This document discusses fluid-structure interactions (FSI) involving moving boundaries, and optimization and control problems governed by FSI models. It presents the governing PDE models coupling the incompressible Navier-Stokes equations for the fluid with nonlinear elasticity equations for the structure. It discusses challenges in analyzing well-posedness and deriving first-order necessary optimality conditions due to the moving boundary and nonlinear coupling between the fluid and structure domains. The goal is to characterize optimal controls by computing the gradient of cost functionals involving the state variables and moving fluid domain.
This talk is an overview of my 30 years or so work in mathematical theory of fluid dynamics, stochastic analysis, control theory, large deviations, ergodicity and nonlinear filtering.
Fluid Dynamics describes the physics of fluids at level of Undergraduate in science (physics, math, engineering). For comments or improvements please contact solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
This talk is an overview of my 30 years or so work in mathematical theory of fluid dynamics, stochastic analysis, control theory, large deviations, ergodicity and nonlinear filtering.
Fluid Dynamics describes the physics of fluids at level of Undergraduate in science (physics, math, engineering). For comments or improvements please contact solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Study of Parametric Standing Waves in Fluid filled Tibetan Singing bowlSandra B
I completed a Summer Project (May-July 2015) in Physics entitled "Study of
Parametric Standing Waves in Fluid filled Tibetan Singing bowl" under the
guidance of Dr. S. Shankaranarayanan at Indian Institute of Science Education
and Research (IISER-TVM). The project was to theoretically analyze and solve the
non-linear equations for the patterns of wave formed on the surface of Tibet
Singing Bowl for ideal and viscous fluid.
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
Unsteady Mhd free Convective flow in a Rotating System with Dufour and Soret ...IOSRJM
Numerical analysis is used to examine the unsteady MHD free convection and mass transfer fluid flow through a porous medium in a rotating system. Impulsively started plate moving its individual plane is considered. Similarity equations of the corresponding momentum, energy, and concentration equations are derived by introducing a time dependent length scale which infect plays the role of a resemblance parameter. The velocity component is taken to be inversely proportional to this parameter. The effects on the velocity, temperature, concentration, local skin-friction coefficients, Nusselt number, Prandl number, Dufour, Soret number and the Sherwood number of the various important parameters entering into the problem separately are discussed with the help of graphs.
Effect Of Elasticity On Herschel - Bulkley Fluid Flow In A TubeIJARIDEA Journal
Abstract— The impact of flexibility on Herschel-Bulkley liquid in a tube is researched. The issue is unraveled scientifically
for two unique sorts: one flux is computed taking the worry of the flexible tube into thought and the other flux is acquired by
considering the weight range relationship. Speed of the inelastic tube is additionally considered. The impact of various
parameters on flux and speed are talked about through charts. The outcomes acquired for the stream attributes uncover
many intriguing practices that warrant additionally contemplate on the non-Newtonian liquid stream wonders, particularly
the shear-diminishing marvels. Shear diminishing decreases the divider shear stretch.
Keywords— Elastic tube, Herschel-Bulkley Fluid, Inlet pressure, Outlet Pressure, Yield Stress.
Chemical Reaction on Heat and Mass TransferFlow through an Infinite Inclined ...iosrjce
The numerical studies are performed to examine the mass transfer flow with thermal diffusion and
diffusion thermo effect past an infinite, inclined vertical plate in a porous medium in the presence of chemical
reaction. First of all, the governing equations are transformed to a system of dimensionless coupled partial
equations. Explicit finite difference method has been used to solve these dimensionless equations for momentum,
concentration and energy equations. During the course of discussion, it is found that various parameters related
to the problem influence the calculated result. Finally, the profiles of velocity, concentration and temperature
are analyzed and illustrated with graphs.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
Study of Parametric Standing Waves in Fluid filled Tibetan Singing bowlSandra B
I completed a Summer Project (May-July 2015) in Physics entitled "Study of
Parametric Standing Waves in Fluid filled Tibetan Singing bowl" under the
guidance of Dr. S. Shankaranarayanan at Indian Institute of Science Education
and Research (IISER-TVM). The project was to theoretically analyze and solve the
non-linear equations for the patterns of wave formed on the surface of Tibet
Singing Bowl for ideal and viscous fluid.
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
Unsteady Mhd free Convective flow in a Rotating System with Dufour and Soret ...IOSRJM
Numerical analysis is used to examine the unsteady MHD free convection and mass transfer fluid flow through a porous medium in a rotating system. Impulsively started plate moving its individual plane is considered. Similarity equations of the corresponding momentum, energy, and concentration equations are derived by introducing a time dependent length scale which infect plays the role of a resemblance parameter. The velocity component is taken to be inversely proportional to this parameter. The effects on the velocity, temperature, concentration, local skin-friction coefficients, Nusselt number, Prandl number, Dufour, Soret number and the Sherwood number of the various important parameters entering into the problem separately are discussed with the help of graphs.
Effect Of Elasticity On Herschel - Bulkley Fluid Flow In A TubeIJARIDEA Journal
Abstract— The impact of flexibility on Herschel-Bulkley liquid in a tube is researched. The issue is unraveled scientifically
for two unique sorts: one flux is computed taking the worry of the flexible tube into thought and the other flux is acquired by
considering the weight range relationship. Speed of the inelastic tube is additionally considered. The impact of various
parameters on flux and speed are talked about through charts. The outcomes acquired for the stream attributes uncover
many intriguing practices that warrant additionally contemplate on the non-Newtonian liquid stream wonders, particularly
the shear-diminishing marvels. Shear diminishing decreases the divider shear stretch.
Keywords— Elastic tube, Herschel-Bulkley Fluid, Inlet pressure, Outlet Pressure, Yield Stress.
Chemical Reaction on Heat and Mass TransferFlow through an Infinite Inclined ...iosrjce
The numerical studies are performed to examine the mass transfer flow with thermal diffusion and
diffusion thermo effect past an infinite, inclined vertical plate in a porous medium in the presence of chemical
reaction. First of all, the governing equations are transformed to a system of dimensionless coupled partial
equations. Explicit finite difference method has been used to solve these dimensionless equations for momentum,
concentration and energy equations. During the course of discussion, it is found that various parameters related
to the problem influence the calculated result. Finally, the profiles of velocity, concentration and temperature
are analyzed and illustrated with graphs.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI)
Similar to QMC: Operator Splitting Workshop, Optimization and Control in Free ans Moving Boundary Fluid-Structure Interactions - Lorena Bociu, Mar 22, 2018
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
The time diffusive boundary layer from rest for Taylor series uniform outer flow is constructed by
integration of the complementary error function of the diffusive similarity variable . The same erfc solution is
also easily “Stokes” transformed to solve harmonic outer flow from rest.
