arXiv subjects

Uwe Naumann

Publications and source records attributed to Uwe Naumann.

At least 19 recordsLinked to original sources

arXiv2025

Seasons's Greetings by AD

We use Algorithmic Differentiation (AD) to implement type-generic tangent and adjoint versions of $$ y=\sum_{i=0}^{n-1} x_{2 i} \cdot x_{2 i+1} $$ in C++. We run an instantiation for char-arithmetic and we print the gradient at $(101~77~114~114~32~121~109~88~115~97)^T$ to std::cout, yielding the output ``Merry Xmas''. Similar instantiations of type-generic second-order tangent and second-order adjoint versions of $$ y=\frac{1}{6} \cdot \sum_{i=0}^{n-1} x^3_{i} $$ yield ``Happy 2026'' at $(72~97~112~112~121~32~50~48~50~54)^T.$ Prepend a sufficiently large number of zeros to the input vector to explore the varying run times of the different derivative codes. The entire source code can be found on https://github.com/un110076/SeasonsGreetings.

cs.CE↗

arXiv2025

Accelerating Deterministic Global Optimization via GPU-parallel Interval Arithmetic

Spatial Branch and Bound (B&B) algorithms are widely used for solving nonconvex problems to global optimality, yet they remain computationally expensive. Though some works have been carried out to speed up B&B via CPU parallelization, GPU parallelization is much less explored. In this work, we investigate the design of a spatial B&B algorithm that involves an interval-based GPU-parallel lower bounding solver: The domain of each B&B node is temporarily partitioned into numerous subdomains, then massive GPU parallelism is leveraged to compute interval bounds of the objective function and constraints on each subdomain, using the Mean Value Form. The resulting bounds are tighter than those achieved via regular interval arithmetic without partitioning, but they remain fast to compute. We implement the method into our open-source solver MAiNGO via CUDA in two manners: wrapping all GPU tasks within one kernel function, or distributing the GPU tasks onto a CUDA graph. Numerical experiments show that using more subdomains leads to significantly tighter lower bounds and thus less B&B iterations. Regarding wall clock time, the proposed spatial B&B framework achieves a speedup of three orders of magnitude compared to applying interval arithmetic on the CPU without domain partitioning. Among the two implementations, the one developed with CUDA graph enables higher efficiency. Moreover, in some case studies, the proposed method delivers competitive or better performance compared to MAiNGO's default solver which is based on McCormick relaxations. These results highlight the potential of GPU-accelerated bounding techniques to accelerate B&B algorithms.

math.OC↗

arXiv2025

Scheduled Jacobian Chaining

This paper addresses the efficient computation of Jacobian matrices for programs composed of sequential differentiable subprograms. By representing the overall Jacobian as a chain product of the Jacobians of these subprograms, we reduce the problem to optimizing the sequence of matrix multiplications, known as the Jacobian Matrix Chain Product problem. Solutions to this problem yield "optimal bracketings", which induce a precedence-constraint scheduling problem. We investigate the inherent parallelism in the solutions and develop a new dynamic programming algorithm as a heuristic that incorporates the scheduling. To assess its performance, we benchmark it against the global optimum, which is computed via a branch-and-bound algorithm.

cs.DM↗

arXiv2024

An Efficient Local Optimizer-Tracking Solver for Differential-Algebriac Equations with Optimization Criteria

A sequential solver for differential-algebraic equations with embedded optimization criteria (DAEOs) was developed to take advantage of the theoretical work done by Deussen et al. Solvers of this type separate the optimization problem from the differential equation and solve each individually. The new solver relies on the reduction of a DAEO to a sequence of differential inclusions separated by jump events. These jump events occur when the global solution to the optimization problem jumps to a new value. Without explicit treatment, these events will reduce the order of convergence of the integration step to one. The solver implements a "local optimizer tracking" procedure to detect and correct these jump events. Local optimizer tracking is much less expensive than running a deterministic global optimizer at every time step. This preserves the order of convergence of the integrator component without sacrificing performance to perform deterministic global optimization at every time step. The newly developed solver produces correct solutions to DAEOs and runs much faster than sequential DAEO solvers that rely only on global optimization.

math.OC↗

arXiv2024

Differential Inversion of the Implicit Euler Method: Symbolic Analysis

The implicit Euler method integrates systems of ordinary differential equations $$\frac{d x}{d t}=G(t,x(t))$$ with differentiable right-hand side $G : {\mathbb R} \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ from an initial state $x=x(0) \in {\mathbb R}^n$ to a target time $t \in {\mathbb R}$ as $x(t)=E(t,m,x)$ using an equidistant discretization of the time interval $[0,t]$ yielding $m>0$ time steps. We present a method for efficiently computing the product of its inverse Jacobian $$(E')^{-1} \equiv \left (\frac{d E}{d x}\right )^{-1} \in {\mathbb R}^{n \times n} $$ with a given vector $v \in {\mathbb R}^n.$ We show that the differential inverse $(E')^{-1} \cdot v$ can be evaluated for given $v \in {\mathbb R}^n$ with a computational cost of $\mathcal{O}(m \cdot n^2)$ as opposed to the standard $\mathcal{O}(m \cdot n^3)$ or, naively, even $\mathcal{O}(m \cdot n^4).$ The theoretical results are supported by actual run times. A reference implementation is provided.

