Iteration | EPFL Graph Search

Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. In mathematics and computer science, iteration (along with the related technique of recursion) is a standard element of algorithms. Mathematics In mathematics, iteration may refer to the process of iterating a function, i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviors and difficult problems – for examples, see the Collatz conjecture and juggler sequences. Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's square root is a c

Uncooperative Rendezvous in Space: Design of an Electronic Architecture for High Performance Avionic with Multi Sensor Input and Intensive Data Rate.

Michaël Yannick Juillard

Active Debris Removal missions consist of sending a satellite in space and removing one or more debris from their current orbit. A key challenge is to obtain information about the uncooperative target. By gathering the velocity, position, and rotation of the desired object, the satellite is able to plan its trajectory and define the sequence of approach. It requires the use of a variety of sensors with often a high data rate. For this task, a dedicated payload computer is envisioned with the responsibility of processing the information from the various rendezvous sensors. This component has the goal to provide meaningful information to the main satellite computer about the targeted debris. The focus of this work is on the data processing, the number of elements, and the electrical energy with constraints due to internal communication.First, an avionic testbench was built at the EPFL Space Center to assess early hardware and software architectures. The goal is to enable Hardware-In-the-Loop testing while developing the payload computer. A communication data bus has been implemented between the Platform and the Payload On-Board Computer. Emphasis was placed on the reliability of the high-level protocol and multiple concepts for improvement were analyzed.In a second phase, the testbench has been extended to support other data buses. The aim was to develop a reliable and efficient backup or fall-back data bus in case of failure in the main link. Four data buses have been tested with extensive analyses on their resilience to error and the efficiency of their data exchange.In parallel, extensive work has been performed to develop a simulation and optimization tool. Its goal is to support the design of payload avionic architectures for ADR missions by providing trade-offs and analyses. In the first iteration, a simulator has been created with instances of various high-level elements modeled.The second iteration introduced the additional dimension of optimization. It is trying to determine the best set of instruments and algorithms to use. With this capability, numerous analyses have been conducted on the influence of various parameters linked to the general optimizer behavior. It allows us to better understand the strength and limitations of the tool.The third step regarding the tool was to start its verification. The task has been to develop an Hardware-In-the-Loop architecture to compare its behavior with the results of the optimizer. The implementation of various mock-up algorithms enables the verification of their models in the tool. These analyses guarantee the feasibility of the outputted solution.The last part was dedicated to the creation and analysis of a realistic payload avionic architecture. The goal was to test the capability of the tool with hardware elements inspired by actual components. This work has shown the procedure to efficiently use the tool in the design phase of a mission.In conclusion, the work on the communication between the Payload and the Platform On-Board Computer has brought valuable lessons and experiences to the projects. They can now be used for the establishment of the high-level protocol into ClearSpace-1 flight software. In addition, the optimizer created allows tackling the design of complex payload avionic architecture where mass saving and processing resources are crucial. It is an essential point to develop a highly efficient payload computer for an Active Debris Removal satellite.

EPFL2022

Large Steps in Inverse Rendering of Geometry

Wenzel Alban Jakob, Baptiste Alexandre Marie Philippe Claude Nicolet

Inverse reconstruction from images is a central problem in many scientific and engineering disciplines. Recent progress on differentiable rendering has led to methods that can efficiently differentiate the full process of image formation with respect to millions of parameters to solve such problems via gradient-based optimization. At the same time, the availability of cheap derivatives does not necessarily make an inverse problem easy to solve. Mesh-based representations remain a particular source of irritation: an adverse gradient step involving vertex positions could turn parts of the mesh inside-out, introduce numerous local self-intersections, or lead to inadequate usage of the vertex budget due to distortion. These types of issues are often irrecoverable in the sense that subsequent optimization steps will further exacerbate them. In other words, the optimization lacks robustness due to an objective function with substantial non-convexity. Such robustness issues are commonly mitigated by imposing additional regularization, typically in the form of Laplacian energies that quantify and improve the smoothness of the current iterate. However, regularization introduces its own set of problems: solutions must now compromise between solving the problem and being smooth. Furthermore, gradient steps involving a Laplacian energy resemble Jacobi's iterative method for solving linear equations that is known for its exceptionally slow convergence. We propose a simple and practical alternative that casts differentiable rendering into the framework of preconditioned gradient descent. Our preconditioner biases gradient steps towards smooth solutions without requiring the final solution to be smooth. In contrast to Jacobi-style iteration, each gradient step propagates information among all variables, enabling convergence using fewer and larger steps. Our method is not restricted to meshes and can also accelerate the reconstruction of other representations, where smooth solutions are generally expected. We demonstrate its superior performance in the context of geometric optimization and texture reconstruction.

ASSOC COMPUTING MACHINERY2021

MATHICSE Technical Report : A Multilevel Stochastic Gradient method for PDE-constrained Optimal Control Problems with uncertain parameters

Matthieu Claude Martin, Fabio Nobile, Panagiotis Tsilifis

In this paper, we present a multilevel Monte Carlo (MLMC) version of the Stochastic Gradient (SG) method for optimization under uncertainty, in order to tackle Optimal Control Problems (OCP) where the constraints are described in the form of PDEs with random parameters. The deterministic control acts as a distributed forcing term in the random PDE and the objective function is an expected quadratic loss. We use a Stochastic Gradient approach to compute the optimal control, where the steepest descent direction of the expected loss, at each iteration, is replaced by independent MLMC estimators with increasing accuracy and computational cost. The refinement strategy is chosen a-priori such that the bias and, possibly, the variance of the MLMC estimator decays as a function of the iteration counter. Detailed convergence and complexity analyses of the proposed strategy are presented and asymptotically optimal decay rates are identified such that the total computational work is minimized. We also present and analyze an alternative version of the multilevel SG algorithm that uses a randomized MLMC estimator at each iteration. Our methodology is validated through a numerical example.

MATHICSE2020