure. In addition, some properties concerning the weak dependence of the POD modes on time and possible parameters in the problem are exploited in order to increase the flexibility and efficiency of the low dimensional model computation. Application of the developed techniques to the approximation of transients in laminar fluid flows and the simulation of attractors in bifurcation problems shows very promising results. The test problems considered to illustrate the various ideas and check the performance of the algorithms are the onedimensional complex Ginzburg-Landau equation and the two-dimensional unsteady liddriven cavity problem

Mixing snapshots and fast time integration of PDEs

A local proper orthogonal decomposition (POD) plus Galerkin projection method was recently developed to accelerate time dependent numerical solvers of PDEs. This method is based on the combined use of a numerical code (NC) and a Galerkin system (GS) in a sequence of interspersed time intervals, INC and IGS, respectively. POD is performed on some sets of snapshots calculated by the numerical solver in the INC intervals. The governing equations are Galerkin projected onto the most energetic POD modes and the resulting GS is time integrated in the next IGS interval. The major computational effort is associated with the snapshots calculation in the first INC interval, where the POD manifold needs to be completely constructed (it is only updated in subsequent INC intervals, which can thus be quite small). As the POD manifold depends only weakly on the particular values of the parameters of the problem, a suitable library can be constructed adapting the snapshots calculated in other runs to drastically reduce the size of the first INC interval and thus the involved computational cost. The strategy is successfully tested in (i) the one-dimensional complex Ginzburg-Landau equation, including the case in which it exhibits transient chaos, and (ii) the two-dimensional unsteady lid-driven cavity problem

The statistical theory of the angiogenesis equations

Angiogenesis is a multiscale process by which a primary blood vessel issues secondary vessel sprouts that reach regions lacking oxygen. Angiogenesis can be a natural process of organ growth and development or a pathological induced by a cancerous tumor. A mean field approximation for a stochastic model of angiogenesis consists of partial differential equation (PDE) for the density of active tip vessels. Addition of Gaussian and jump noise terms to this equation produces a stochastic PDE that defines an infinite dimensional L\'evy process and is the basis of a statistical theory of angiogenesis. The associated functional equation has been solved and the invariant measure obtained. The results are compared to a direct numerical simulation of the stochastic model of angiogenesis and invariant measure multiplied by an exponentially decaying factor. The results of this theory are compared to direct numerical simulations of the underlying angiogenesis model. The invariant measure and the moments are functions of the Korteweg-de Vries soliton which approximates the deterministic density of active vessel tips.Comment: 29 pages, 1 figure, 1 tabl

Ensemble Averages, Soliton Dynamics and Influence of Haptotaxis in a Model of Tumor-Induced Angiogenesis

In this work, we present a numerical study of the influence of matrix degrading enzyme (MDE) dynamics and haptotaxis on the development of vessel networks in tumor-induced angiogenesis. Avascular tumors produce growth factors that induce nearby blood vessels to emit sprouts formed by endothelial cells. These capillary sprouts advance toward the tumor by chemotaxis (gradients of growth factor) and haptotaxis (adhesion to the tissue matrix outside blood vessels). The motion of the capillaries in this constrained space is modelled by stochastic processes (Langevin equations, branching and merging of sprouts) coupled to continuum equations for concentrations of involved substances. There is a complementary deterministic description in terms of the density of actively moving tips of vessel sprouts. The latter forms a stable soliton-like wave whose motion is influenced by the different taxis mechanisms. We show the delaying effect of haptotaxis on the advance of the angiogenic vessel network by direct numerical simulations of the stochastic process and by a study of the soliton motion.We thank Vincenzo Capasso, Bjorn Birnir and Boris Malomed for fruitful discussions. This work has been supported by the Ministerio de Economía y Competitividad grant MTM2014-56948-C2-2-P

Uncovering spatio-temporal patterns in semiconductor superlattices by efficient data processing tools

Time periodic patterns in a semiconductor superlattice, relevant to microwave generation, are obtained upon numerical integration of a known set of drift-diffusion equations. The associated spatiotemporal transport mechanisms are uncovered by applying (to the computed data) two recent data processing tools, known as the higher order dynamic mode decomposition and the spatiotemporal Koopman decomposition. Outcomes include a clear identification of the asymptotic self-sustained oscillations of the current density (isolated from the transient dynamics) and an accurate description of the electric field traveling pulse in terms of its dispersion diagram. In addition, a preliminary version of a data-driven reduced order model is constructed, which allows for extremely fast online simulations of the system response over a range of different configurations.The authors are indebted to two anonymous referees for some useful comments and suggestions on an earlier version of the paper. This work has been supported by the Fondo Europeo de Desarrollo Regional Ministerio de Ciencia, Innovación y Universidades–Agencia Estatal de Investigación, under Grants No. TRA2016-75075-R, No. MTM2017-84446-C2-2-R, and No. PID2020-112796RB-C22, and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23) and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation)