If the outer flow vector varies weakly along the boundary, the perturbation pressure gradient accelerates
the slowest fluid nearest the wall the most to disproportionately vary the shear stress. So centripetal
acceleration causes secondary crossflow inside the boundary layer with strong wall shear towards the center of
curvature. But longitudinal acceleration also implies an inflow towards the wall which thins the outer boundary
layer with a weak further increase of the wall shear, but the strongest perturbation steady streaming.
Simple new particular solutions for these two perturbations are easily constructed in terms of products of
integrals and derivatives of any primary diffusive solution. For an outer flow as a series in the square root of
time, all homogeneous time coefficients remain just iterated error functions. Each systolic pulse in the aortic
arch was considered as a Taylor series flow from rest to calculate the wall shear vectors.
When the outer flow oscillates forever more, its primary diffusive boundary layer asymptotes to the Stokes
oscillatory exponential decay with distance from the wall. The particular perturbations are exactly evaluated
and also confined near the wall but with mean slip. The mean slip homogenous perturbations diffuse outside the
Stokes layer into steady streaming as complementary error functions with inverse time correction functions.
Extended Taylor series computations provide more detail of the perturbation transients.
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014Cristina Staicu
Cristobal Bertoglio, David Barber, Nicholas Gaddum, Israel Valverde, Marcel Rutten, et al.. Identification of artery wall stiffness: in vitro validation and in vivo results of a data assimilation procedure applied to a 3D fluid-structure interaction model. Journal of Biomechanics, Elsevier, 2014, 47 (5),
pp.1027-1034. 10.1016/j.jbiomech.2013.12.029 . hal-00925902v2
I received explicit thank you from the INRIA team for my support in the Sheffield team.
Thermal instability of incompressible non newtonian viscoelastic fluid with...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Thermal instability of incompressible non newtonian viscoelastic fluid with...eSAT Journals
Abstract The thermal convection of incompressible Walters’B′ Rotating viscoelastic fluid is considered in the presence of uniform vertical magnetic field. By applying normal mode analysis method, the dispersion relation has been derived and solved analytically. For stationary convection, Walters’B′ viscoelastic fluid behaves like an ordinary (Newtonian) fluid. It is studied that rotation has a stabilizing effect, whereas the magnetic field has both stabilizing and destabilizing effects. The rotation and magnetic field are found to introduce oscillatory mode in the system which were non – existent in their absence. The results are presented through graphs. Key words: Walters’B′ Fluid, thermal convection, viscoelasticity, Rotation, Magnetic field.
Persistence of power-law correlations in nonequilibrium steady states of gapp...Jarrett Lancaster
The existence of quasi-long range order is demonstrated in nonequilibrium steady states in isotropic XY spin chains including of two types of additional terms that generate a gap in the energy spectrum. The system is driven out of equilibrium by initializing a domain-wall magnetization profile through application of external magnetic field and switching off the magnetic field at the same time the energy gap is activated. An energy gap is produced by either applying a staggered magnetic field in the transverse direction or introducing a modulation to the XY coupling. The magnetization, spin current and spin-spin correlation functions are computed in the thermodynamic limit at long times after the quench. For both types of systems, we find the persistence of power-law correlations despite the ground state correlation functions exhibiting exponential decay. It is discussed how these power-law correlations appear related to the periodic nature of the perturbation which generates the energy gap.
Analytic Solution Of Stokes Second Problem For Second-Grade FluidSimar Neasy
Academic Paper Writing Service
http://StudyHub.vip/Analytic-Solution-Of-Stokes-Second-Prob 👈
Similar to QMC: Operator Splitting Workshop, Optimization and Control in Free ans Moving Boundary Fluid-Structure Interactions - Lorena Bociu, Mar 22, 2018 (20)
Recently, the machine learning community has expressed strong interest in applying latent variable modeling strategies to causal inference problems with unobserved confounding. Here, I discuss one of the big debates that occurred over the past year, and how we can move forward. I will focus specifically on the failure of point identification in this setting, and discuss how this can be used to design flexible sensitivity analyses that cleanly separate identified and unidentified components of the causal model.
I will discuss paradigmatic statistical models of inference and learning from high dimensional data, such as sparse PCA and the perceptron neural network, in the sub-linear sparsity regime. In this limit the underlying hidden signal, i.e., the low-rank matrix in PCA or the neural network weights, has a number of non-zero components that scales sub-linearly with the total dimension of the vector. I will provide explicit low-dimensional variational formulas for the asymptotic mutual information between the signal and the data in suitable sparse limits. In the setting of support recovery these formulas imply sharp 0-1 phase transitions for the asymptotic minimum mean-square-error (or generalization error in the neural network setting). A similar phase transition was analyzed recently in the context of sparse high-dimensional linear regression by Reeves et al.
Many different measurement techniques are used to record neural activity in the brains of different organisms, including fMRI, EEG, MEG, lightsheet microscopy and direct recordings with electrodes. Each of these measurement modes have their advantages and disadvantages concerning the resolution of the data in space and time, the directness of measurement of the neural activity and which organisms they can be applied to. For some of these modes and for some organisms, significant amounts of data are now available in large standardized open-source datasets. I will report on our efforts to apply causal discovery algorithms to, among others, fMRI data from the Human Connectome Project, and to lightsheet microscopy data from zebrafish larvae. In particular, I will focus on the challenges we have faced both in terms of the nature of the data and the computational features of the discovery algorithms, as well as the modeling of experimental interventions.
Bayesian Additive Regression Trees (BART) has been shown to be an effective framework for modeling nonlinear regression functions, with strong predictive performance in a variety of contexts. The BART prior over a regression function is defined by independent prior distributions on tree structure and leaf or end-node parameters. In observational data settings, Bayesian Causal Forests (BCF) has successfully adapted BART for estimating heterogeneous treatment effects, particularly in cases where standard methods yield biased estimates due to strong confounding.
We introduce BART with Targeted Smoothing, an extension which induces smoothness over a single covariate by replacing independent Gaussian leaf priors with smooth functions. We then introduce a new version of the Bayesian Causal Forest prior, which incorporates targeted smoothing for modeling heterogeneous treatment effects which vary smoothly over a target covariate. We demonstrate the utility of this approach by applying our model to a timely women's health and policy problem: comparing two dosing regimens for an early medical abortion protocol, where the outcome of interest is the probability of a successful early medical abortion procedure at varying gestational ages, conditional on patient covariates. We discuss the benefits of this approach in other women’s health and obstetrics modeling problems where gestational age is a typical covariate.
