NUMERICAL ASPECTS OF LARGE-TIME OPTIMAL CONTROL OF BURGERS EQUATION N. ALLAHVERDI †‡ , A. POZO ‡ , AND E. ZUAZUA ‡§ Abstract. In this paper, we discuss the efficiency of various numerical methods for the inverse design of the Burgers equation, both in the viscous and in the inviscid case, in long time-horizons. Roughly, the problem consists in, given a final desired target, to identify the initial datum that leads to it along the Burgers dynamics. This constitutes an ill-posed backward problem. We highlight the importance of employing a proper discretization scheme in the numerical ap- proximation of the equation under consideration to obtain an accurate approximation of the optimal control problem. Convergence in the classical sense of numerical analysis does not suffice since nu- merical schemes can alter the dynamics of the underlying continuous system in long time intervals. As we shall see, this may end up affecting the efficiency on the numerical approximation of the inverse design, that could be polluted by spurious high frequency numerical oscillations. To illustrate this, two well-known numerical schemes are employed: the modified Lax-Friedrichs scheme (MLF) and the Engquist-Osher (EO) one. It is by now well-known that the MLF scheme, as time tends to infinity, leads to asymptotic profiles with an excess of viscosity, while EO captures the correct asymptotic dynamics. We solve the inverse design problem by means of a gradient descent method and show that EO performs robustly, reaching efficiently a good approximation of the minimizer, while MLF shows a very strong sensitivity to the selection of cell and time-step sizes, due to excess of numerical viscosity. The achieved numerical results are confirmed by numerical experiments run with the open source nonlinear optimization package (IPOPT). Key words. Burgers equation, inverse design, numerical approximation, long time horizons, optimal control, Engquist-Osher, Lax-Friedrichs. 2010 AMS subject classifications. 49M (primary); 35Q35, 65M06 (secondary). 1. Introduction. We analyze the numerical approximation of the inverse design problem for the Burgers equation, both in the viscous and in the inviscid case: (1.1) ∂ t u + ∂ x u 2 2 = ν∂ xx u, x ∈ R,t> 0, u(x, 0) = u 0 (x), x ∈ R. Given a time T> 0 and a target function u * the aim is to identify the initial datum u 0 so that the solution, at time t = T , reaches the target u * or gets as close as possible to it. In particular, we focus on problems with large values of T , for which convergence of the numerical schemes in the classical sense of numerical analysis might not suffice to obtain accurate results. Essentially, the question consists in solving the Burgers equation backwards, a problem that is ill-posed. In the viscous case ν> 0, this is due to the intrinsic strong time-irreversibility of the parabolic Burgers equation that is enhanced by the nonlinear phenomena governing the hyperbolic dynamics. In the inviscid hyperbolic case, the nonlinearity of the model that produces, in particular, the emergence of shock discontinuities, makes the problem to be ill-posed too, having, in some cases, multiple solutions. † New York City College of Technology. 300 Jay Street, Brooklyn, NY 11201 (USA). ‡ BCAM - Basque Center for Applied Mathematics. Alameda Mazarredo 14, 48009 Bilbao (Basque Country - Spain). § Ikerbasque - Basque Foundation for Science. Alameda Urquijo 36-5, Plaza Bizkaia, 48011 Bilbao (Basque Country - Spain). ¶ E-mails: [email protected], [email protected], [email protected]. 1
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“A FUNDAMENTAL PART OF OUR GROWTH STRATEGY.”SPRINTING AHEAD WITH CENTRIC PLM
“When you’re talking about introducing 600 styles for each season in approximately 30 different factories, the process becomes very complex. There were many errors and communication failures between different areas, which generated quality issues and delays.”
Luis Hernández, Chief Operating Officer at Capa de Ozono, recalls that everyday operations at the Mexican fashion footwear brand became difficult to manage several years ago. Seeking ways to reduce time to market, develop a more robust product development process and centralize product information, Capa de Ozono began looking for a Product Lifecycle Management (PLM) solution.
“We focus on being a fast fashion company, which means we have to get on board with new developments and designs that are appropriate for the market in the fastest possible way,” explains Stefan Hromatka, CEO of Grupo Acritus, Capa de Ozono’s parent company. “The biggest challenge we have, is to identify new trends on time and drop the trends that no longer work.”
Today, with Centric PLM in place, Capa de Ozono has improved product development time by 20%, been able to launch new products on schedule 90% of the time and has doubled its number of collections, without needing to increase design and development staff.
Why did Capa de Ozono decide to make the leap with Centric PLM and how has it changed the way people work every day at the company?
We chose Centric because it was the solution that we could configure to best fit our needs. It’s
like a tailored suit.
Delia Macias, capa de ozono
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CHALLENGES
+ Rapid growth in numbers of styles, factories and stores
+ Addition of two new brands
+ Keeping up with the pace of fast fashion
+ Reducing time to market
+ Communication problems, errors and version control issues
RAPID GROWTHCapa de Ozono was founded in 1992 and is headquartered in León, Guanajuato, the capital of the Mexican footwear industry. Capa de Ozono designs, manufactures and sells footwear, mainly for the teenage and young adult market, and is one of Mexico’s largest fashion footwear retailers.