A mathematical model of thyroid disease response to radiotherapy

We present a mechanistic biomathematical model of molecular radiotherapy of thyroid disease. The general model consists of a set of differential equations describing the dynamics of different populations of thyroid cells with varying degrees of damage caused by radiotherapy (undamaged cells, sub-lethally damaged cells, doomed cells, and dead cells), as well as the dynamics of thyroglobulin and antithyroglobulin autoantibodies, which are important surrogates of treatment response. The model is presented in two flavours: on the one hand, as a deterministic continuous model, which is useful to fit populational data, and on the other hand, as a stochastic Markov model, which is particularly useful to investigate tumor control probabilities and treatment individualization. The model was used to fit the response dynamics (tumor/thyroid volumes, thyroglobulin and antithyroglobulin autoantibodies) observed in experimental studies of thyroid cancer and Graves’ disease treated with 131I-radiotherapy. A qualitative adequate fitting of the model to the experimental data was achieved. We also used the model to investigate treatment individualization strategies for differentiated thyroid cancer, aiming to improve the tumor control probability. We found that simple individualization strategies based on the absorbed dose in the tumor and tumor radiosensitivity (which are both magnitudes that can potentially be individually determined for every patient) can lead to an important raise of tumor control probabilities.This project has received funding from the Instituto de Salud Carlos III (PI17/01428 grant, FEDER co-funding). This project has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie grant agreement No 839135. This project has received funding from FEDER/Ministerio de Ciencia, Innovación y Universidades;Agencia Estatal de Investigación, under grant MTM2017-84446-C2-2-R

Mixing Snapshots and Fast Time Integration of PDEs

A local proper orthogonal decomposition (POD) plus Galerkin projection method was recently developed to accelerate time dependent numerical solvers of PDEs. This method is based on the combined use of a numerical code (NC) and a Galerkin sys- tem (GS) in a sequence of interspersed time intervals, INC and IGS, respectively. POD is performed on some sets of snapshots calculated by the numerical solver in the INC inter- vals. The governing equations are Galerkin projected onto the most energetic POD modes and the resulting GS is time integrated in the next IGS interval. The major computa- tional e®ort is associated with the snapshots calculation in the ¯rst INC interval, where the POD manifold needs to be completely constructed (it is only updated in subsequent INC intervals, which can thus be quite small). As the POD manifold depends only weakly on the particular values of the parameters of the problem, a suitable library can be con- structed adapting the snapshots calculated in other runs to drastically reduce the size of the ¯rst INC interval and thus the involved computational cost. The strategy is success- fully tested in (i) the one-dimensional complex Ginzburg-Landau equation, including the case in which it exhibits transient chaos, and (ii) the two-dimensional unsteady lid-driven cavity proble

Reduced order modeling of some fluid flows of industrial interest

Some basic ideas are presented for the construction of robust, computationally efficient reduced order models amenable to be used in industrial environments, combined with somewhat rough computational fluid dynamics solvers. These ideas result from a critical review of the basic principles of proper orthogonal decomposition-based reduced order modeling of both steady and unsteady fluid flows. In particular, the extent to which some artifacts of the computational fluid dynamics solvers can be ignored is addressed, which opens up the possibility of obtaining quite flexible reduced order models. The methods are illustrated with the steady aerodynamic flow around a horizontal tail plane of a commercial aircraft in transonic conditions, and the unsteady lid-driven cavity problem. In both cases, the approximations are fairly good, thus reducing the computational cost by a significant factor

On the mathematical modelling of tumor-induced angiogenesis

An angiogenic system is taken as an example of extremely complex ones in the field of Life Sciences, from both analytical and computational points of view, due to the strong coupling between the kinetic parameters of the relevant branching-growth-anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. To reduce this complexity, for a conceptual stochastic model we have explored how to take advantage of the system intrinsic multiscale structure: one might describe the stochastic dynamics of the cells at the vessel tip at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. But the outcomes of relevant numerical simulations show that the proposed model, in presence of anastomosis, is not self-averaging, so that the "propagation of chaos" assumption cannot be applied to obtain a deterministic mean field approximation. On the other hand we have shown that ensemble averages over many realizations of the stochastic system may better correspond to a deterministic reaction-diffusion system.The authors acknowledge the contribution of the anonymous Referees for a significant improvement of the manuscript. This work has been supported by the Spanish Ministerio de Econom´ıa y Competitividad grant MTM2014- 56948-C2-2-P. VC has been supported by a Chair of Excellence UC3M-Santander at the Universidad Carlos III de Madrid. It is a great pleasure to acknowledge fruitful discussions with Daniela Morale of the Department of Mathematics of the University of Milan

Fast time integration of PDEs combining POD and Galerkin projection based on a limited set of mesh points

Recently, an adaptive method to accelerate time dependent numerical solvers of systems of PDEs that require a high cost in computational time and memory has been proposed [3] (see also [1, 2]). The method combines on the fly such numerical solver with a proper orthogonal decomposition, from which we identify modes, a Galerkin projection (that provides a reduced system of equations), and the integration of the reduced system. The strategy is based on a truncation error estimate and a residual estimate, designed to control the truncation error and the mode truncation instability, respectively. These estimates support the selection of the appropiate time intervals in which the numerical solver is run to first construct and then update, on demand, the POD modes. Moreover, to reduce the computational effort needed at the outset to generate the initial POD subspace, information from former simulations or generic libraries (e.g. trigonometric functions or orthogonal polynomials) were also used. To improve the computational efficiency of the method presented in [3] a crucial step is to use a limited number of points (instead of the whole computational mesh used in the spatial discretization) to both perform POD and to Galerkin?project the equations. In this work we will discuss and compare several alternatives in representative examples illustrating that a suitable point selection can make the cost of the reduced order model (associated with POD, Galerkin projection and the integration of the resulting Galerkin system) negligible compared to that of the reference numerical solver