Adaptive multilevel trust-region methods for time-dependent PDE-constrained optimization

On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D

The topic for the current Mini-Workshop organized by Karl Kunisch (Univ. Graz), Angela Kunoth (Univ. Bonn) and Rolf Rannacher (Univ. Heidelberg) emerged from the Oberwolfach Workshop “Numerical Techniques for Optimization Problems with PDE Constraints” which was held February 16–22, 2003. It was realized that numerically solving control problems which are constrained by time-dependent nonlinear PDEs (Partial Differential Equations) are particularly challenging with respect to the complexity of the problem. Mathematically, one has to minimize a functional under PDE constraints and possibly additional constraints on the state and the control. Standard discretizations on uniform grids in space and time will only yield solutions where the inherent structures of the problem (nonlinearity, constraints) are not sufficiently captured. Certain optimization problems for large coupled systems of partial differential equations are currently not yet numerically treatable or do not satisfy the time constraints required in practice. Overcoming this barrier can only be achieved by designing new mathematically founded algorithmic approaches. The road towards this goal leads to many interesting problems in optimization, linear algebra, numerics, analysis, and approximation theory. The conference had 21 participants which represented continuous optimization, numerical analysis and scientific computing. 18 talks were given. The 10 longer, overview-style talks were on optimization with PDEs, focussing on These talks were intended to bridge the gap between the different research fields Optimization and Numerics. They were complemented by 8 shorter talks on more specialized research topics, ranging from efficiency indices for optimization over iterative methods for the coupled systems and multigrid acceleration to modelling issues in optimization, and data compression by means of proper orthogonal decompositions and central Voronoi tesselations. Different modern approaches to overcome the complexity issues in numerical simulations for PDE-constrained optimization have been presented and discussed. One of the approaches is to employ fast iterative solvers like multigrid on uniform grids. The methodology which conceptually provides the largest potential is to introduce adaptivity. This drastically reduces complexity but depending on the context may require solving an additional problem. Wavelet approaches particularly allow to resolve each of the variables separately and in addition provide a built-in preconditioning. Yet another approach uses compressed information in order to efficiently solve the primal-dual system. All these issues are currently very active research areas. The extensive discussions held during this workshop have produced a number of new ideas and connections. It was agreed upon that a mere black-box-style matching of efficient PDE codes with optimization tool boxes would on one hand remain much below its potential and on the other hand not help overcoming complexity barriers. Some concepts from automatic differentiation seem to carry over to adaptive methods. Even combining adaptivity with proper orthogonal decompositions may be a very promising direction. The many new ideas discussed during this workshop will have to be further elaborated in future.