math.NA↗

arXiv2024

Matrix-Free Jacobian Chaining

The efficient computation of Jacobians represents a fundamental challenge in computational science and engineering. Large-scale modular numerical simulation programs can be regarded as sequences of evaluations of in our case differentiable subprograms with corresponding elemental Jacobians. The latter are typically not available. Tangent and adjoint versions of the individual subprograms are assumed to be given as results of algorithmic differentiation instead. The classical (Jacobian) Matrix Chain Product problem is reformulated in terms of matrix-free Jacobian-matrix (tangents) and matrix-Jacobian products (adjoints), subject to limited memory for storing information required by latter. All numerical results can be reproduced using an open-source reference implementation.

cs.LG↗

arXiv2023

Towards Sobolev Pruning

The increasing use of stochastic models for describing complex phenomena warrants surrogate models that capture the reference model characteristics at a fraction of the computational cost, foregoing potentially expensive Monte Carlo simulation. The predominant approach of fitting a large neural network and then pruning it to a reduced size has commonly neglected shortcomings. The produced surrogate models often will not capture the sensitivities and uncertainties inherent in the original model. In particular, (higher-order) derivative information of such surrogates could differ drastically. Given a large enough network, we expect this derivative information to match. However, the pruned model will almost certainly not share this behavior. In this paper, we propose to find surrogate models by using sensitivity information throughout the learning and pruning process. We build on work using Interval Adjoint Significance Analysis for pruning and combine it with the recent advancements in Sobolev Training to accurately model the original sensitivity information in the pruned neural network based surrogate model. We experimentally underpin the method on an example of pricing a multidimensional Basket option modelled through a stochastic differential equation with Brownian motion. The proposed method is, however, not limited to the domain of quantitative finance, which was chosen as a case study for intuitive interpretations of the sensitivities. It serves as a foundation for building further surrogate modelling techniques considering sensitivity information.

cs.LG↗

arXiv2023

Differentiable Programming: Efficient Smoothing of Control-Flow-Induced Discontinuities

We want to obtain derivatives in discontinuous program code, where default Algorithmic Differentiation may not perform well. Specifically, we consider discontinuities induced by control flow statements, where meaningful derivatives should ideally be capable of representing the resulting kinks in the trajectory. To achieve this, one can interpolate the trajectory at the control flow statements before taking the derivative. We formulate a method to efficiently interpolate between all boundaries induced by control flow in program code. Theoretically, code can be viewed as a series of piecewise continuous functions applied in succession. These functions are nested inside one another and result in a function composition with several cases. We interpret this function composition as a tree and devise a heuristic to identify paths that are relevant to the interpolation. This allows us to conceive a language that smoothly interpolates control-flow statements automatically and efficiently, making it fully differentiable.

cs.PL↗

arXiv2023

Numerical simulation of differential-algebraic equations with embedded global optimization criteria

We are considering differential-algebraic equations with embedded optimization criteria (DAEOs) in which the embedded optimization problem is solved by global optimization. This actually leads to differential inclusions for cases in which there are multiple global optimizer at the same time. Jump events from one global optimum to another result in nonsmooth DAEs and thus reduction of the convergence order of the numerical integrator to first-order. Implementation of event detection and location as introduced in this work preserves the higher-order convergence behavior of the integrator. This allows to compute discrete tangents and adjoint sensitivities for optimal control problems.

math.OC↗

arXiv2023

A Matrix-Free Newton Method

A modification of Newton's method for solving systems of $n$ nonlinear equations is presented. The new matrix-free method relies on a given decomposition of the invertible Jacobian of the residual into invertible sparse local Jacobians according to the chain rule of differentiation. It is motivated in the context of local Jacobians with bandwidth $2m+1$ for $m\ll n$. A reduction of the computational cost by $\mathcal{O}(\frac{n}{m})$ can be observed. Supporting run time measurements are presented for the tridiagonal case showing a reduction of the computational cost by $\mathcal{O}(n).$ Generalization yields the combinatorial Matrix-Free Newton Step problem. We prove NP-completeness and we present algorithmic components for building methods for the approximate solution. Inspired by adjoint Algorithmic Differentiation, the new method shares several challenges for the latter including the DAG Reversal problem. Further challenges are due to combinatorial problems in sparse linear algebra such as Bandwidth or Directed Elimination Ordering.

math.NA↗

arXiv2023

Seven open problems in applied combinatorics

We present and discuss seven different open problems in applied combinatorics. The application areas relevant to this compilation include quantum computing, algorithmic differentiation, topological data analysis, iterative methods, hypergraph cut algorithms, and power systems.

