Related papers: Algorithmic aspects of elliptic bases in finite fi…
We define a generalized finite element method for the discretization of elliptic partial differential equations in heterogeneous media. An adaptive local finite element basis (AL basis) on a coarse mesh which does not resolve the matrix of…
This is a book about computational aspects of modular forms and the Galois representations attached to them. The main result is the following: Galois representations over finite fields attached to modular forms of level one can, in almost…
We study the complexity (that is, the weight of the multiplication table) of the elliptic normal bases introduced by Couveignes and Lercier. We give an upper bound on the complexity of these elliptic normal bases, and we analyze the weight…
We consider a class of parameter-dependent optimal control problems of elliptic PDEs with constraints of general type on the control variable. Applying the concept of variational discretization, [4], together with techniques from the…
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
Elliptic curves over finite fields with predefined conditions in the order are practically constructed using the theory of complex multiplication. The stage with longest calculations in this method reconstructs some polynomial with integer…
We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans…
The study of parameter-dependent partial differential equations (parametric PDEs) with countably many parameters has been actively studied for the last few decades. In particular, it has been well known that a certain type of parametric…
Binary field extensions are fundamental to many applications, such as multivariate public key cryptography, code-based cryptography, and error-correcting codes. Their implementation requires a foundation in number theory and algebraic…
We give a deterministic polynomial time $2^{O(r)}$-approximation algorithm for the number of bases of a given matroid of rank $r$ and the number of common bases of any two matroids of rank $r$. To the best of our knowledge, this is the…
The elliptic curve discrete logarithm problem is of fundamental importance in public-key cryptography. It is in use for a long time. Moreover, it is an interesting challenge in computational mathematics. Its solution is supposed to provide…
We consider representations of algebraic tori $T_n(F_q)$ over finite fields. We make use of normal elliptic bases to show that, for infinitely many squarefree integers $n$ and infinitely many values of $q$, we can encode $m$ torus elements,…
We provide an algorithm for computing an effective basis of homology of elliptic surfaces over the complex projective line on which integration of periods can be carried out. This allows the heuristic recovery of several algebraic…
It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…
When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer…
We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE.…
An accurate approximation of solutions to elliptic problems in infinite domains is challenging from a computational point of view. This is due to the need to replace the infinite domain with a sufficiently large and bounded computational…
In this paper, we develop and study algorithms for approximately solving the linear algebraic systems: $\mathcal{A}_h^\alpha u_h = f_h$, $ 0< \alpha <1$, for $u_h, f_h \in V_h$ with $V_h$ a finite element approximation space. Such problems…
We give a brief report on our computations of linear determinantal representations of smooth plane cubics over finite fields. After recalling a classical interpretation of linear determinantal representations as rational points on the…