Contents Statements of the problems Surrogate optimization Methods

Contents • • Statements of the problems Surrogate optimization Methods of optimizations Technological Stages of Simulation Technical requirements Structure of ICE System environment support 2

Formal Statements of the Direct Problems Initial Boundary Value Problem Boundary & Initial Conditions Example of Differential Operator Example of Boundary Conditions Computational Domain 3

Optimization Statements for Inverse Problems Objective functioal Linear Constraints Nonlinear Constraints State Equation Global minimization 4

Methods of Optimization. 1 Lagrange Multipliers: Lagrangian 5

Methods of Optimization. 2 First Order KKT Conditions 6

Methods of Optimization. 3 Interior Point Methods Modified Lagrangian with penalty vector Saddle point problem 7

Methods of Optimization. 4 Modified KKT conditions 8

Methods of Optimization. 5 Equations for increments in Newton method 9

Methods of Optimization. 6 Consequent Quadratic Programming H-approximation of Gessian Trust Region Method Global minimization 10

Differentiation of the Functionals. Boundary Permutations. Sensitive Functions 11

Sensitive to Variations of Coefficients 12

Technological Stages of Mathematical Modeling • • geometric and functional modeling grid generation approximations (FEM, FVM, DGM) algebraic solutions optimization & inverse problems postprocessing and vizualization control of computational process decision making 13

Surrogate-Based Optimization • • • Initial Sampling & Cross-Validation Polynomials & Radial Basis Functions (RBF) Moving Least-Squares Kriging (universal, blind, noisy data) Vector Regression Multiple Objectives 14

KANTOROVICH : integrated computational environment for optimization methods to solve inverse problems Statements for constrained loccal and global optimization problems Computation of the model sensitive functions Classical optimization methods Interior points methods Consequent quadratic programming Trust region methods Surrogate optimization Parallel technologies 15

Technological Stages of Mathematical Modeling 16

Technical Requirements • • • Adaptation to the computer evolution Extendability of models & algorithms Flexible Data Structures High Performance of Scientific Software Component Object Technologies Multi-language & Cross-Platform Features 17

System Environment Support • Automatic Construction of Algorithms, Validation, Verification, and Testing • Mapping of Algorithms on Computer Architecture • Automatic Analytical Transformations • Configuration of Particular Applications • Creating Domain Specific languages and Interfaces, Data control • Extended Knowledge Base 18

THANKS FOR ATTENTION ! 19