Plenary Lectures

**Prof. H. Thomas Banks**

Center for Research in Scientific Computation, North Carolina State University, USA

**Lecture Title**:** The prohorov metric framework and aggregate data inverse problems for random PDEs**

**Abstract**: We consider nonparametric estimation of probability measures for parameters in problems where only **aggregate** (population level) **data** are available. We summarize an existing computational method for the estimation problem which has been developed over the past several decades [1-6]. Theoretical results are presented which establish the existence and consistency of very general (ordinary, generalized and other) least squares estimates and estimators for the measure estimation problem with speciﬁc application to random PDEs.

[1] HT Banks, WC Thompson. Existence and consistency of a nonparametric estimator of probability measures in the Prohorov Metric Framework. International Journal of Pure and Applied Mathematics, 2015, 103: 819-843.

[2] HT Banks, S Hu, WC Thompson. Modeling and Inverse Problems in the Presence of Uncertainty, CRC Press, Chapman & Hall, 2014.

[3] HT Banks, ZR Kenz, WC Thompson. A review of selected techniques in inverse problem nonparametric probability distribution estimation. Journal of Inverse and Ill-Posed Problems, 2012, 429-460.

[4] HT Banks, D Bortz, G Pinter, L Potter. Modeling and imaging techniques with potential for application in bioterrorism. Bioterrorism: Mathematical Modeling Applications in Homeland Security. SIAM, 2003, 129-154.

[5] HT Banks and KL Bihari. Modelling and estimating uncertainty in parameter estimation. Inverse Problems, 2001, 17 (1): 95-111.

[6] HT Banks and BG Fitzpatrick. Estimation of growth rate distributions in size structured population models. Quarterly of Applied Mathematics. 1991, 49: 215-235.

**Prof. Gengdong Cheng**

State Key Laboratory of Structural Analysis for Industrial Equipment,Dalian University of Technology, China

**Lecture Title**: **Topology Optimization via Sequential Integer Programming and Discrete Sensitivity Analysis
**

**Abstract**: The mathematical essence of structural topology optimization is large-scale nonlinear integer programming. To overcome its huge computational burden, the popular way is to relax the 0-1 variable constraints and transform the integer programming to continuous variable programming. To cope with the variable transformation, the well-know SIMP (Solid Isotropic Material with Penalty) method introduces the interpolation schemes for the material properties versus design variables with penalty and achieves great success and popularity. However, there is no doubt that directly tackling the large-scale nonlinear integer programing is very important. This paper solves the structural topology optimization problems with single or multiple constraints by applying the Canonical Dual Theory together with Sequential Approximate Programming approach under the classic structural topology optimization formulations.

The present paper firstly present a new study on the discrete sensitivity analysis, with which the explicit and separable approximate Sequential Quadratic Integer Programming (SQIP) or Sequential Linear Integer Programming (SLIP) subproblems are constructed. And then, the subproblems are solved by applying the Canonical relaxation algorithm based on CDT theory. Their special mathematical structures are exploited to develop analytic solution of Khun-Tucker condition of the dual programming. Numerical experiments of two linear and quadratic integer programming problems with random coefficients show that the Canonical relaxation algorithm can get approximate solutions with good properties very efficiently and the dual gap is negligible when the number of design variables increases.

Two different move limit strategies within the new method are presented. The new method first solves a set of classic topology optimization problems with only material usage constraint, including minimum structural compliance design under constant load, maximum heat transfer efficiency for the heat conduction problem. And then we apply the method to the topology optimization problems with multiple constraints, including minimum structural compliance design under an additional local displacement constraint and minimum structural compliance design under infill constraints. The results of these problems demonstrate that the new method can efficiently solve the discrete variable structural topology optimization problems with multiple nonlinear constraints or many local linear constraints in a unified and systematic way and can get integer solutions when combined with the move limit strategy of controlling the volume fraction parameter. It can deal with much more design variables than the general branch and bound method.

**Prof. Kalyanmoy Deb**

Electrical and Computer Engineering, Michigan State University, USA

**Lecture Title**: **Customized Optimization for Practical Problem solving**

**Abstract**: Practitioners are often reluctant to use a formal optimization method for routine design and other practical applications, mainly due to the general perception of requiring a large computational time and ending up with a specialized and often "brittle" solution. Optimization methods have come a long way and are made flexible to handle various practicalities including reduction of solution time, handle large dimensions, search for robust and reliable solutions, and discover useful knowledge understanding intricacies of the problem. In this talk, we shall emphasize the importance of customized optimization algorithms in handling various practicalities. A few case studies from industries involving an extreme scale (billion-dimensional) problem and computationally expensive (consuming two days per evaluation) will be presented to demonstrate the usefulness of computational intelligence methods.

**Prof. Jari Kaipio**

Department of Mathematics, The University of Auckland, New Zealand

**Lecture Title**: *Modelling of boundary uncertainties in inverse problems*

**Abstract**: A large class of inverse problems are induced by partial differential equations and the related initial-boundary value problems. In many cases, the exact shape of the domains is only approximately known[1]. Furthermore, domain truncation is also often carried out for computational reasons. On these truncation boundaries, the boundary conditions are unknown [2,3]. In this talk, we consider the modelling of such boundary uncertainties [4].

[1] A Nissinen, V Kolehmainen, JP Kaipio. Compensation ofmodelling errors due to unknown domain boundary in electricalimpedance tomography. IEEE Transactions on Medical Imaging, 2011, 30: 231-242.

[2] D Calvetti, PJ Hadwin, JMJ Huttunen, D Isaacson, JP Kaipio, D McGivney, E Somersalo, J Volzer. Artificial boundary conditionsand domain truncation in electrical impedance tomography, Part I: Theory and preliminary results. Inverse Problem Imaging, 2015, 9: 749-766.

[3] D Calvetti, PJ Hadwin, JMJ Huttunen, JP Kaipio, E Somersalo. Artificial boundary conditions and domain truncation inelectrical impedance tomography. Part II: Computational results. Inverse Problem Imaging, 2015, 9: 767-789, .

[4] JP Kaipio, V Kolehmainen. Approximate marginalization overmodeling errors and uncertainties in inverse problems. Bayesian Theory and Applications, Oxford University Press, 2013.

**Important Dates**

- Feb. 28, 2019,
- Submission of mini-symposium proposals
- Mar. 31, Apr. 30, 2019,
- Two-page abstract due
- Apr. 23, May 10, 2019, Abstract acceptance
- Jun. 15, 2019, Full paper (optional) due
- Jun. 1 - Aug. 15, 2019, Early registration

**Previous IPDO Symposia**