PurposeThis paper aims to examine the thermal transmission and entropy generation of hybrid nanofluid filled containers with solid body inside. The solid body is seen as being both isothermal and capable of producing heat. A time-dependent non-linear partial differential equation is used to represent the transfer of heat through a solid body. The current study’s objective is to investigate the key properties of nanoparticles, external forces and particular attention paid to the impact of hybrid nanoparticles on entropy formation. This investigation is useful for researchers studying in the area of cavity flows to know features of the flow structures and nature of hybrid nanofluid characteristics. In addition, a detailed entropy generation analysis has been performed to highlight possible regimes with minimal entropy generation rates. Hybrid nanofluid has been proven to have useful qualities, making it an attractive coolant for an electrical device. The findings would help scientists and engineers better understand how to analyse convective heat transmission and how to forecast better heat transfer rates in cutting-edge technological systems used in industries such as heat transportation, power generation, chemical production and passive cooling systems for electronic devices.Design/methodology/approachThermal transmission and entropy generation of hybrid nanofluid are analysed within the enclosure. The domain of interest is a square chamber of size L, including a square solid block. The solid body is considered to be isothermal and generating heat. The flow driven by temperature gradient in the cavity is two-dimensional. The governing equations, formulated in dimensionless primitive variables with corresponding initial and boundary conditions, are worked out by using the finite volume technique with the SIMPLE algorithm on a uniformly staggered mesh. QUICK and central difference schemes were used to handle convective and diffusive elements. In-house code is developed using FORTRAN programming to visualize the isotherms, streamlines, heatlines and entropy contours, which are handled by Tecplot software. The influence of nanoparticles volume fraction, heat generation factor, external magnetic forces and an irreversibility ratio on energy transport and flow patterns is examined.FindingsThe results show that the hybrid nanoparticles concentration augments the thermal transmission and the entropy production increases also while the augmentation of temperature difference results in a diminution of entropy production. Finally, magnetic force has the significant impact on heat transfer, isotherms, streamlines and entropy. It has been observed that the external magnetic force plays a good role in thermal regulations.Research limitations/implicationsHybrid nanofluid is a desirable coolant for an electrical device. Various nanoparticles and their combinations can be analysed. Ferro-copper hybrid nanofluid considered with the help of prevailing literature review. The research would benefit scientists and engineers by improving their comprehension of how to analyses convective heat transmission and forecast more accurate heat transfer rates in various fields.Practical implicationsDue to its helpful characteristics, ferrous-copper hybrid nanofluid is a desirable coolant for an electrical device. The research would benefit scientists and engineers by improving their comprehension of how to analyse convective heat transmission and forecast more accurate heat transfer rates in cutting-edge technological systems used in sectors like thermal transportation, cooling systems for electronic devices, etc.Social implicationsEntropy generation is used for an evaluation of the system’s performance, which is an indicator of optimal design. Hence, in recent times, it does a good engineering sense to draw attention to irreversibility under magnetic force, and it has an indispensable impact on investigation of electronic devices.Originality/valueAn efficient numerical technique has been developed to solve this problem. The originality of this work is to analyse convective energy transport and entropy generation in a chamber with internal block, which is capable of maintaining heat and producing heat. Effects of irreversibility ratio are scrutinized for the first time. Analysis of convective heat transfer and entropy production in an enclosure with internal isothermal/heat generating blocks gives the way to predict enhanced heat transfer rate and avoid the failure of advanced technical systems in industrial sectors.

We discuss a linear-quadratic optimal control problem with pointwise control and state constraints. The state constraints are regularized by a Lavrentiev type regularization. The main results of the paper are estimates for the regularization error and the stability with respect to noisy data.

Advances in Adaptive Computational Methods in Mechanics

The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on finer grids. A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, whichused the CG method as basic iteration method on each level. In [18] it has been proven, that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasi-uniform triangulations. The present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non-uniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and robustness of the cascadic multigrid method.

We study parametric optimal control problems governed by a system of time-dependent partial differential equations (PDE) and subject to additional control and state constraints. An approach is presented to compute the optimal control functions and the so-called sensitivity differentials of the optimal solution with respect to perturbations. This information plays an important role in the analysis of optimal solutions as well as in real-time optimal control. The method of lines is used to transform the perturbed PDE system into a large system of ordinary differential equations. A subsequent discretization then transcribes parametric ODE optimal control problems into perturbed nonlinear programming problems (NLP), which can be solved efficiently by SQP methods. Second-order sufficient conditions can be checked numerically and we propose to apply an NLP-based approach for the robust computation of the sensitivity differentials of the optimal solutions with respect to the perturbation parameters. The numerical method is illustrated by the optimal control and sensitivity analysis of the Burgers equation.

Combination of an adaptive multilevel SQP method and a space-time adaptive PDAE solver for optimal control problems

In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples.

An abstract linear-quadratic optimal control problem with pointwise control constraints is investigated. This article is concerned with the discretization of the control by piecewise linear functions. Under the assumption that the optimal control and the optimal adjoint state are Lipschitz continuous and piecewise of class C 2, an approximation of order h 3/2, is proved for the solution of the control discretized problem with respect to the solution of the continuous one. The theoretical results are illustrated by numerical tests.

