Theoretical analysis, construction and implementation of efficient algorithms for computational problems of continuous mathematics, such as: approximation and integration of multivariate functions, problems of linear algebra including large systems of linear equations, ordinary and partial differential equations, optimization. Computer graphics and computer added geometric design. Computational complexity and tractability of continuous problems. Approximation theory and applications.

- Approximation theory: wavelets and geometry of Banach spaces and their applications in theoretical questions of numerical methods and image processing
- Numerical methods of solving PDE, discretization of PDEs by finite element and other methods, error bounds of these discretizations, designing and analysis of parallel algorithms for discretizations of PDEs by finite element method
- Computer graphics image synthesis, visualization, illumination models, rendering algorithms and computer aided geometric design (geometric continuity of curves and surfaces, shape optimization)
- Numerical methods for solving partial differential equations, particularly the finite element method for kinetic equations; numerical methods in finance; scientific computing and computer simulations
- Numerical PDEs, numerical linear algebra, scientific computing, parallel algorithms, numerical simulations
- Numerical methods of solving of PDEs, in particular domain decomposition methods for solving elliptic equations, mainly based on the abstract framework of additive Schwarz method (ASM); development of the methods on nonmatching meshes
- Approximate solving of ordinary and partial differential equations, numerical linear algebra
- Computational complexity, tractability, and construction of optimal algorithms for continuous problems, where available information is partial, priced, and contaminated with deterministic or random noise; numerical integration and approximation of scalar and multivariate functions.
- Złożoność obliczeniowa i podatność zagadnień ciągłych, takich jak aproksymacja i całkowanie funkcji wielu zmiennych, w oparciu o informację częściową
- Complexity of continuous problems: computational complexity and algorithms for continuous problems such as approximation and integration in many dimensions, based on partial and noisy information