Difference-in-differences is a widely used evaluation strategy that draws causal inference from observational panel data. Its causal identification relies on the assumption of parallel trends, which is scale-dependent and may be questionable in some applications. A common alternative is a regression model that adjusts for the lagged dependent variable, which rests on the assumption of ignorability conditional on past outcomes. In the context of linear models, Angrist and Pischke (2009) show that the difference-in-differences and lagged-dependent-variable regression estimates have a bracketing relationship. Namely, for a true positive effect, if ignorability is correct, then mistakenly assuming parallel trends will overestimate the effect; in contrast, if the parallel trends assumption is correct, then mistakenly assuming ignorability will underestimate the effect. We show that the same bracketing relationship holds in general nonparametric (model-free) settings. We also extend the result to semiparametric estimation based on inverse probability weighting.
We develop sensitivity analyses for weak nulls in matched observational studies while allowing unit-level treatment effects to vary. In contrast to randomized experiments and paired observational studies, we show for general matched designs that over a large class of test statistics, any valid sensitivity analysis for the weak null must be unnecessarily conservative if Fisher's sharp null of no treatment effect for any individual also holds. We present a sensitivity analysis valid for the weak null, and illustrate why it is conservative if the sharp null holds through connections to inverse probability weighted estimators. An alternative procedure is presented that is asymptotically sharp if treatment effects are constant, and is valid for the weak null under additional assumptions which may be deemed reasonable by practitioners. The methods may be applied to matched observational studies constructed using any optimal without-replacement matching algorithm, allowing practitioners to assess robustness to hidden bias while allowing for treatment effect heterogeneity.
The world of health care is full of policy interventions: a state expands eligibility rules for its Medicaid program, a medical society changes its recommendations for screening frequency, a hospital implements a new care coordination program. After a policy change, we often want to know, “Did it work?” This is a causal question; we want to know whether the policy CAUSED outcomes to change. One popular way of estimating causal effects of policy interventions is a difference-in-differences study. In this controlled pre-post design, we measure the change in outcomes of people who are exposed to the new policy, comparing average outcomes before and after the policy is implemented. We contrast that change to the change over the same time period in people who were not exposed to the new policy. The differential change in the treated group’s outcomes, compared to the change in the comparison group’s outcomes, may be interpreted as the causal effect of the policy. To do so, we must assume that the comparison group’s outcome change is a good proxy for the treated group’s (counterfactual) outcome change in the absence of the policy. This conceptual simplicity and wide applicability in policy settings makes difference-in-differences an appealing study design. However, the apparent simplicity belies a thicket of conceptual, causal, and statistical complexity. In this talk, I will introduce the fundamentals of difference-in-differences studies and discuss recent innovations including key assumptions and ways to assess their plausibility, estimation, inference, and robustness checks.
We present recent advances and statistical developments for evaluating Dynamic Treatment Regimes (DTR), which allow the treatment to be dynamically tailored according to evolving subject-level data. Identification of an optimal DTR is a key component for precision medicine and personalized health care. Specific topics covered in this talk include several recent projects with robust and flexible methods developed for the above research area. We will first introduce a dynamic statistical learning method, adaptive contrast weighted learning (ACWL), which combines doubly robust semiparametric regression estimators with flexible machine learning methods. We will further develop a tree-based reinforcement learning (T-RL) method, which builds an unsupervised decision tree that maintains the nature of batch-mode reinforcement learning. Unlike ACWL, T-RL handles the optimization problem with multiple treatment comparisons directly through a purity measure constructed with augmented inverse probability weighted estimators. T-RL is robust, efficient and easy to interpret for the identification of optimal DTRs. However, ACWL seems more robust against tree-type misspecification than T-RL when the true optimal DTR is non-tree-type. At the end of this talk, we will also present a new Stochastic-Tree Search method called ST-RL for evaluating optimal DTRs.
A fundamental feature of evaluating causal health effects of air quality regulations is that air pollution moves through space, rendering health outcomes at a particular population location dependent upon regulatory actions taken at multiple, possibly distant, pollution sources. Motivated by studies of the public-health impacts of power plant regulations in the U.S., this talk introduces the novel setting of bipartite causal inference with interference, which arises when 1) treatments are defined on observational units that are distinct from those at which outcomes are measured and 2) there is interference between units in the sense that outcomes for some units depend on the treatments assigned to many other units. Interference in this setting arises due to complex exposure patterns dictated by physical-chemical atmospheric processes of pollution transport, with intervention effects framed as propagating across a bipartite network of power plants and residential zip codes. New causal estimands are introduced for the bipartite setting, along with an estimation approach based on generalized propensity scores for treatments on a network. The new methods are deployed to estimate how emission-reduction technologies implemented at coal-fired power plants causally affect health outcomes among Medicare beneficiaries in the U.S.
Laine Thomas presented information about how causal inference is being used to determine the cost/benefit of the two most common surgical surgical treatments for women - hysterectomy and myomectomy.
We provide an overview of some recent developments in machine learning tools for dynamic treatment regime discovery in precision medicine. The first development is a new off-policy reinforcement learning tool for continual learning in mobile health to enable patients with type 1 diabetes to exercise safely. The second development is a new inverse reinforcement learning tools which enables use of observational data to learn how clinicians balance competing priorities for treating depression and mania in patients with bipolar disorder. Both practical and technical challenges are discussed.
The method of differences-in-differences (DID) is widely used to estimate causal effects. The primary advantage of DID is that it can account for time-invariant bias from unobserved confounders. However, the standard DID estimator will be biased if there is an interaction between history in the after period and the groups. That is, bias will be present if an event besides the treatment occurs at the same time and affects the treated group in a differential fashion. We present a method of bounds based on DID that accounts for an unmeasured confounder that has a differential effect in the post-treatment time period. These DID bracketing bounds are simple to implement and only require partitioning the controls into two separate groups. We also develop two key extensions for DID bracketing bounds. First, we develop a new falsification test to probe the key assumption that is necessary for the bounds estimator to provide consistent estimates of the treatment effect. Next, we develop a method of sensitivity analysis that adjusts the bounds for possible bias based on differences between the treated and control units from the pretreatment period. We apply these DID bracketing bounds and the new methods we develop to an application on the effect of voter identification laws on turnout. Specifically, we focus estimating whether the enactment of voter identification laws in Georgia and Indiana had an effect on voter turnout.