Cardiac arrhythmias affect millions of adults in the U.S. each year. This irregularity in the beating of the heart is often caused by dysregulation of calcium in cardiomyocytes, the cardiac muscle cell. Cardiomyocytes function through the interplay between electrical excitation, calcium signaling, and mechanical contraction, an overall process known as calcium-induced calcium release (CICR). A system of seven coupled nonlinear time-dependent partial differential equations (PDEs), which model physiological variables in a cardiac cell, link the processes of cardiomyocytes. Through parameter studies for each component system at a time, we create a set of values for critical parameters that connect the calcium store in the sarcoplasmic reticulum, the effect of electrical excitation, and mechanical contraction in a physiologically reasonable manner. This paper shows the design process of this set of parameters and then shows the possibility to study the influence of a particular problem parameter using the overall model.

Eliminating grid-orientation errors in alternating-direction implicit schemes

: Complex physical phenomena involving chemically reacting systems or the transport of heat or fluids are often modeled by coupled systems of time-dependent, nonlinear partial differential equations. The difficulties in understanding the stability of the differential equation systems and in designing efficient, accurate numerical methods for their solution are widely recognized and were the focus of this research. We have worked on four general aspects of the analysis and numerical approximation of systems of partial differential equations. These areas of research are: (1) modeling aspects and stability analysis for nonlinear time-dependent partial differential equations; (2) use and analysis of finite element or finite difference methods to discretize coupled systems of nonlinear differential equations; (3) development of adaptive or local grid refinement capabilities to resolve local phenomena in large-scale applications; and (4) development of data structures, preconditioners, and efficient solution algorithms for large-scale problems on new computer architectures. Emphasis has been placed upon multiphase or multicomponent, transport-dominated flow processes with dynamic local phenomena. The research also involved a mix of analysis, algorithm development, and large-scale computation using newer computer architectures.

The Use of Homotopy Analysis Method to Solve the Time-Dependent Nonlinear Eikonal Partial Differential Equation

The movements of the prostate relative to the pelvic lymph nodes during intensity-modulated radiation therapy treatment can limit margin reduction and affect the protection of the organs at risk (OAR). In this study, the authors performed an analysis of three adaptive treatment strategies that combine information from both bony and gold marker registrations. The robustness of those treatments against the interfraction prostate movements was evaluated. A retrospective study was conducted on five prostate cancer patients with 7-13 daily cone-beam CTs (CBCTs). The clinical target volumes (CTVs) consisting of pelvic lymph nodes, prostate, and seminal vesicles as well as the OARs were delineated on each CBCT and the initial CT. Three adaptive strategies were analyzed. Two of these methods relied on a two-step patient positioning at each fraction. First step: a bony registration was used to deliver the nodal CTV prescription. Second step: a gold marker registration was then used either to (1) complete the dose delivered to the prostate (complement); (2) or give almost the entire prescription to the prostate with a weak dose gradient between the targets to compensate for possible motions (gradient). The third method (COR) used a pool of precalculated plans based on images acquired at previous treatment fractions. At each new fraction, a plan is selected from that pool based on the daily position of prostate center-of-mass. The dosimetric comparison was conducted and results are presented with and without the systematic shift in the prostate position on the CT planning. The adaptive strategies were compared to the current clinical standard where all fractions are treated with the initial nonadaptive plan. The minimum daily prostate D95% is improved by 2%, 9%, and 6% for the complement, the gradient, and the COR approaches, respectively, compared to the nonadaptive method. The average nodal CTV D95% remains constant across the strategies, except for the gradient approach where a reduction of 7% is observed. However, a correction of the systematic shift reduced the problem, and the adaptive strategies remain robust against the prostate movement across the fraction. The bladder V55Gy is reduced by 35% on average for the adaptive strategies. Because they offer increased CTV coverage and OAR sparing, adaptive methods may be suitable candidates for simple and efficient adaptive treatment strategies for prostate cancer. Margin reduction and systematic error correction in the prostate position improve the protection of the OAR and the dose coverage. A cumulative dose to simulate a complete treatment would show real effects and allow a better comparison between each method.