Capa de Ozono sells through a network of 235 branded stores in 50 cities and has over 300 wholesale clients as well as a popular online channel. The company has grown significantly in recent years.
“When I arrived, we had 80 stores and today we have 180 owned stores and 50 franchises,” says Stefan Hromatka. “We almost doubled the business in the last five years. Last year we acquired two other brands: a footwear brand called Dorothy Gaynor and a swimsuit brand, Zíngara.”
Together, Capa de Ozono, Dorothy Gaynor and Zíngara make up the Grupo Acritus group. Acquiring the new brands expanded the business by approximately 46%.
KEEPING UP THE PACEIn order to keep producing affordable, fashionable footwear, Capa de Ozono is pursuing a fast fashion strategy. The company began looking for ways to reduce time to market and develop a more robust product development process that would ensure quality in high volume production.
Managing large amounts of product-related information using spreadsheets, email and various documents became overly complex and errors crept in. Communication sometimes broke down, resulting in delays and quality control issues.
As Delia Macias, Senior Design Manager at Capa de Ozono explains, “The product information was created and passed on to many different departments that would make their own changes. Version control became very difficult and complicated.”
“As the company grew, we saw the need to adapt to new processes,” she continues. “It was then that we realized that we needed a PLM solution.”
RESULTS
+ Improved product development time by 20%
+ New products launched on schedule 90% of the time
+ Doubled number of collections
+ Comprehensive visibility and control of collections
IF THE SHOE FITSCapa de Ozono made contact with Centric Software, who had established a Latin American office in Mexico City in 2015. Following positive references from other customers in the region, successful product demonstrations and a PLM workshop, Capa de Ozono decided that Centric PLM was the perfect fit.
Capa de Ozono selected Centric PLM on the basis of its footwear expertise, the solution’s flexibility and its ability to completely cover the entire product lifecycle.
“We chose Centric because it was the solution that we could configure to best fit our needs,” explains Delia Macias. “It’s like a tailored suit.”
As Luis Hernández adds, “We thought that the ability to manage new product development and introduction projects using the calendar management module was a very interesting feature. It’s extremely useful for managing the implementation schedules of new products.”
STRIDING INTO THE FUTURECapa de Ozono implemented Centric PLM within budget and on schedule and uses it to run the entire product development process.
Capa de Ozono now has a firm foundation for integrating existing processes and systems with PLM, as well as plans to connect Centric PLM with their in-house ERP system to further aid the seamless delivery of products from design conception, right through to manufacturing and distribution. In order to make sure that designers could take advantage of Centric’s Adobe Connect capability, the company standardized their design software to Adobe Illustrator.
“My whole team uses it,” says Delia Macias. “We have reduced the average amount of time we spend on product development by 20 percent. Centric PLM gives me comprehensive and extensive visibility of each collection and better control, since we have everything we need to develop a product on a single platform. The process for designers is made so much simpler.”
Capa de Ozono is expanding the use of Centric’s Calendar module to enhance specific quality control measures for footwear. Production departments are now able to use information generated by design and development teams to validate products in advance, assess the feasibility of manufacturing new products and improve product quality. The company plans to add a group of manufacturing suppliers to Centric PLM so that they can share technical information and improve the quoting process.
“We can now reach levels of 90 percent compliance with the planned dates for launching new products to the market,” says Luis Hernández. “We have also been able to double the number of collections we launch.”
As Stefan Hromatka concludes, “Centric PLM helps us as managers to gain more visibility and a much more controlled process and users can free up time to focus on their designated roles - speeding up product development. As a crucial part of our omnichannel plans, Centric will be the database for the master data of all our brands in the future.”
“We consider Centric PLM to be a fundamental part of our growth strategy.”
ABOUT CENTRIC SOFTWAREFrom its headquarters in Silicon Valley and offices in trend capitals around the world, Centric Software provides a Digital Transformation Platform for the most prestigious names in fashion, retail, footwear, luxury, outdoor and consumer goods. Centric Visual Innovation Platform (VIP) is a visual, fully digital collection of boards for touch-based devices like iPad, iPhone and large-scale, touch-screen televisions. Centric VIP transforms decision making and automates execution to truly collapse time to market and distance to trend. Centric’s flagship product lifecycle management (PLM) platform, Centric 8, delivers enterprise-class merchandise planning, product development, sourcing, business planning, quality, and collection management functionality tailored for fast-moving consumer industries. Centric SMB packages extended PLM including innovative technology and key industry learnings tailored for small businesses.
Centric Software has received multiple industry awards, including the Frost & Sullivan Global Product Differentiation Excellence Award in Retail, Fashion, and Apparel PLM in 2018 and Frost & Sullivan’s Global Retail, Fashion, and Apparel PLM Product Differentiation Excellence Award in 2016 and 2012. Red Herring named Centric to its Top 100 Global list in 2013, 2015, and 2016.