We study experimental design in large-scale stochastic systems with substantial uncertainty and structured cross-unit interference. We consider the problem of a platform that seeks to optimize supply-side payments p in a centralized marketplace where different suppliers interact via their effects on the overall supply-demand equilibrium, and propose a class of local experimentation schemes that can be used to optimize these payments without perturbing the overall market equilibrium. We show that, as the system size grows, our scheme can estimate the gradient of the platform’s utility with respect to p while perturbing the overall market equilibrium by only a vanishingly small amount. We can then use these gradient estimates to optimize p via any stochastic first-order optimization method. These results stem from the insight that, while the system involves a large number of interacting units, any interference can only be channeled through a small number of key statistics, and this structure allows us to accurately predict feedback effects that arise from global system changes using only information collected while remaining in equilibrium.
We discuss a general roadmap for generating causal inference based on observational studies used to general real world evidence. We review targeted minimum loss estimation (TMLE), which provides a general template for the construction of asymptotically efficient plug-in estimators of a target estimand for realistic (i.e, infinite dimensional) statistical models. TMLE is a two stage procedure that first involves using ensemble machine learning termed super-learning to estimate the relevant stochastic relations between the treatment, censoring, covariates and outcome of interest. The super-learner allows one to fully utilize all the advances in machine learning (in addition to more conventional parametric model based estimators) to build a single most powerful ensemble machine learning algorithm. We present Highly Adaptive Lasso as an important machine learning algorithm to include.
In the second step, the TMLE involves maximizing a parametric likelihood along a so-called least favorable parametric model through the super-learner fit of the relevant stochastic relations in the observed data. This second step bridges the state of the art in machine learning to estimators of target estimands for which statistical inference is available (i.e, confidence intervals, p-values etc). We also review recent advances in collaborative TMLE in which the fit of the treatment and censoring mechanism is tailored w.r.t. performance of TMLE. We also discuss asymptotically valid bootstrap based inference. Simulations and data analyses are provided as demonstrations.
We describe different approaches for specifying models and prior distributions for estimating heterogeneous treatment effects using Bayesian nonparametric models. We make an affirmative case for direct, informative (or partially informative) prior distributions on heterogeneous treatment effects, especially when treatment effect size and treatment effect variation is small relative to other sources of variability. We also consider how to provide scientifically meaningful summaries of complicated, high-dimensional posterior distributions over heterogeneous treatment effects with appropriate measures of uncertainty.
Climate change mitigation has traditionally been analyzed as some version of a public goods game (PGG) in which a group is most successful if everybody contributes, but players are best off individually by not contributing anything (i.e., “free-riding”)—thereby creating a social dilemma. Analysis of climate change using the PGG and its variants has helped explain why global cooperation on GHG reductions is so difficult, as nations have an incentive to free-ride on the reductions of others. Rather than inspire collective action, it seems that the lack of progress in addressing the climate crisis is driving the search for a “quick fix” technological solution that circumvents the need for cooperation.
This seminar discussed ways in which to produce professional academic writing, from academic papers to research proposals or technical writing in general.
Machine learning (including deep and reinforcement learning) and blockchain are two of the most noticeable technologies in recent years. The first one is the foundation of artificial intelligence and big data, and the second one has significantly disrupted the financial industry. Both technologies are data-driven, and thus there are rapidly growing interests in integrating them for more secure and efficient data sharing and analysis. In this paper, we review the research on combining blockchain and machine learning technologies and demonstrate that they can collaborate efficiently and effectively. In the end, we point out some future directions and expect more researches on deeper integration of the two promising technologies.
In this talk, we discuss QuTrack, a Blockchain-based approach to track experiment and model changes primarily for AI and ML models. In addition, we discuss how change analytics can be used for process improvement and to enhance the model development and deployment processes.
More from The Statistical and Applied Mathematical Sciences Institute (20)
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
How libraries can support authors with open access requirements for UKRI fund...
QMC: Operator Splitting Workshop, Optimization and Control in Free ans Moving Boundary Fluid-Structure Interactions - Lorena Bociu, Mar 22, 2018
1. Optimization and Control in Free and
Moving Boundary Fluid-Structure
Interactions
Lorena Bociu 1
SAMSI - Operator Splitting Methods in Data Analysis
March 21, 2018
1
The research was supported by NSF-DMS 1312801 and NSF CAREER 1555062
2. Fluid-Structure Interactions (FSI)
Interaction of some movable and/or deformable structure with
an internal or surrounding fluid flow
industrial processes, aero-elasticity, and biomechanics
The boundary of the domain is not known in advance,
but has to be determined as part of the solution.
Free boundary: steady-state problem.
Moving boundary: time dependent problems and the position of the
boundary is a function of time and space.
3. Lagrangian vs Eulerian
Coupling of incompressible Navier-Stokes equations with
an elastic solid.
Solid: displacements are often relatively small
computational domain: Ω
Lagrangian formulation: focus on the material particle x and its
evolution
Fluid: displacements are large and usually irrelevant
we are mostly interested in the velocity field
Eulerian framework: observe what happens at a given point x in the
physical space.
4. Lagrangian vs Eulerian
Coupling of incompressible Navier-Stokes equations with
an elastic solid.
Solid: displacements are often relatively small
computational domain: Ω
Lagrangian formulation: focus on the material particle x and its
evolution
Fluid: displacements are large and usually irrelevant
we are mostly interested in the velocity field
Eulerian framework: observe what happens at a given point x in the
physical space.
FSI: Fluid + elasticity + interface conditions between solid and fluid
5. Lagrangian vs Eulerian
Coupling of incompressible Navier-Stokes equations with
an elastic solid.
Solid: displacements are often relatively small
computational domain: Ω
Lagrangian formulation: focus on the material particle x and its
evolution
Fluid: displacements are large and usually irrelevant
we are mostly interested in the velocity field
Eulerian framework: observe what happens at a given point x in the
physical space.
FSI: Fluid + elasticity + interface conditions between solid and fluid
match these two different frameworks
6. Fluid - Elasticity Interaction: PDE model
Configuration: the elastic body moves and deforms inside the fluid.
Elastic body located at time t ≥ 0 in a domain Ω(t) ⊂ R3
with
boundary Γ(t).
The fluid occupies domain Ωf
(t) = D ¯Ω(t), with smooth boundary
Γ(t) ∪ Γf .