math.CO↗

arXiv2023

Elimination Techniques for Algorithmic Differentiation Revisited

All known elimination techniques for (first-order) algorithmic differentiation (AD) rely on Jacobians to be given for a set of relevant elemental functions. Realistically, elemental tangents and adjoints are given instead. They can be obtained by applying software tools for AD to the parts of a given modular numerical simulation. The novel generalized face elimination rule proposed in this article facilitates the rigorous exploitation of associativity of the chain rule of differentiation at arbitrary levels of granularity ranging from elemental scalar (state of the art) to multivariate vector functions with given elemental tangents and adjoints. The implied combinatorial Generalized Face Elimination problem asks for a face elimination sequence of minimal computational cost. Simple branch and bound and greedy heuristic methods are employed as a baseline for further research into more powerful algorithms motivated by promising first test results. The latter can be reproduced with the help of an open-source reference implementation.

math.OC↗

arXiv2022

Reduction of the Random Access Memory Size in Adjoint Algorithmic Differentiation by Overloading

Adjoint algorithmic differentiation by operator and function overloading is based on the interpretation of directed acyclic graphs resulting from evaluations of numerical simulation programs. The size of the computer system memory required to store the graph grows proportional to the number of floating-point operations executed by the underlying program. It quickly exceeds the available memory resources. Naive adjoint algorithmic differentiation often becomes infeasible except for relatively simple numerical simulations. Access to the data associated with the graph can be classified as sequential and random. The latter refers to memory access patterns defined by the adjacency relationship between vertices within the graph. Sequentially accessed data can be decomposed into blocks. The blocks can be streamed across the system memory hierarchy thus extending the amount of available memory, for example, to hard discs. Asynchronous i/o can help to mitigate the increased cost due to accesses to slower memory. Much larger problem instances can thus be solved without resorting to technically challenging user intervention such as checkpointing. Randomly accessed data should not have to be decomposed. Its block-wise streaming is likely to yield a substantial overhead in computational cost due to data accesses across blocks. Consequently, the size of the randomly accessed memory required by an adjoint should be kept minimal in order to eliminate the need for decomposition. We propose a combination of dedicated memory for adjoint $L$-values with the exploitation of remainder bandwidth as a possible solution. Test results indicate significant savings in random access memory size while preserving overall computational efficiency.

cs.MS↗

arXiv2021

Differentiable Scripting

In Computational Science, Engineering and Finance (CSEF) scripts typically serve as the "glue" between potentially highly complex and computationally expensive external subprograms. Differentiability of the resulting programs turns out to be essential in the context of derivative-based methods for error analysis, uncertainty quantification, optimization or training of surrogates. We argue that it should be enforced by the scripting language itself through exclusive support of differentiable (smoothed) external subprograms and differentiable intrinsics combined with prohibition of nondifferentiable branches in the data flow. Illustration is provided by a prototype adjoint code compiler for a simple Python-like scripting language.

cs.MS↗

arXiv2021

Hessian Chain Bracketing

Second derivatives of mathematical models for real-world phenomena are fundamental ingredients of a wide range of numerical simulation methods including parameter sensitivity analysis, uncertainty quantification, nonlinear optimization and model calibration. The evaluation of such Hessians often dominates the overall computational effort. The combinatorial {\sc Hessian Accumulation} problem aiming to minimize the number of floating-point operations required for the computation of a Hessian turns out to be NP-complete. We propose a dynamic programming formulation for the solution of {\sc Hessian Accumulation} over a sub-search space. This approach yields improvements by factors of ten and higher over the state of the art based on second-order tangent and adjoint algorithmic differentiation.

cs.MS↗

arXiv2021

Numerical Stability of Tangents and Adjoints of Implicit Functions

We investigate errors in tangents and adjoints of implicit functions resulting from errors in the primal solution due to approximations computed by a numerical solver. Adjoints of systems of linear equations turn out to be unconditionally numerically stable. Tangents of systems of linear equations can become instable as well as both tangents and adjoints of systems of nonlinear equations, which extends to optima of convex unconstrained objectives. Sufficient conditions for numerical stability are derived.

math.NA↗

arXiv2021

On the Computational Complexity of the Chain Rule of Differential Calculus

Many modern numerical methods in computational science and engineering rely on derivatives of mathematical models for the phenomena under investigation. The computation of these derivatives often represents the bottleneck in terms of overall runtime performance. First and higher derivative tensors need to be evaluated efficiently. The chain rule of differentiation is the fundamental prerequisite for computing accurate derivatives of composite functions which perform a potentially very large number of elemental function evaluations. Data flow dependences amongst the elemental functions give rise to a combinatorial optimization problem. We formulate {\sc Chain Rule Differentiation} and we prove it to be NP-complete. Pointers to research on its approximate solution are given.

cs.CC↗

arXiv2021

Differential Invariants

Validation is a major challenge in differentiable programming. The state of the art is based on algorithmic differentiation. Consistency of first-order tangent and adjoint programs is defined by a well-known first-order differential invariant. This paper generalizes the approach through derivation of corresponding differential invariants of arbitrary order.

math.NA↗