Let D ⊂ R3
be the control volume. D contains the solid and the
fluid at each time t ≥ 0, i.e. D = Ω(t) ∪ Ωf
(t), with smooth
boundary ∂D = Γf .
7. Navier-Stokes - Eulerian Framework
Fluid: Newtonian viscous, homogeneous, and incompressible.
Its behavior is described by its velocity w and pressure p.
The viscosity of the fluid is ν > 0, and the fluid strain and
stress tensors are given by
ε(w) =
1
2
[Dw + (Dw)∗
], σ(p, w) = −pI + 2νε(w),
where Dw is the gradient matrix of w, and (Dw)∗ represents
the transpose of Dw.
The fluid state satisfies the following Navier-Stokes equations:
wt − ν ∆w + Dw · w + p = v1 on Ωf (t)
div w = 0 on Ωf (t)
w = 0 on Γf
8. Structural Deformation: Lagrangian formulation
The evolution of the fluid domain Ωf
(t) is induced by the
structural deformation through the common interface Γ(t).
O ⊂ D: reference configuration for the solid; ∂O = S
Of
= D ¯O: reference fluid configuration. T
D is described by a smooth, injective map:
ϕ : ¯D × R+
−→ ¯D, (x, t) → ϕ = ϕ(x, t).
For x ∈ O, ϕ(x, t): the position at time t of the material point x.
On Of
, ϕ(x, t) is defined as an arbitrary extension of the restriction
of ϕ to S, which preserves the boundary Γf , i.e. ϕ = IΓf
on Γf .
J(ϕ) > 0: Jacobian of the deformation ϕ(t)
9. Nonlinear elasticity
St. Venant - Kirchhoff equations: large displacement, small
deformation elasticity. Green-St. Venant nonlinear strain tensor:
σ(ϕ) =
1
2
[(Dϕ)∗
Dϕ − I].
Piola transform of the Cauchy stress tensor:
P(x) = Dϕ(x)[λTr[σ(ϕ)]I + 2µσ(ϕ)])
Equilibrium equations for elasticity :
Jρ∂ttϕ − DivP = Jρv2 on O
10. FSI - PDE model
wt − ν ∆w + Dw · w + p = v1 on Ωf
(t)
div w = 0 on Ωf
(t)
w = 0 on Γf
Jρ∂ttϕ − DivP = Jρv2 on O
w ◦ ϕ = ϕt on S
Pn = J(ϕ)(σ(p, w) ◦ ϕ)(Dϕ)−∗
n on S
ϕ = IΓf
on Γf ,
IC:ϕ(·, 0) = ϕ0
, ϕt(·, 0) = ϕ1
, w(·, 0) = w0
, p(·, 0) = p0
on (O)2
× (Oc
)2
.
11. PDE-constrained Optimization Problems governed by FSI
In most of the applications, the goal is the
optimization or optimal control of the considered process,
related sensitivity analysis (with respect to relevant physical parameters).
minimize turbulence in the fluid
optimize fluid velocity or pressure
optimize the deformation of the structure
minimize wall shear stresses
12. PDE-constrained Optimization Problems governed by FSI
In most of the applications, the goal is the
optimization or optimal control of the considered process,
related sensitivity analysis (with respect to relevant physical parameters).
minimize turbulence in the fluid
optimize fluid velocity or pressure
optimize the deformation of the structure
minimize wall shear stresses
Control problems in FSI: most of the literature is focused on the
assumption of small but rapid oscillations of the solid body, so that
the common interface may be assumed fixed:
Lasiecka and Bucci ’05, ’10, Lasiecka, Triggiani, and Zhang ’11,
Lasiecka and Tuffaha, ’08-’09, Avalos-Triggiani ’08-’12.
13. PDE-constrained Optimization Problems governed by FSI
In most of the applications, the goal is the
optimization or optimal control of the considered process,
related sensitivity analysis (with respect to relevant physical parameters).
minimize turbulence in the fluid
optimize fluid velocity or pressure
optimize the deformation of the structure
minimize wall shear stresses
Recently, PDE constrained optimization problems governed by free
boundary interactions have been considered, with most research
studies mainly addressed in the context of the numerical analysis
of the finite element methods [Antil-Nochetto-Sodre ’14,
Richter-Wick ’13, Van Der Zee et al ’10]
14. Steady State Navier-Stokes and Elasticity
−ν ∆w + Dw · w + p = v|Ωf
on Ωf
divw = 0 on Ωf
w = 0 on Γ := ϕ(S)
−DivP = v|Ωe
on O
Pn = J(ϕ)(σ(p, w) ◦ ϕ)(Dϕ)−∗
n on S
w = 0, ϕ = IΓf
on Γf
2
P.G. Ciarlet, Mathematical Elasticity Vol. I: Three-dimensional Elasticity, North-Holland Publishing Co.,
15. Steady State Navier-Stokes and Elasticity
−ν ∆w + Dw · w + p = v|Ωf
on Ωf
divw = 0 on Ωf
w = 0 on Γ := ϕ(S)
−DivP = v|Ωe
on O
Pn = J(ϕ)(σ(p, w) ◦ ϕ)(Dϕ)−∗
n on S
w = 0, ϕ = IΓf
on Γf
Cauchy Stress Tensor T : Ωe → S3
, T = [J−1
P · (Dϕ)∗
] ◦ ϕ−1
[2
]
−ν ∆w + Dw · w + p = v|Ωf
on Ωf
divw = 0 on Ωf
w = 0 on Γ := ϕ(S)
−DivT = v|Ωe
on Ωe = ϕ(O)
T n = σ(p, w)n on Γ
w = 0, ϕ = IΓf
on Γf .
2
P.G. Ciarlet, Mathematical Elasticity Vol. I: Three-dimensional Elasticity, North-Holland Publishing Co.,
16. OCP
We consider the optimal control problem:
min J(w, v) = 1/2 w − wd
2
L2(Ωf ) + 1/2 v 2
H3(D) (1)
subject to
−ν ∆w + Dw · w + p = v|Ωf
on Ωf
divw = 0 on Ωf
w = 0 on Γ := ϕ(S)
−DivT = v|Ωe
on Ωe = ϕ(O)
T n = σ(p, w)n on Γ
w = 0, ϕ = IΓf
on Γf .
distributed control v ∈ H3
(D)
wd ∈ L2
(Ωf ) is a desired fluid velocity.
17. OCP
min J(w, v) = 1/2 w − wd
2
L2(Ωf ) + 1/2 v 2
H3(D) (2)
subject to
(E)
−ν ∆w + Dw · w + p = v|Ωf
on Ωf
divw = 0 on Ωf
w = 0 on Γ := ϕ(S)
−DivT = v|Ωe on Ωe = ϕ(O)
T n = σ(p, w)n on Γ
w = 0, ϕ = IΓf
on Γf .
Goals:
1. Existence of an optimal control
2. First-order necessary conditions of optimality (NOC)
18. Well-posedness Analysis
FSI: parabolic-hyperbolic coupled system
regularity gap of the fluid and structure velocities on the common
interface: the traces of the elastic component at the energy level are
not defined via the standard trace theory, and this induces a loss of
regularity at the boundary of the coupled system.
Coutand-Shkoller ’05-’06: Existence of strong solutions for the case of a linear and then quasi-linear elastic
body flowing within a viscous, incompressible fluid, under the assumptions of smooth initial data (i.e., the
initial fluid velocity w0
belongs to H5
, and the initial data for elasticity (ϕ0
, ϕ1
) belong to H3
× H2
).
Due to the incompressibility condition of the fluid, uniqueness of solution for the model required higher
regularity for the initial data (i.e., (w0
, ϕ0
, ϕ1
) ∈ H7
× H5
× H4
).
Kukavica-Tuffaha-Ziane ’09-’11, Ignatova-Kukavica-Lasiecka-Tuffaha ’12-’14, Raymond-Vanninathan ’15
for N-S coupled with linear elasticity/wave equation.
The authors of [Ignatova-Kukavica-Lasiecka-Tuffaha] also prove global in time well-posedness for small
initial data of the Navier–Stokes-elasticity model involving a wave equation with frictional damping, and
they show that the energy associated with smooth and sufficiently small solutions of the damped model
decay exponentially to zero.
Canic-Muha ’13-’14: dynamical coupling (which is of great interest in the modeling and analysis of the
cardiovascular system) - 2D
Grandmont’02, Wick-Wollner’14: steady state NS-St. Venant elasticity equations.
19. OCP
min J(w, v) = 1/2 w − wd
2
L2(Ωf ) + 1/2 v 2
H3(D) (3)
subject to
(E)
−ν ∆w + Dw · w + p = v|Ωf
on Ωf
divw = 0 on Ωf
w = 0 on Γ := ϕ(S)
−DivT = v|Ωe on Ωe = ϕ(O)
T n = σ(p, w)n on Γ
w = 0, ϕ = IΓf
on Γf .
Goals:
1. Existence of an optimal control (EOC)
2. First-order necessary conditions of optimality (NOC)
Compute the gradient of the functional J.
Characterization of the optimal control will pave the way for a
numerical study of the problem.
20. NOC: Main Challenge
Lagrangian: L = J − (weak form of the system)
Not convex-concave, due to the nonlinearity of the
control-to-state map.
Min-Max theory does not apply, i.e., one can not reduce the
cost function gradient to the derivative of the Lagrangian with
respect to the control, at its saddle point [ Delfour-Zolesio ’86]
Optimality conditions must be derived from differentiability
arguments on the cost functional J with respect to the control v.
Main challenge: dependence of the cost integrals in J on the
unknown domain Ωf , which also depends on the control v.
min J(w, v) = 1/2 w − wd
2
L2(Ωf ) + 1/2 v 2
H3(D)
Directional derivative of J with respect to v in the direction of v :
for small parameter s ≥ 0, consider the perturbed functional
J(v + sv ) and then calculate the derivative at s = 0 of the function
s → J(v + sv ).
21. With the following notation for the s-derivatives at s = 0,
ϕ =
∂
∂s
ϕs
s=0
, U = ϕ ◦ϕ−1
, w =
∂
∂s
ws
s=0
, and p =
∂
∂s
ps
s=0
,
we can compute the directional derivative of J as
∂J(v; v ) = lim
s→0
J(v + sv ) − J(v)
s
=
∂
∂s
J(v + sv )
s=0
=
∂
∂s
1
2 (Ωf )s
|ws − wd |2
+
1
2
v + sv 2
H3(D)
s=0
=
Ωf
(w − wd ) · w +
1
2 Γ
|w − wd |2
U · nf + (v, v )H3(D)
22. With the following notation for the s-derivatives at s = 0,
ϕ =
∂
∂s
ϕs
s=0
, U = ϕ ◦ϕ−1
, w =
∂
∂s
ws
s=0
, and p =
∂
∂s
ps
s=0
,
we can compute the directional derivative of J as
∂J(v; v ) = lim
s→0
J(v + sv ) − J(v)
s
=
∂
∂s
J(v + sv )
s=0
=
∂
∂s
1
2 (Ωf )s
|ws − wd |2
+
1
2
v + sv 2
H3(D)
s=0
=
Ωf
(w − wd ) · w +
1
2 Γ
|w − wd |2
U · nf + (v, v )H3(D)
The challenge of applying optimization tools to free boundary FSI is
the proper derivation of the sensitivity and adjoint sensitivity
information with correct balancing conditions on the common
interface.
As the interaction is a coupling of Eulerian and Lagrangian
quantities, sensitivity analysis on the system falls into the framework
of shape analysis.
23. Sensitivity System [LB - J.-P. Zolesio]
−ν∆w + (Dw’) w + (Dw) w’ + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U’ = 0 on Γ
−Div T(U’) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U’) on Γ
w = 0, U = 0 on Γf
24. Sensitivity System [LB - J.-P. Zolesio]
−ν∆w + (Dw’) w + (Dw) w’ + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U’ = 0 on Γ
−Div T(U’) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U’) on Γ
w = 0, U = 0 on Γf
Θ = Dϕ ◦ ϕ−1
DU := Θ∗
(DU )Θ,
T(U ) := (DU )T +
1
det Θ
Θ · {λTr(DU )I + µ[DU + (DU )∗
]}Θ∗
,
25. Sensitivity System [LB - J.-P. Zolesio]
−ν∆w + (Dw’) w + (Dw) w’ + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U’ = 0 on Γ
−Div T(U’) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U’) on Γ
w = 0, U = 0 on Γf
Θ = Dϕ ◦ ϕ−1
DU := Θ∗
(DU )Θ,
T(U ) := (DU )T +
1
det Θ
Θ · {λTr(DU )I + µ[DU + (DU )∗
]} Θ∗
,
Σ(U )
26. Sensitivity System [LB - J.-P. Zolesio]
−ν∆w + (Dw’) w + (Dw) w’ + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U’ = 0 on Γ
−Div T(U’) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U’) on Γ
w = 0, U = 0 on Γf
Θ = Dϕ ◦ ϕ−1
DU := Θ∗
(DU )Θ,
T(U ) := (DU )T +
1
det Θ
Θ · {λTr(DU )I + µ[DU + (DU )∗
]} Θ∗
,
Σ(U )
˜Σ(U )
27. Sensitivity System [LB - J.-P. Zolesio]
−ν∆w + (Dw’) w + (Dw) w’ + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U’ = 0 on Γ
−Div T(U’) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U’) on Γ
w = 0, U = 0 on Γf
Θ = Dϕ ◦ ϕ−1
DU := Θ∗
(DU )Θ,
T(U ) := (DU’)T +
1
det Θ
Θ · {λTr(DU )I + µ[DU + (DU )∗
]} Θ∗
,
Σ(U )
˜Σ(U )
28. Sensitivity System [LB - J.-P. Zolesio ]
−ν∆w + (Dw’) w + (Dw) w’ + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U’ = 0 on Γ
−DivT(U ) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U’) on Γ
w = 0, U = 0 on Γf
29. Sensitivity System [LB - J.-P. Zolesio ]
−ν∆w + (Dw’) w + (Dw) w’ + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U’ = 0 on Γ
−DivT(U ) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U’) on Γ
w = 0, U = 0 on Γf
Γ U , n
B(U ) =(T + pI − 2νε(w)) · [(DΓU )∗
n + (D2
bΩe )UΓ] +(DT )U · n
+ div(U )T · n − T · (DU )∗
· n−
− U , n (−DivΓ(T ) + [∂νpI − 2ν∂νε(w)] · n).
30. Notation
(Df)ij = ∂j fi ∈ M3
is the gradient matrix at a ∈ X of any vector
field f = (fi ) : X ⊂ R3
→ R3
.
div f = ∂i fi ∈ R is the divergence of f : X ⊂ R3
→ R3
.
Div T = ∂j Tij ∈ R3
is the divergence of any second-order tensor
field T = (Tij ) : X ⊂ R3
→ M3
.
A∗
= transpose of A, for any A ∈ M3
.
dΩ(x) =
infy∈Ω |y − x| Ω = ∅
∞ Ω = ∅
is the distance function
bΩ(x) = dΩ(x) − dΩc (x) , ∀x ∈ Rn
is the oriented distance fn.
from x to Ω, for any Ω ⊂ Rn
.
H = ∆bΩ = Tr(D2
bΩ) is the additive curvature of Γ = ∂Ω. [3
]
3
M.C. Delfour and J.P. Zolesio, Shapes and Geometries: Analysis, Differential Calculus and Optimization,
SIAM 2001.
31. Sensitivity System [LB - J.-P. Zolesio]
−ν∆w + (Dw )w + (Dw)w + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U = 0 on Γ
−DivT(U ) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U’) on Γ
w = 0, U = 0 on Γf
Γ U , n
B(U ) =(T + pI − 2νε(w)) · [(DΓU )∗
n + (D2
bΩe
)UΓ] +(DT )U · n
+ div(U )T · n − T · (DU )∗
· n−
− U , n (−DivΓ(T ) + [∂νpI − 2ν∂νε(w)] · n).
32. Sensitivity System [LB - J.-P. Zolesio]
−ν∆w + (Dw )w + (Dw)w + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U = 0 on Γ
−DivT(U ) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U’) on Γ
w = 0, U = 0 on Γf
Γ U , n
B(U ) =(T + pI − 2νε(w)) · [(DΓU )∗
n + (D2
bΩe
)UΓ] +(DT )U · n
+ div(U )T · n − T · (DU )∗
· n−
− U , n (−DivΓ(T ) + [∂νpI − 2ν∂νε(w)] · n).
= B1 · Γ U , n − U , n B2 + (DT )U · n + div(U )T · n − T · (DU )∗
· n
33. Connection to Shape Analysis
As vs = v + sv , the geometry of the problem moves with the
flow of a vector field that depends on the deformation ϕs.
The perturbation Γs of the boundary is built by the flow of the
vector field V (s, x) = ∂
∂s ϕs ◦ ϕ−1
s , i.e.,
Γs = Ts(V )(S), where Ts(V ) : Ωe → (Ωe)s, Ts(V ) = ϕs◦ϕ−1
.
34. Connection to Shape Analysis
As vs = v + sv , the geometry of the problem moves with the
flow of a vector field that depends on the deformation ϕs.
The perturbation Γs of the boundary is built by the flow of the
vector field V (s, x) = ∂
∂s ϕs ◦ ϕ−1
s , i.e.,
Γs = Ts(V )(S), where Ts(V ) : Ωe → (Ωe)s, Ts(V ) = ϕs◦ϕ−1
.
(ϕ , w , p ): ‘shape’ derivatives with respect to the speed V , which
is a vector field that depends on ϕs and is not given a priori.
Standard theory on shape derivatives: the domain is perturbed
by an a priori given vector field and then the speed method is
applied.
s-derivatives: ‘pseudo-shape derivatives’, in the sense that
much of the theory of shape calculus remains applicable.
35. Goal: find the gradient of J at v: J (v; v )
∂J(v; v ) =
Ωf
(w − wd ) · w +
1
2 Γ
|w − wd |2
U · nf + (v, v )H3(D)
Sensitivity system provides the characterization for (U , w , p ):
−ν∆w + (Dw )w + (Dw)w + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U = 0 on Γ
−DivT(U ) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U ) on Γ
w = 0, U = 0 on Γf
v does not appear in the chain rule computation, since it is hidden
in the sensitivity equations for w , p , and U .
36. Goal: find the gradient of J at v: J (v; v )
∂J(v; v ) =
Ωf
(w − wd ) · w +
1
2 Γ
|w − wd |2
U · nf + (v, v )H3(D)
Sensitivity system provides the characterization for (U , w , p ):
−ν∆w + (Dw )w + (Dw)w + p = v Ωf
in Ωf
divw = 0 in Ωf
w + (Dw)U = 0 on Γ
−DivT(U ) = v Ωe
in Ωe
T(U ) · n = (−p I + 2νε(w )) · n + B(U ) on Γ
w = 0, U = 0 on Γf
v does not appear in the chain rule computation, since it is hidden
in the sensitivity equations for w , p , and U .
Idea: Introduce a suitable adjoint problem that eliminates the
s-derivatives and provides an explicit representation for J (v; v ).
37. Theorem (LB - K. Martin)
For the optimal control problem:
min J(w, v) = 1/2 w − wd
2
L2(Ωf ) + 1/2 v 2
H3(D),
subject to FSI, the gradient of the cost functional is given by
J (v; v ) = (v , v)D + (v |Ωf
, Q) + (v |Ωe
, R),
where Q, P, and R solve the following adjoint sensitivity problem:
−ν∆Q + (Dw)∗
Q − (DQ)w + P = w − wd Ωf
div(Q) = 0 Ωf
−Div ¯T (R) = 0 Ωe
Q = R Γ
¯T (R)n + (Dw)∗
σ(P, Q)n + divΓ[B1R]n − (DT ∆
· n)∗
R
−H T n, R n + Γ T n, R
−DivΓ(n ⊗ T R) + B2, R n = 1
2 |w − wd |2
nf Γ
Q = 0 Γf
(4)
38. Matching of Normal Stress Tensors
¯T (R)n+(Dw)∗
σ(P, Q)n+divΓ[B1R]n−(DT ∆
·n)∗
R−H T n, R n+ Γ T n, R
−DivΓ(n ⊗ T R) + B2, R n =
1
2
|w − wd |2
nf
B1 = T + pI − 2νε(w) and B2 = −DivΓ(T ) + [∂νpI − 2ν∂νε(w)] · n
(DT ∆
· f )ik := ∂k Tij fj
39. Matching of Normal Stress Tensors
¯T (R)n+(Dw)∗
σ(P, Q)n+divΓ[B1R]n−(DT ∆
·n)∗
R−H T n, R n+ Γ T n, R
−DivΓ(n ⊗ T R) + B2, R n =
1
2
|w − wd |2
nf
B1 = T + pI − 2νε(w) and B2 = −DivΓ(T ) + [∂νpI − 2ν∂νε(w)] · n
(DT ∆
· f )ik := ∂k Tij fj
DT is defined as (DT .e)ij = (∂k Tij )ek . With the above notation,
we can IBP
˜Γc
{(DT )γ} · ne, R =
˜Γc
(∂k Tij γk )(ne)j Ri
=
˜Γc
γk (∂k Tij (ne)j Ri ) =
˜Γc
γ, (DT ∆
· ne)∗
R .
40. Matching of Normal Stress Tensors
¯T (R)n+(Dw)∗
σ(P, Q)n+divΓ[B1R]n−(DT ∆
·n)∗
R−H T n, R n+ Γ T n, R
−DivΓ(n ⊗ T R) + B2, R n =
1
2
|w − wd |2
nf
B1 = T + pI − 2νε(w) and B2 = −DivΓ(T ) + [∂νpI − 2ν∂νε(w)] · n
(DT ∆
· f )ik := ∂k Tij fj
DT is defined as (DT .e)ij = (∂k Tij )ek . With the above notation,
we can IBP
˜Γc
{(DT )γ} · ne, R =
˜Γc
(∂k Tij γk )(ne)j Ri
=
˜Γc
γk (∂k Tij (ne)j Ri ) =
˜Γc
γ, (DT ∆
· ne)∗
R .
˜B(R) = divΓ[B1R]n − (DT ∆
· n)∗
R − H T n, R n + Γ T n, R
−DivΓ(n ⊗ T R) + B2, R n
41. Theorem (LB - K. Martin)
For the optimal control problem:
min J(w, v) = 1/2 w − wd
2
L2(Ωf ) + 1/2 v 2
H3(D),
subject to FSI, the gradient of the cost functional is given by
J (v; v ) = (v , v)D + (v |Ωf
, Q) + (v |Ωe
, R),
where Q, P, and R solve the following adjoint sensitivity problem:
−ν∆Q + (Dw)∗
Q − (DQ)w + P = w − wd Ωf
div(Q) = 0 Ωf
−Div ¯T (R) = 0 Ωe
Q = R Γ
¯T (R)n + (Dw)∗
σ(P, Q)n + ˜B(R) = 1
2 |w − wd |2
nf Γ
Q = 0 Γf
(5)
42. References
1. F. Abergel and R. Temam, On Some Control Problems in Fluid Mechanics, Theoret. Comput. Fluid
Dynamics 1, (1990), 303-325.
2. H. Antil, R. H. Nochetto, and P. Sodr´e, Optimal Control of a Free Boundary Problem: Analysis with
Second-Order Sufficient Conditions, SIAM J. Control Optim. 52, 5, (2014), 2771-2799.
3. L. Bociu, L. Castle, K. Martin, and D. Toundykov, Optimal Control in a Free Boundary Fluid-Elasticity
Interaction, AIMS Proceedings, (2015), 122-131.
4. L. Bociu, D. Toundykov, and J.-P. Zol´esio, Well-Posedness Analysis for a Linearization of a Fluid-Elasticity
Interaction, SIAM J. Math. Anal., 47, 3, (2015), 1958-2000.
5. L. Bociu, J.-P. Zol´esio, Linearization of a coupled system of nonlinear elasticity and viscous fluid, “Modern
Aspects of the Theory of Partial Differential Equations”, in the series “Operator Theory: Advances and
Applications”, 216, (Springer, Basel, 2011), 93-120.
6. L. Bociu, J.-P. Zol´esio, Existence for the linearization of a steady state fluid - nonlinear elasticity
interaction, DCDA-S, (2011), 184-197.
7. L. Bociu, J.-P. Zol´esio, Sensitivity analysis for a free boundary fluid-elasticity interaction, EECT 2, (2012),
55-79.
8. P.G. Ciarlet, Mathematical Elasticity Volume I: Three-dimensional Elasticity, North-Holland Publishing Co.,
Amsterdam, 1988.
9. M.C. Delfour and J.P. Zolesio, Shapes and Geometries: Analysis, Differential Calculus and Optimization,
SIAM 2001.
10. L. Formaggia, A. Quarteroni, A. Veneziani Eds., Cardiovascular Mathematics. Modeling and simulation of
the circulatory system. Vol I., MS & A, (Springer-Verlag Italia, Milano, 2009).
11. T. Richter and T. Wick, Optimal Control and Parameter Estimation for Stationary Fluid-Structure
Interaction Problems, SIAM J. Sci. Comput., 35, 5, (2013), 1085-1104.
12. T. Wick and W. Wollner, On the differentiability of fluid-structure interaction problems RICAM-Report,
16, (2014).